Acknowledgement I would like to begin by thanking my collaborators without whom the work presented in this thesis would not be possible: Alina Chertoc

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6 Acknowledgement I would like to begin by thanking my collaborators without whom the work presented in this thesis would not be possible: Alina Chertock (North Carolina State University), Alexander Kurganov (Tulane University), Eitan Tadmor (CSCAMM, University of Maryland), Şeyma Nur Özcan (North Carolina State University) and Tong Wu (Tulane University). I am particularly thankful for my advisor Dr. Alexander Kurganov, who has taught me everything I know about scientific research, who has been always full of energy, ideas, supports and insights, and also a good consultant for life. I feel very grateful to Dr. Alina Chertock, who has been very reliable and organized, and from whom I have learned many great lessons on Mathematical writing. I wish to thank my collaborator and academic brother, Tong Wu, with whom I have shared countless working nights and a-ha moments. In particular, I would like to thank Dr. Tewodros Amdeberhan for his great insight in the stability proof in Chapter 3 and I would like to thank Dr. James Mac Hyman who always cheered me up and encouraged me to pursue my passion in research. ii

7 Also, I would like to give many thanks to my peers: Dr. Franz Hoffmann, Lin Jiu, Yuanzhen Cheng, Zhuolin Qu, Yu Liu, Dmitry Kurochkin, Jianjun Huang and Kui Zhang. Finally, I wish to thank Drs. Lisa Fauci, Ricardo Cortez, Xuefeng Wang and David DaGang Yang for their services on my dissertation committee. iii

8 List of Tables. Example.3: L -errors for the non-well-balanced CU scheme Example 3.: The errors and convergence rates of the second-order SI-RK3 method Example 4.: Errors in computing the steady-state solution for different sets of data Example 4.3: Friction coefficients and slopes Example 4.4: The L - and L -errors and convergence rates for h and q Example 4.6: The L -errors and convergence rates for h and q Example 4.7: The weighted L -differences between the solutions computed on consecutive grids and the corresponding convergence rates. 97 iv

9 List of Figures. Graph of numerical diffusion modifier Example.: Solution (ρ(y, 0.), v(y, 0.), E(y, 0.) and p(y, 0.)) computed using N = 00 and N = 000 cells Example.: Pressure perturbation computed by the well-balanced and non-well-balanced CU schemes Example.3: Pressure perturbation computed by the well-balanced and non-well-balanced CU schemes using uniform cells Example.3: Contour plot of the pressure perturbation computed by well-balanced and non-well-balanced using different resolutions Example.4: Density and velocity computed by the well-balanced CU scheme Example.4: Density and velocity computed by the non-well-balanced CU scheme Stability region D SSP contained in C Surface plot of log 0 ( K(y, z ) + ); the stability domains D SSP3 and D SI-RK3( ) Conjecture 3.3.: max R SI-RK3(z, z ) over z such that R SSP3 (z ) α and z 0 as a function of α Example 3.: Convergence toward the equilibrium for the SI-RK3 and IMEX-SSP3(3,3,) methods v

10 3.5 Example 3.3: Solutions computed by the SI-RK3 and IMEX-SSP3(3,3,) methods Example 3.4: Solution obtained using the SI-RK3 method Example 3.4: Solution obtained using IMEX-SSP3(3,3,) method Example 3.5: Solutions obtained using the SI-RK3 method with fixed time step restrictions Example 3.5: Solutions obtained using the IMEX-SSP3(3,3,) method with fixed time step restrictions The bottom setting of numerical examples Example 4.: Evolution of the solution in the supercritical case (Test in Table 4.) Example 4.: Evolution of the solution in the subcritical case (Test 4 in Table 4.) Example 4.: Evolution of the solution with large bottom slope (Test 5 in Table 4.) Example 4.3: Outlet discharge as a function of time Example 4.4: Solution (w(x, 00) and q(x, 00)) computed using N = 50 and the exact steady-state solution Example 4.5: Snapshots of w, q and u computed with ε = 0 4, 0 8 and Example 4.6: Water profiles at t = Example 4.6: Zoom at the wet/dry fronts of the computed and reference (N = 5600) solutions Example 4.7: Solution computed using the fine grid Example 4.8: Three house configurations and contour plot of the urban area structure vi

11 4. Example 4.8: Outlet discharge (the experimental data vs. the computed results) as a function of time Example 4.8a: Special treatment of a house region Example 4.8a: Outlet discharge (the experimental data vs. the computed results) as a function of time Example 4.8a: Outlet discharge as a function of time: convergence to a steady state Example 4.8a: Water height snapshots at the steady state achieved when the rain source is not switched off Example 4.8a: Water height snapshots at times t = 5, 45 and Example 4.8a: Significance of the bottom friction vii

12 Contents Acknowledgement ii List of Tables iv List of Figures v Introduction. Euler Equations with Gravitation Shallow Water System with Friction Well-Balanced Central-Upwind Schemes for the Euler Equations with Gravitation 7. Introduction One-Dimensional Numerical Method Second-Order Semi-Discrete Central-Upwind Scheme Well-Balanced Central-Upwind Scheme Two-Dimensional Numerical Method Well-Balanced Central-Upwind Scheme Numerical Examples One-Dimensional Examples Two-Dimensional Examples viii

13 3 Steady State Preserving Semi-Implicit Runge-Kutta Methods Introduction New Semi-Implicit Methods Absolute Stability of Two SSP-Based SI-RK Methods Numerical Examples Scalar ODEs Systems of ODEs Arising from Semi-Discretizations of PDEs Well-Balanced Central Upwind Scheme for the Shallow Water System with Friction Terms Introduction Steady-State Solutions Numerical Method One-Dimensional Central-Upwind Scheme Two-Dimensional Central-Upwind Scheme Time Integration Method Numerical Examples One-Dimensional Examples Two-Dimensional Examples Conclusion References 08 ix

14 Chapter Introduction In this dissertation, we are interested in developing numerical methods for hyperbolic systems of balance law, which in the one-dimensional (-D) case take the following form: q t + f(q, x) x = ψ(q, x, t), (.) where q is the vector of unknown function, f is the flux function and ψ denotes the source term. The system (.) admits steady-state solutions, in which the flux gradient is exactly balanced with the source term. Many physical waves are in fact small perturbations of steady-state solutions, like flows in lakes, rivers, oceans and atmospheric activities. Capturing those solutions with numerical methods is a non-trivial task, since the size of the perturbation is so small that they may be dominated by the truncation errors, especially when a coarse computational grid is employed. The truncation errors can be limited by using a very fine grid, however, it is not feasible in many physically relevant applications that have large scales in both space and time. Therefore, it is important to design a well-balanced scheme, which can exactly preserve the steady-state solutions on the discrete level. Then the perturbations in the solutions will be properly resolved.

15 In this dissertation, we develop well-balanced schemes for two systems of balance laws: the Euler equations of gas dynamics with gravitation and the shallow water equations with friction terms. Due to their distinct natures, we construct the wellbalanced schemes for them using different sets of numerical techniques, which are outlined separately in the following two sections.. Euler Equations with Gravitation The first system we consider in this dissertation is the Euler equations with gravitational terms, which in the -D case can be written as q t + G(q) y = S(q), (.) where is a vector of conservative variables, ρ q = ρv E (.3) ρv G(q) = ρv + p v(e + p) (.4) is the flux function, and 0 S(q) = ρg ρvg (.5) is the gravitational source term. Here, ρ is the density, v denotes the velocity, E is the total energy and p represents the pressure. In the gravitational source term S, g

16 3 denotes the gravitational constant. The system (.) (.4) is closed by the following equation of state (EOS): E = p γ + ρ v, (.6) where γ stands for the specific heat ratio. The Euler equations with gravitation are used to model astrophysical and atmospheric phenomena in many fields including supernova explosions, climate modeling and weather forecasting. Similar to the shallow water model in., many physical relevant solutions of system (.) (.5) are small perturbations of steady states and it is a challenging task to numerically capture such solutions. The steady state solutions of (.) (.5) can be obtained by solving the timeindependent system G(q) y = S(q), which can be rewritten as (ρv) y = 0, (ρv + w) y = 0, (v(e + p)) y = ρvg, where w := p + R, y R(y) := g y 0 ρ(ξ)dξ. (.7) The simplest steady state is the motionless one, for which v 0 and w Const. (.8) Based on the central-upwind schemes, we aim to construct a well-balanced scheme which is capable of preserving such a steady state. In the central-upwind schemes, the numerical solution is realized in terms of cell averages of the conservative variables (q = (ρ, ρv, E) T for the -D case). The cell averages are used to reconstruct a global

17 4 piecewise polynomial approximation of the numerical solution, which is then used to evolve the computed solution in time. Unfortunately, the central-upwind schemes implemented using the above reconstruction procedure do not possess the well-balanced property. In, we introduce a special reconstruction based on the equilibrium variables (ρ, ρv, w) T rather than the conservative ones. This results in a well-balanced central-upwind scheme for system (.) (.5) which is capable of preserving steady state (.0). We apply the well-balanced scheme to various test problems and the numerical results clearly demonstrate the importance of the well-balance property and the improved performance of our new numerical scheme compared to the nonwell-balanced one. The detailed description of the second-order well-balanced central-upwind schemes for the Euler equations with gravitation is given in.. Shallow Water System with Friction The second example of well-balanced central-upwind schemes is concerned with shallow water system with friction terms, which in the -D case reads: h t + (hu) x = 0, (hu) t + (hu + g ) h x (.9) = ghb x F. Here, h(x, t) denotes the water depth, u(x, t) is the average velocity, B(x) represents the bottom topography function and g is the gravitational constant. In the second equation of (.9), the term ghb x is the geometric source and F represents the bottom friction which can be defined, for instance, using a classical Manning formulation: F = g n u u, (.0) h/3 with n being the Manning coefficient.

18 5 The system (.9) admits steady state solutions, i.e., the solutions which are independent of time variable t. Many physical phenomena correspond to small perturbations of these steady state solutions. From a practical point of view, one of the most important steady-state solutions is the lake at rest state: u 0, h + B Const. (.) The numerical methods that exactly preserve such solutions are called well-balanced. Unlike (.), where u 0 and the friction term F is inactive, the system (.9) also admits the following family of moving steady states: hu q 0, B x C, h ( ) n q0 3/0, Cq 0 > 0, (.) C which corresponds to the situation when the water flows over a slanted surface with a constant slope and the geometric source keeps in balance with the bottom friction. Solving the system (.9) numerically is a challenging task due to several reasons. First, many physically relevant steady states, like (.) and (.), are characterized by a delicate balance between the flux gradients and source terms. If the method does not respect this balance, the numerical errors may lead to spurious oscillations, in which the magnitude of artificial waves may be larger than the magnitude of the exact solution itself. The second major difficulty is related to the computation of solutions when the water depth is very small or even zero, which happens near the wet/dry front. In such a case, small numerical oscillations may lead to negative values of water depth, which is unphysical. Also, the bottom friction term (.0) becomes very stiff when the water depth h is very small. As a result, an explicit discretization of friction term (.0) can only be adopted together with a severe time step restriction, which is needed for the stability of the overall numerical scheme. For hyperbolic system with stiff terms, the implicit-explicit (IMEX) time dis-

19 6 cretization schemes [, ] are commonly used as alternatives to the explicit ones. With the stiff terms implicitly/semi-implicitly treated, the undesirable time step restriction is no longer imposed. However, such methods cannot accurately maintain the balance between the flux gradients and source terms. Our goal is to develop an efficient well-balanced positivity preserving numerical scheme for (.9), which is capable of preserving both steady states (.) and (.). A second-order well-balanced and positivity preserving central-upwind scheme was presented in [3] for solving the system (.9) without friction. It can be directly extended to the friction case, but the efficiency may greatly suffer when almost dry areas are present. In [4] and also presented in 4, we have introduced a semi-implicitexplicit version of the central-upwind scheme. The proposed scheme is based on a new time discretization method, which we have derived from the explicit strong stability preserving (SSP) approach [5 9]. The new method has second-order accuracy and inherits the desired properties from the explicit SSP methods: It is well-balanced and positivity preserving. Since the stiff friction term is treated semi-implicitly, no extra time restriction is needed and our scheme can be used with the time steps only based on the CFL condition, which is determined by the nonstiff part of the system (.9). Recently in [0], we generalized the aforementioned semi-implicit treatment on the stiff terms into a family of second-order semi-implicit time integration methods for systems of ODEs with stiff damping term (presented in 3). The obtained numerical results clearly demonstrate that the ability of the introduced ODE solver to exactly preserve equilibria plays an important role in achieving high resolution when a coarse grid is used in the discretization of PDEs.

20 7 Chapter Well-Balanced Central-Upwind Schemes for the Euler Equations with Gravitation The contents of this chapter have been submitted to SIAM Journal on Scientific Computing. In this chapter, we develop a second-order well-balanced central-upwind scheme for the Euler equations of gas dynamics with gravitation, which is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a wellbalanced piecewise linear reconstruction of equilibrium variables, which is combined with a well-balanced evolution in time, achieved by reducing the amount of numerical viscosity (present at the central-upwind scheme) in the areas where the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces on a number of one- and two-dimensional examples.

21 8. Introduction We consider the Euler equations of gas dynamics with gravitation, which can be written in the two-dimensional (-D) case as q t + F (q) x + G(q) y = S(q), (.) where ρ ρu q := ρv E (.) is a vector of conservative variables, and ρu ρu + p F (q) = ρuv u(e + p) ρv ρuv and G(q) := ρv + p v(e + p) (.3) are the fluxes in the x- and y-directions, and 0 ρφ x S(q) = ρφ y ρuφ x ρvφ y (.4) is the source term. Here, ρ is the density, u and v are the x- and y-velocities, E is the total energy, p is the pressure and φ is the time-independent linear gravitational potential.

22 9 The system (.) (.4) is closed using the following equation of state (EOS): E = p γ + ρ (u + v ), (.5) where γ stands for the specific heat ratio. Here, we consider a physically relevant case, in which the gravitational potential is taken in the y-direction only, that is, φ x = 0 and φ y = g. The system of balance laws (.) (.5) is used to model astrophysical and atmospheric phenomena in many fields including supernova explosions [], (solar) climate modeling and weather forecasting []. In many physical applications, solutions of the system (.) (.5) are small perturbations of the steady states. As mentioned in., the well-balance property of the numerical scheme plays a very important role in resolving those small perturbations using a coarse grid, which is the feasible option in the many large-scale applications. Well-balanced schemes were introduced in [3] and mainly developed in the context of shallow water equations, for details, see, e.g., [3,4 9]. Some of these schemes have been extended for the Euler equations with gravitational fields. In [30], quasi-steady wave-propagation methods were developed for models with a static gravitational field. In [], well-balanced finite-volume methods, which preserve a certain class of steady states, were derived for nearly hydrostatic flows. In [3 33], gas-kinetic schemes were extended to the multidimensional gas dynamic equations and well-balanced numerical methods were developed for problems, in which the gravitational potential was modeled by a piecewise step function. More recently, higher order finite-difference methods for the gas dynamics with gravitation were introduced in [34]. Our goal is to develop a well-balanced numerical method capable of exactly preserving the steady state solutions, which can be derived as follows. Consider, for

23 0 simplicity, a one-dimensional (-D) version of the system (.) (.5): q t + G(q) y = S(q), (.6) where ρ ρv 0 q := ρv, G(q) := ρv + p, S(q) := ρg, E = p γ + ρv. E v(e + p) ρvg (.7) The steady-state solutions of (.6), (.7) can be obtained by solving the time-independent system G(q) y = S(q). To this end, we first incorporate the source term ρg into the flux, introduce a new global variable w, w := p + R, y R(y, t) := g ρ(ξ, t) dξ, (.8) and rewrite the system G(q) y = S(q) as (ρv) y = 0, (ρv + w) y = 0, (v(e + p)) y = ρvg. (.9) It then immediately follows that the simplest steady state of (.9), (.8) is the motionless one, for which v 0 and w Const. (.0) The corresponding -D steady state is u = v 0 and w Const. (.)

24 In this chapter, we develop a new well-balanced central-upwind (CU) scheme for the Euler equations with gravitation. CU schemes were initially introduced in [35] for hyperbolic systems of conservation laws, further developed in [36 38] and extended to systems of balance laws in [3, 4, 5 7, 3, 39]. The CU schemes are Godunovtype finite-volume methods that are efficient, highly accurate and do not require any (approximate) Riemann problem solver (the latter makes the CU schemes applicable in a black-box manner to a wide variety of multidimensional hyperbolic systems of conservation and balance laws). In the CU schemes, the numerical solution is realized in terms of cell averages of the conservative variables (q := (ρ, ρv, E) T or q := (ρ, ρu, ρv, E) T for the -D and -D Euler equations, respectively). The cell averages are used to construct a global piecewise polynomial approximation of the numerical solution, which is then used to evolve the computed solution in time. Unfortunately, when a CU scheme is applied on the Euler equations using a reconstruction procedure of the conservative variables, the resulting numerical method does not posses the well-balanced property. We therefore modify the reconstruction step and introduce a special reconstruction based on the equilibrium variables, (ρ, ρv, w) T (or (ρ, ρu, ρv, w) T in -D) rather than the conservative ones. This results in a well-balanced CU scheme for the Euler equations with gravitation. The strategy of reconstructing equilibrium variables (rather than conservative ones) has been adopted in developing well-balanced CU schemes for shallow water equations with source terms in [3, 4, 5 7, 3, 39]. However, the equilibrium variable of Euler equations with gravitation is not locally defined, that is, the computation of which requires a delicate quadrature of gas density ρ (the quadrature will be introduced in. and.3 for -D and -D cases, respectively). Also, the effect of numerical viscosity jeopardizes the preservation of steady-state solutions, when the Euler equations are considered. We therefore proposed a way to limit the amount of

25 numerical viscosity when the solution is in the steady state regime. The chapter is organized as follows. In. and.3, we develop the well-balanced CU schemes for -D and -D Euler equations with gravitation. Special -D and -D well-balanced reconstructions are presented in.. and.3., respectively. In.4, we present a number of -D and -D numerical examples.. One-Dimensional Numerical Method In this section, we first (..) briefly describe the semi-discrete CU scheme from [37] and then (..) derive its well-balanced modification for the -D Euler equations with gravitation... Second-Order Semi-Discrete Central-Upwind Scheme For simplicity, we partition the computational domain into finite-volume cells C k := [y k, y k+ ] of size C k = y centered at y k = k y, k = k L,..., k R, and the cell interfaces are denoted by y k± cell averages of the numerical solution, q k (t) := y := (k ± /) y. We assume that at time level t, the C k q(y, t) dy, are available. A semi-discrete CU scheme from [37] applied to (.6), (.7) is the following system of ODEs: d dt q k = G k+ G k +S k, (.) y where G k+ := b + G(q k+ k N) b G(q k+ k+ S ) b + k+ b k+ ( ) + β k+ q S k+ qk N, βk+ := b + b k+ k+ b + k+, b k+ (.3) are numerical fluxes, and S k = (0, gρ k, g(ρv) k ) T

26 3 are approximations of the cell averages of the source term. In (.3), q N k and qs k+ are the one-sided point values of the computed solution at cell interfaces y = y k+. To construct a second-order scheme, these variables are to be calculated using the piecewise linear reconstruction q(y) = k ( qk + (q y ) k (y y k ) ) χ Ck (y), (.4) where χ Ck is a characteristic function of the interval C k. We then obtain qk N := q(y k+ 0) = q k + y (q y) k, qk+ S := q(y k+ +0) = q k+ y (q y) k+. (.5) To avoid oscillations, the vertical slopes in (.5), (q y ), are to be computed using a nonlinear limiter applied to the cell averages {q k }. In all of the numerical experiments presented in.4, we have used a generalized minmod limiter (see, e.g., [40 43]) applied in the component-wise manner: ( (q y ) k = minmod θ q k+ q k y, q k+ q k, θ q ) k q k, (.6) y y where the minmod function is defined by minmod(z, z,...) := min(z, z,...), if z i > 0 i, max(z, z,...), if z i < 0 i, 0, otherwise, (.7) and the parameter θ [, ] controls the amount of numerical dissipation: The use of larger values of θ typically leads to less dissipative, but more oscillatory scheme. Finally, the one-sided local speeds of propagation, b ±, are estimated using the k+

27 4 smallest and largest eigenvalues of the Jacobian G q : b + k+ = max ( v N k + c N k, v S k+ + c S k+, 0 ), b k+ = min ( v N k c N k, v S k+ c S k+, 0 ), (.8) where the velocities, vk N and vs k+, are obtained using the identity v (ρv)/ρ, cn k and c S k+ are the speeds of sound defined by c = γp/ρ, and the pressures, p N k and ps k+, are obtained using the EOS (.7). Unfortunately, the CU scheme (.) (.8) is not capable of exactly preserving the steady-state solution (.0). Indeed, substituting (.0) into (.) (.3) and noting that b + k+ = b, k, we obtain the ODE system k+ dρ k dt = β k+ (ρ S k+ ρn k ) β k (ρ S k ρn k ), y d(ρv) k dt de k dt = (ps k+ + pn k ) (ps k + pn k ), y = β k+ (p S k+ pn k ) β k (p S k pn k ), (γ ) y (.9) whose RHS does not necessarily vanish and hence the steady state would not be preserved at the discrete level. We would like to stress that even for the first-order version of the CU scheme (.) (.8), that is, when (q y ) k 0 in (.4), (.5), the RHS of (.9) does not vanish. This means that the lack of balance between the numerical flux and source terms is a fundamental problem of the scheme. We also note that for smooth solutions, the balance error in (.9) is expected to be of order ( y), but a coarse grid solution may contain large spurious waves... Well-Balanced Central-Upwind Scheme In this section, we present a well-balanced modification of the CU scheme from... The new scheme will be developed by first introducing well-balanced reconstruction

28 5 performed on the equilibrium variables rather than the conservative ones and then deriving modified formulae for the numerical fluxes and sources. To this end, we once again incorporate the source term ρg into the flux and rewrite the system (.6) (.7) as follows: ρ t + (ρv) y = 0, (ρv) t + (ρv + w) y = 0, E t + (v(e + p)) y = ρvg, (.0) which can be put into the vector form (.6) with ρ ρv 0 q := ρv, G(q) := ρv + w, S(q) := 0, E v(e + p) ρvg where w is given by (.8). Well-Balanced Reconstruction We now describe a special reconstruction, which is used to derive a well-balanced CU scheme. The main idea is to reconstruct equilibrium variables (ρ, ρv, w) rather than (ρ, ρv, E). For the first two components we still use formula (.4) to obtain the same piecewise linear reconstructions as before, ρ(y) and ( ρv)(y), and compute the corresponding point values of ρ N,S and (ρv) N,S, and then obtain v N,S = (ρv) N,S /ρ N,S. To reconstruct the third equilibrium variable w, we first compute the point values of R by integrating the piecewise linear reconstruction of ρ, ρ(y) = k ( ρk + (ρ y ) k (y y k ) ) χ Ck (y),

29 6 which results in the piecewise quadratic approximation of R: R(y) = g y ρ(ξ) dξ y kl = g k [ k y ρ i +ρ k (y y k ) + (ρ ] y) k (y y k )(y y k+ ) χ Ck (y). i=k L Then, the point values of R at the cell interfaces and cell centers are R k+ k k = g y ρ i and R k = g y ρ i + g y ρ k g( y) (ρ y ) k, (.) 8 i=k L i=k L respectively, and the values of w at the cell centers are set as w k = p k + R k, (.) ( where p k = (γ ) E k ρ ) k vk is obtained from the corresponding EOS (.5) and v k = (ρv) k /ρ k. Equipped with (.), we then apply the minmod reconstruction procedure to {w k } and obtain the point values of w at the cell interfaces: w N k = w k + y (w y) k, w S k+ = w k+ y (w y) k+, where ( (w y ) k = minmod θ w k+ w k y, w k+ w k, θ w ) k w k. y y Finally, the point values of p and E needed for computation of numerical fluxes are p N k = wk N R k+, p S k = wk S R k

30 7 and E N k = pn k γ + ( (ρv) N k ) ρ N k, E S k = ps k γ + ( ) (ρv) S k, ρ S k respectively. Remark... In practice, it is convenient to compute the point values of R k+ R k recursively, that is, replacing (.) with and R kl = 0, R k+ = R k + g yρ k, R k = R k + g y ρ k g( y) (ρ y ) k, 8 k = k L,..., k R. (.3) Well-Balanced Evolution The cell-averages of q are evolved in time according to the following system of ODEs: d dt q k = G k+ G k +S k. (.4) y Here, the second and third components of the numerical fluxes G are computed the same way as in (.3): G () k+ := b + k+ ( ρ N k (v N k ) + w N k ) ( b k+ ρ S k+ (vk+ S ) + wk+) S b k+ b + k+ (.5) ( ) + β k+ (ρv) S k+ (ρv) N k, G (3) k+ b + v k+ k N(EN k + pn k ) b v k+ k+ S (ES k+ + ps k+ ) := b + b k+ k+ (.6) ( ) + β k+ E S k+ Ek N,

31 8 while the first component should be modified in order to preserve the steady state (.0): G () k+ = b + (ρv) N k+ k b (ρv) S k+ k+ + β k+ H b + b k+ k+ ( wk+ w k y yk R + y kl max k {w k } (.7) ) (ρ ) S k+ ρ N k. Notice that the last term in (.7) is now multiplied by a smooth function H, designed to be very small when the computed solution is locally (almost) at steady state, that is, at the cell interfaces where w k+ w k y 0, and to be very close to elsewhere. This is done in order to guarantee the well-balanced property of the scheme as we show in Theorem.. proved in... On the other hand, the modification of the original CU flux is quite minor since H(ψ) is very close to unless ψ is very small. A sketch of a typical function H is shown in Figure.. In all of our numerical experiments, we have used H(ψ) = (Cψ)m + (Cψ) m, (.8) with C = 00 and m = 6. To reduce the dependence of the computed solution on the choice of particular values of C and m, the argument of H in (.7) is normalized by a factor y k R + y kl max k {w k, which makes H(ψ) dimensionless. } H ψ Figure.: Sketch of H(ψ). Finally, the cell averages of the source term are approximated using the midpoint

32 9 quadrature rule as follows: S k = (0, 0, g(ρv) k ) T. (.9) Proof of the Well-Balanced Property Theorem... The semi-discrete CU scheme (.4) (.9) coupled with the reconstruction described in.. is well-balanced in the sense that it preserves the steady state (.0). Proof. Assume that at certain time level, we have v N k v k v S k 0 and w N k w k w S k ŵ, (.30) where ŵ is a constant. To show that the proposed scheme is well-balanced, we need to show that the right-hand side (RHS) of (.4) is identically equal to zero for the data in (.30). Since the source term (.9) vanishes for v k = 0, it is enough to prove that the numerical fluxes are constant for the data in (.30). Indeed, the first components of the numerical flux, (.7), vanish since vk N = ( vk+ S = 0 and w k = w k+ = ŵ (the latter implies H wk+ w k yk R + y ) kl y max k w k = H(0) = 0). The second components of the numerical flux, (.5), are constant and equal to ŵ since v N k = vs k+ = 0 and wn k = ws k+ = ŵ. Finally, the third components of the numerical flux, (.6), also vanish: G (3) k+ ( ) β = β k+ E S k+ Ek N k+ = γ = β [ k+ (w S γ = 0, k+ R k+ ( p S k+ p N k ) ) ] ) (wk N R k+ since w N k = ws k+ = ŵ.

33 0.3 Two-Dimensional Numerical Method In this section, we describe the well-balanced semi-discrete CU scheme for the -D Euler equations with gravitation. Similarly to the -D case, we rewrite the system (.) (.5) as follows: ρ t + (ρu) x + (ρv) y = 0, (ρu) t + (ρu + p) x + (ρuv) y = 0, (ρv) t + (ρuv) x + (ρv + w) y = 0, E t + (u(e + p)) x + (v(e + p)) y = ρvg. (.3) This system can also be written in the vector form (.) with ρ ρu ρv 0 ρu ρu + p ρuv 0 q :=, F (q) :=, G(q) :=, S(q) :=, ρv ρuv ρv + w 0 E u(e + p) v(e + p) ρvg where w := p + R, y R(x, y, t) := g ρ(x, ξ, t) dξ. (.3).3. Well-Balanced Central-Upwind Scheme We consider a rectangular computational domain and partition it into the uniform Cartesian cells C j,k := [x j, x j+ ] [y k, y k+ ] of size C j,k = x y centered at (x j, y k ) = (j x, k y), j = j L,..., j R, k = k L,..., k R. Similarly to the -D case, we assume that at a certain time level t, the cell averages of the computed numerical solution, q j,k (t) := x y q(x, y, t) dx dy, (.33) C j,k

34 are available. Well-Balanced Reconstruction Similarly to the -D case, we reconstruct only the first three components of the conservative variables q (ρ, ρu and ρv): q (i) (x, y) = q (i) j,k + (q(i) x ) j,k (x x j ) + (q (i) y ) j,k (y y k ), (x, y) C j,k, i =,, 3, (.34) and compute the corresponding point values at the cell interfaces (x j±, y k ) and (x j, y k± ): (q (i) ) E j,k := q (i) (x j+ 0, y k ) = q (i) j,k + x (q(i) x ) j,k, (q (i) ) W j,k := q (i) (x j + 0, y k ) = q (i) j,k x (q(i) x ) j,k, (q (i) ) N j,k := q (i) (x j, y k+ 0) = q (i) j,k + y (q(i) y ) j,k, (q (i) ) S j,k := q (i) (x j, y k + 0) = q (i) j,k y (q(i) y ) j,k, i =,, 3, where the slopes (q (i) x ) j,k and (q (i) y ) j,k are computed using a nonlinear limiter, for example, the generalized minmod limiter: (q (i) x ) j,k = minmod (q y (i) ) j,k = minmod ( ( θ q (i) j+,k x θ q (i) j,k+ y (i) q j,k (i) q j,k, q (i) (i) j+,k q x j,k, q (i) (i) j,k+ q y j,k, θ q ) (i) (i) j,k q j,k, x, θ q ) (i) (i) j,k q j,k, y i =,, 3. We then estimate the one-sided local speeds of propagation in the x- and y- directions, respectively, using the smallest and largest eigenvalues of the Jacobians

35 F q G and : q a + = max ( u E j+,k j,k + c E j,k, u W j+,k + c W j+,k, 0 ), a j+,k = min ( u E j,k c E j,k, u W j+,k c W j+,k, 0 ), b + j,k+ b j,k+ = max ( v N j,k + c N j,k, v S j,k+ + c S j,k+, 0 ), = min ( v N j,k c N j,k, v S j,k+ c S j,k+, 0 ), where the velocities u E j,k, uw j+,k, vn j,k and vs j,k+ are obtained from the identities u (ρu)/ρ and v (ρv)/ρ and the speeds of sound c E j,k, cw j+,k, cn j,k and cs j,k+ are computed from the definition c = γp/ρ. The calculation of the point values for the forth conservative variable E requires a special treatment, which is different in the horizontal (x) and vertical (y) directions. In the x-direction, we first compute the point values of p at the cell centers using the EOS (.5): ( p j,k = (γ ) E j,k (ρu) j,k + ) (ρv) j,k, ρ j,k and then compute the cell interface values of p using a nonlinear limiter, for example, the generalized minmod one: p E j,k = p j,k + x (p x) j,k, p W j,k = p j,k x (p x) j,k, where ( (p x ) j,k = minmod θ p j+,k p j,k x, p j+,k p j,k, θ p ) j,k p j,k. x x Equipped with these values, we then compute the required cell interface values of E: E E j,k = pe j,k γ + ρe ( j,k (u ) E ( ) ) j,k + v E j,k, Ej,k W = pw j,k γ + ρw j,k ( (u ) W ( ) ) j,k + v W j,k. Remark.3.. As an alternative approach, one can compute the point values E E j,k and

36 3 E W j,k using a piecewise linear reconstruction of the conservative variable E rather than p and still obtain a well-balanced reconstruction. However, our numerical experiments (not reported in this chapter for the sake of brevity) indicate that reconstructing E in the x-direction leads to the loss of symmetry in the computed solution. In the y-direction, we follow the same idea as in the -D case. First, we compute the values of R at the cell interfaces and cell centers in a complete analogy with (.3): R j,k+ = R j,k + g yρ j,k, R j,kl = 0, R j,k = R j,k + g y ρ j,k g( y) (ρ y ) j,k, 8 We then compute w j,k as follows: j = j L,..., j R, k = k L,..., k R. w j,k = p j,k + R j,k. Next, reconstructing w in the y-direction yields w N j,k = w j,k + y (w y) j,k, w S j,k = w j,k y (w y) j,k, where ( (w y ) j,k = minmod θ w j,k+ w j,k y, w j,k+ w j,k, θ w ) j,k w j,k. y y Finally, the obtained point values of w are used to evaluate the corresponding point values p from (.3): p N j,k = wj,k N R j,k+, p S j,k = wj,k S R j,k,

37 4 and E from (.5): E N j,k = pn j,k γ + ( (ρu) N j,k ) + ( (ρv) N j,k ) ρ N j,k, E S j,k = ps j,k γ + ( ) (ρu) S ( ) j,k + (ρv) S j,k. ρ S j,k Well-Balanced Evolution The cell-averages of q are evolved in time according to the following system of ODEs: d dt q j,k = F j+,k F j,k G j,k+ G j,k +S j,k. (.35) x y Here, F and G are numerical fluxes. Introducing the notations α j+ a +,k := j+,ka j+,k a + and β j,k+ j+,k a j+,k := b + b j,k+ j,k+ b + j,k+ b j,k+,

38 5 we write the components of F and G as a + = j+,k(ρu)e j,k a j+,k(ρu)w j+,k j+,k a + j+,k a j+,k F () F () ( wj+,k + α j+,kh w j,k x a + = j+ j+,k F (3),k ( ρ E j,k (u E j,k ) + p E j,k ( ) + α j+ (ρu),k W j+,k (ρu) E j,k, a + = j+ j+,k F (4) ) (ρ W j+,k ρ E j,k), xk R + x kl max j,k {w j,k } ) a j+,k(ρw j+,k (uw j+,k ) + p W j+,k ) a + j+,k a j+,k ( ) a + + α j+ (ρv) j+,k a j+,k,k W j+,k (ρv) E j,k,,kρe j,k ue j,k ve j,k a j+,kρw j+,k uw j+,k vw j+,k a + = j+,kue j,k (EE j,k + pe j,k ) a j+,kuw j+,k (EW j+,k + pw j+,k ) j+,k a + j+,k a j+,k + α j+ ( ) E,k W j+,k Ej,k E, G () j,k+ G () j,k+ G (3) j,k+ = b + (ρv) N j,k+ j,k b (ρv) S j,k+ j,k+ + β j,k+ H = = b j,k+ ( wj,k+ w j,k b + j,k+ y yk R + y kl max j,k {w j,k } b + ρ N j,k+ j,k un j,k vn j,k b ρ S j,k+ j,k+ us j,k+ vs j,k+ b + j,k+ b + j,k+ ) (ρ S j,k+ ρ N j,k), ( ) + β j,k+ (ρu) S j,k+ (ρu) N j,k, b j,k+ ( ( ρ N j,k (vj,k N ) + wj,k) N b j,k+ ρ S j,k+ (vj,k+ S ) + wj,k+) S b + b j,k+ j,k+ G (4) j,k+ ( ) + β j,k+ (ρv) S j,k+ (ρv) N j,k, =b + v j,k+ j,k N (EN j,k + pn j,k ) b v j,k+ j,k+ S (ES j,k+ + ps j,k+ ) ( ) b + b + β j,k+ E S j,k+ j,k+ j,k+ Ej,k N, where the function H in the first components of the x and y numerical fluxes is as before defined in (.8). The cell averages of the source term in (.35) are approximated

39 6 using the midpoint quadrature rule as follows: S j,k = (0, 0, 0, g(ρv) j,k ) T. Finally, we state the following well-balanced property of the proposed -D CU scheme. Theorem.3.. The -D semi-discrete CU scheme described in.3. and.3. above is well-balanced in the sense that it preserves the steady state (.). Proof. The proof is similar to the proof of Theorem....4 Numerical Examples In this section, we present a number of -D and -D numerical examples, in which we demonstrate the performance of the proposed well-balanced semi-discrete CU scheme. In all of the examples below, we have used a third-order strong stability preserving (SSP) Runge-Kutta method (see, e.g., [5,6,44]) to solve the ODE systems (.4) and (.35). The CFL number has been set to 0.4. Also, we have used the following constant values: the minmod parameter θ =.3 and the specific heat ratio γ = One-Dimensional Examples Example. Shock Tube Problem. The first example is a modification of the Sod shock tube problem taken from [3,34]. As pointed out in [3], one can explicitly impose the well-balanced condition, however, such treatment will damage the ability of the resulting numerical scheme to capture the shocks. By solving the following benchmark problem, we would to demonstrate that our proposed scheme possesses the shock-capturing property.

40 7 We solve the -D system (.6), (.7) with g = in the computational domain [0, ] using the following initial data: (, 0, ), if y 0.5, (ρ(y, 0), v(y, 0), p(y, 0)) = (0.5, 0, 0.), if y > 0.5, and reflecting boundary conditions at the both ends of the computational domain. These boundary conditions are implemented using the ghost cell technique as follows: ρ kl := ρ kl, v kl := v kl, w kl := w kl, ρ kr + := ρ kr, v kr + := v kr, w kr + := w kr, where N := k R k L + is a total number of grid cells. We compute the solution using N = 00 uniformly placed grid cells and compare it with the reference solution obtained using N = 000 uniform cells. In Figure., we plot both the coarse and fine grid solutions at time T = 0.. As one can see, the proposed CU scheme captures the solutions on coarse mesh quite well showing a good agreement with both the reference solution and the results obtained in [3, 34]. Example. Isothermal Equilibrium Solution. In the second example, taken from [34] (see also [30 3]), we test the ability of the proposed CU scheme to accurately capture small perturbations of the steady state ρ(y) = e y, v(y) 0, p(y) = ge y, (.36) which satisfies (.0). We take the computational domain [0, ] and use a zero-order extrapolation at the

41 Density N=00 N= x N=00 N=000 Velocity x Energy N=00 N= Pressure N=00 N= x x Figure.: Example.: Solution (ρ(y, 0.), v(y, 0.), E(y, 0.) and p(y, 0.)) computed using N = 00 and N = 000 cells. boundaries: ρ kl := ρ kl, v kl := v kl, w kl := w kl, ρ kr + := ρ kr, v kr + := v kr, w kr + := w kr. Note that the boundary conditions on w can be recast in terms of p and ρ as p kl = p kl + g yρ kl, p kr + = p kr g yρ kr. We first numerically verify the well-balanced property of the proposed CU scheme by solving the -D system (.6), (.7) with g = subject to the initial conditions corresponding to the steady state (.36). We use several uniform grids and observe

42 9 that the initial conditions are preserved within the machine accuracy. Next, we introduce a small initial pressure perturbation and consider the system (.6), (.7) subject to the following initial data: ρ(y, 0) = e y, v(y, 0) 0, p(y, 0) = ge y + ηe 00(y 0.5), where η is a small positive number. In the numerical experiments, we use larger (η = 0 ) and smaller (η = 0 4 ) perturbations. We first apply the proposed well-balanced CU scheme to this problem and compute the solution at time T = 0.5. The obtained pressure perturbation (p(y, 0.5) ge y ) computed using N = 00 and N = 000 (reference solution) uniform grid cells are plotted in Figure.3 for both η = 0 and η = 0 4. As one can see, the scheme accurately captures both small and large perturbations on a relatively coarse mesh with N = 00. In order to demonstrate the importance of the well-balanced property, we apply the non-well-balanced CU scheme described in.. to the same initialboundary value problem. The obtained results are shown in Figure.3 as well. It should be observed that while the larger perturbation is quite accurately computed by both schemes, the non-well-balanced CU scheme fails to capture the smaller one. x 0 3 x initial state WB, N=00 WB, N=000 Non WB, N= initial state WB, N=00 WB, N=000 Non WB, N= x x Figure.3: Example.: Pressure perturbation (p(y, 0.5) ge y ) computed by the well-balanced (WB) and non-well-balanced (Non-WB) CU schemes with N = 00 and N = 000 for η = 0 (left) and η = 0 4 (right).

43 30.4. Two-Dimensional Examples Example.3 Isothermal Equilibrium Solution. The first -D example was studied in [34]. We consider the system (.3), (.3) with g = subject to the initial data that are in an isothermal equilibrium: ρ(x, y, 0) = ρ 0 e ρ 0 gy p 0, p(x, y, 0) = p 0 e ρ 0 gy p 0, u(x, y, 0) v(x, y, 0) 0, (.37) where ρ 0 =. and p 0 =, and the solid wall boundary conditions imposed at the edges of the unit square [0, ] [0, ]. We compute the solution until the final time T = using the proposed wellbalanced CU scheme on 50 50, and uniform cells. On all of these grids, the initial data are preserved within the machine accuracy. On contrary, the non-well-balanced CU scheme preserves the initial equilibrium within the accuracy of the scheme only, as can be seen in Table., where we present the L -errors for both ρ, ρu, ρv and E components of the non-well-balanced solution. N N ρ ρu ρv E E E E E E E+00.07E E E E+00 7.E-06.57E-05 Table.: Example.3: L -errors for the non-well-balanced CU scheme. Next, we add a small perturbation to the initial pressure (compare with (.37)): p(x, y, 0) = p 0 e ρ 0 gy p 0 + ηe 00ρ 0 g p 0 ((x 0.3) +(y 0.3) ), η = 0 3. In Figures.4 and.5 (upper row), we plot the pressure computed by both the well-balanced and non-well-balanced CU schemes at time T = 0.5 using uniform cells. As one can clearly see, the well-balanced CU scheme can capture the small pressure perturbation much more accurately than the non-well-balanced one.

44 3 When the mesh is refined to uniform cells, the non-well-balanced solution becomes better, but still less accurate than the well-balanced one, see Figure.5 (lower row). Figure.4: Example.3: Pressure perturbation computed by the well-balanced (left) and non-wellbalanced (right) CU schemes using uniform cells. Example.4 Explosion. In the second -D example, we compare the performance of well-balanced and non-well-balanced CU schemes in an explosion setting and demonstrate nonphysical shock waves generated by non-well-balanced scheme. We solve the system (.3), (.3) with g = 0.8 in the computational domain [0, 3] [0, 3], subject to the following initial data: ρ(x, y, 0), u(x, y, 0) 0, 0.005, (x.5) + (y.5) < 0.0, p(x, y, 0) = gy + 0, otherwise. Zero-order extrapolation is used as the boundary conditions in all of the directions. We use a uniform grid with 0 0 cells and compute the solution by both the well-balanced and non-well-balanced CU schemes until the final time T =.4. At first, a circular shock wave is developed and later on it transmits through the boundary. Due to the heat generated by the explosion, the gas at the center expands

45 Figure.5: Example.3: Contour plot of the pressure perturbation computed by well-balanced (left column) and non-well-balanced (right column) CU schemes using (upper row) and (lower row) uniform cells. and its density decreases generating a positive vertical momentum at the center of the domain. In Figures.6 and.7, we plot the solution (ρ and u + v at times t =.,.8 and.4) computed by the well-balanced and non-well-balanced schemes, respectively. As one can see, the well-balanced scheme captures the behavior of the solution at all stages, while the non-well-balanced scheme produces significant oscillations at the smaller time t =., which totally dominate the solution, especially its velocity field, by the final time T =.4.

46 33 Figure.6: Example.4: Density (ρ) and velocity ( u + v ) computed by the well-balanced CU scheme. Figure.7: Example.4: Density (ρ) and velocity ( u + v ) computed by the non-well-balanced CU scheme.

47 34 Chapter 3 Steady State Preserving Semi-Implicit Runge-Kutta Methods The contents of this chapter have been submitted to SIAM Journal on Numerical Analysis. In this chapter, we develop a family of second-order semi-implicit time integration methods for systems of ordinary differential equations (ODEs) with stiff damping term. The important feature of the new methods resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only. The new semi-implicit methods are based on the modification of explicit strong stability preserving Runge-Kutta (SSP-RK) methods and are proven to have a formal second order of accuracy, A(α)-stability and stiff decay. We illustrate the performance of the proposed SSP-RK based semi-implicit methods on both a scalar ODE example and a system of ODEs arising from the semi-discretization of the shallow water equations with stiff friction term. The obtained numerical results

48 35 clearly demonstrate that the ability of the introduced ODE solver to exactly preserve equilibria plays an important role in achieving high resolution when a coarse grid is used. 3. Introduction In this chapter, we consider the numerical integration of ordinary differential equations (ODEs) of the form u = f(u, t) + G(u, t)u, (3.) where u = u(t) R N is an unknown vector function, f : R N R N is a given vector field and G : R N N R N N is a diagonal non-positive definite matrix representing a (stiff) damping term. Systems like (3.) may arise from semi-discretizations of a time-dependent partial differential equations (PDEs) by the method of lines. In such case, the vector u usually corresponds to the spatial discretization of an unknown quantity and functions f and G correspond to the spatial discretization of the terms of different types in the given PDE (e.g., nonlinear hyperbolic fluxes, friction/source terms, etc). Development of numerical methods for the system (3.) is a challenging task due to the presence of the stiff damping term, which can lead to a great loss in accuracy and efficiency of the method. In many applications, the explicit treatment of the stiff damping term imposes a severe time step restriction which is several orders of magnitude smaller than a typical time step used for the corresponding damping-free version of the studied system. An attractive alternative to explicit methods is implicit-explicit (IMEX) Runge- Kutta (RK) schemes, which treat the stiff part of (3.) implicitly and thus typically have the stability domains based on the nonstiff term only, see, e.g., [45 50]. Modern IMEX-RK methods are based on a combination of explicit strong-stability preserving

49 36 Runge Kutta (SSP-RK) methods [5, 6, 5] for the nonstiff terms and an L-stable implicit RK methods [5, 53] for the stiff terms. SSP methods are popular since when solving an ODE system obtained as a semi-discretization of a nonlinear timedependent PDE, a stronger (nonlinear) stability is required to resolve discontinuous solutions of the underlying PDE in a non-oscillatory manner, see, e.g. [9, 44]. Another remedy to avoid severe time step and accuracy limitations is to use a semiimplicit method, in which the stiff term is discretized in a semi-implicit approach, that is, only a portion of the stiff term is implicitly treated. Such semi-implicit methods have been widely used in shallow water equations (see, e.g., [54, 55]) and other applications (see, e.g., [56,57]). One of the strategies is to let the first factor in in (3.), G(u, t), be treated explicitly and the second factor u be treated implicitly. For such methods, the extra time step restriction introduced by the stiff terms are much milder than their fully explicit counterpart and thus semi-implicit methods are typically much more efficient. In addition, compared to the time integration methods with a fully implicit treatment of the damping terms, the evolution equation for the semi-implicit treatment is very easy to solve and implement. In many practical applications, some special properties of the ODE solver are demanded to reflect the critical characters of the solution of the system (3.) that represent significant physical features of the underlying model. In this case, in order to preserve the physical meaning of the numerical solution, it is important to maintain these properties with both the spatial discretization and time integration. In particular, we are interested in problems whose solutions are small perturbations of steady states, u(t) û s.t. f(û, t) G(û, t)û t, (3.) and thus it is very important to derive a numerical method that preserves such steady state solutions exactly. Most of the IMEX-RK methods, however, do not to satisfy this requirement. Another key property of a numerical scheme for (3.) is to preserve

50 37 the sign of the numerical solution when the exact solution is either positive or negative. When, for instance, the initial condition u(0) and function f satisfy {u(0) 0, f 0} or {u(0) 0, f 0}, (3.3) the exact solution of (3.), u(t) maintains the same sign as u(0). Though the nonpositive stiff damping term G(u, t)u may be dominating, the numerical discretization of this term should not alter the sign of the numerical solution. The violation of such requirement may result in unphysical solution. Examples of a positivity preserving IMEX-RK methods can be found in [46, 58], but the use of a more restrictive than usual time step may be required to maintain the sign preserving property of these schemes. Some of the IMEX-RK methods can preserve the steady-state solutions, for example, the IMEX methods in [45]. However, the explicit parts of these methods do not belong to SSP family, and more importantly, the implicit parts of the methods in [45] contain negative entries in their Butcher tables of their implicit parts. These negative entries are dangerous in the sense that they introduce stability problems when a stiff problem is considered. In this chapter, we propose new second-order semi-implicit time integration methods for the system (3.), which are capable of preserving the steady state (3.) as well as maintaining the sign of solution under the condition (3.3). Our semi-implicit methods are based on the modification of explicit SSP-RK methods as shown in 3., where we also prove their formal second order of accuracy, A(α)-stability with α = π/4 and stiff decay, steady state and sign preserving properties under the time step restriction determined by the nonstiff part of the system (3.) only. The remaining part of the chapter is organized as follow. In 3.3, we study stability of the semi-implicit methods that are based on two popular SSP-RK explicit solvers. In 3.4, we illustrate the

51 38 performance of the proposed SSP-RK based semi-implicit methods on both a simple scalar ODE example and a system of ODEs arising from the spatial discretization of the shallow water equations with (stiff) friction term. 3. New Semi-Implicit Methods In this section, we develop a new class of second-order semi-implicit RK methods for the system (3.). A unique feature of the developed methods is their ability to exactly preserve the steady states (3.) and maintain the sign of the computed solution under the condition (3.3). For the simplicity of presentation, we consider here a scalar ODE with a nonstiff term f(u, t) and stiff damping term g(u, t)u such that g(u, t) 0: u = f(u, t) + g(u, t)u. (3.4) A general explicit m-stage RK method for (3.4) reads (see, e.g., [6]) u (0) = u n, i [ u (i) = α i,k u (k) + β i,k t(f (k) + g (k) u (k) ) ], i =,..., m, k=0 (3.5) u n+ = u (m), where f (k) := f(u (k), t (k) ) and g (k) := g(u (k), t (k) ). Here, t (k) := t n + D k t, where D k is given by i D 0 = 0, D i = α i,k (D k + β i,k ). (3.6) k=0 The RK method defined in (3.5) is fully determined by its coefficients {α i,k, β i,k }, which can be used to describe its properties. In particular, the consistency require-

52 39 ment of the i th intermediate solution u (i) can be written as i α i,k =, i =,..., m, (3.7) k=0 and for the final solution u n+ as D m =. (3.8) Note that the RK method (3.5) is in fact a linear combination of the first-order forward Euler (FE) steps, namely, we can rewrite where i u (i) = α i,k u FE i,k, (3.9) k=0 u FE i,k := u (k) + β i,k t(f (k) + g (k) u (k) ) (3.0) is, in fact, one-step FE evolution with the time step equal to β i,k t. According to [5, 6], the RK method is SSP provided the linear combination in (3.9) is a convex combination, that is, α i,k 0, for all i, k, and an appropriate time step restriction (based on the time step restriction for the FE method) is imposed. Also notice that negative time increments (which are undesirable when time irreversible equations are solved) are avoided if β i,k 0 for all i, k. Unfortunately, when the SSP-RK or any other explicit RK methods are applied to ODEs of the form (3.4), the stiffness of damping term will impose a severe time step restriction which will greatly impair the efficiency of the method. The aim of this work is to derive a class of efficient semi-implicit methods with a time step requirement, which depends on the nonstiff part only. To this end, we first replace the FE evolution steps (3.0), which are used as components in the RK method (3.5),

53 40 by the semi-implicit (SI) ones: u SI i,k := u (k) + β i,k t(f (k) + g (k) u SI i,k) u SI i,k = u(k) + β i,k tf (k) β i,k tg (k). (3.) This leads to the following semi-implicit scheme: u (0) = u n, i ( ) u u (i) (k) + β i,k tf (k) = α i,k, i =,..., m, β i,k tg (k) k=0 (3.) u n+ = u (m). However, the scheme (3.) with the coefficients {α i,k, β i,k } directly borrowed from (3.5) is at most first-order accurate (see Remark 3..). We, therefore, propose a correction step which rectifies the final stage solution u (m) : u n+ = u(m) C m ( t) f (m) g (m) + C m ( tg (m) ), (3.3) where the constant C m can be recursively computed by i C 0 = 0, C i = α i,k (C k + βi,k), i =,..., m. (3.4) k=0 Combining the semi-implicit evolution formula (3.) with the correction step (3.3), we introduce a new class of second-order semi-implicit Runge-Kutta (SI-RK) methods for (3.4): u (0) = u n, i ( ) u u (i) (k) + β i,k tf (k) = α i,k, i =,..., m, β i,k tg (k) k=0 u n+ = u(m) C m ( t) f (m) g (m) + C m ( tg (m) ), (3.5)

54 4 where the set of coefficients {α i,k, β i,k } is taken directly from the explicit SSP-RK method of an appropriate order. Remark 3... Note that in the degenerate case of g 0, the SI-RK method (3.5) is identical to the corresponding explicit RK method (3.5). In the remaining part of this section, we provide proofs of the second-order accuracy, A(α)-stability with α = π/4 and stiff decay, steady state and sign preserving properties of the new SI-RK methods. We begin with proving the following lemma, where we measure the difference between the intermediate solutions computed by the RK method (3.5) and SI-RK method (3.5). Lemma 3... Let us assume that the RK (3.5) and SI-RK (3.5) methods with β i,k 0 i, k are applied to equation (3.4) to evolve the solution u n for one time step from t n to t n+ = t n + t. We denote the obtained intermediate solutions by u (i) RK and u (i) SI-RK, respectively. Then, u (i) SI-RK u(i) RK = C i( t) g n (f n + g n u n ) + O(( t) 3 ), i = 0,..., m, (3.6) where C i are defined in (3.4). Proof. For the sake of simplicity, we define f (i) SI-RK := f(u(i) SI-RK, t(i) ), g (i) SI-RK := g(u(i) SI-RK, t(i) ), f (i) RK := f(u(i) RK, t(i) ) and g (i) RK := g(u(i) RK, t(i) ) for i = 0,..., m. We will prove the lemma by induction. First, for i = 0 equation (3.6) is true because u (0) SI-RK u(0) RK = un u n = 0, and C 0 = 0 as defined in (3.4). For i = l, 0 < l m, let us assume that (3.6) is true for all i l. The l th

55 4 intermediate solutions in (3.5) and (3.5) are l u (l) SI-RK = l u (l) RK = k=0 k=0 ( ) u (k) SI-RK α + β l,k tf (k) SI-RK l,k, (3.7) β l,k tg (k) SI-RK )] α l,k [ u (k) RK + β l,k t ( f (k) RK + g(k) RK u(k) RK The Taylor expansion of (3.7) with respect to t gives l u (l) SI-RK = [ α l,k u (k) SI-RK + β l,k t ( f (k) SI-RK + ) g(k) SI-RK u(k) SI-RK k=0 + β l,k( t) g (k) SI-RK ( f (k) SI-RK + g(k) SI-RK u(k). (3.8) SI-RK) ] + O(( t) 3 ). (3.9) By the induction assumption u (k) SI-RK = u(k) RK + C k( t) g n (f n + g n u n )) + O(( t) 3 ), 0 k l, and thus the second term in the summation on the right-hand side (RHS) of (3.9) can be written as follows: β l,k t ( f (k) SI-RK + ) g(k) SI-RK u(k) SI-RK = βl,k t ( f (k) RK + g(k) RK RK) u(k) + O(( t) 3 ). (3.0) By consistency of the SI-RK method (3.5) u (k) SI-RK = un + O( t), (3.) and hence the third term in the summation on the RHS of (3.9) is βl,k( t) g (k) ( (k) SI-RK f SI-RK + g(k) SI-RK SI-RK) u(k) = β l,k ( t) g n( f n + g n u n) + O(( t) 3 ). (3.) Finally, substituting (3.0) and (3.) into (3.9) and subtracting (3.8) from (3.9),

56 43 we obtain the desired estimate: u (l) SI-RK u(l) RK l ( = α l,k Ck ( t) g n (f n + g n u n ) + βl,k( t) g n (f n + g n u n ) ) + O(( t) 3 ) = k=0 ( l ) α l,k (C k + βl,k) ( t) g n (f n + g n u n ) + O(( t) 3 ) k=0 (3.4) = C l ( t) g n (f n + g n u n ) + O(( t) 3 ). Remark 3... As it immediately follows from Lemma 3.., the scheme (3.) is at most first-order accurate. The correction step introduced in (3.3) is needed to increase the order to the second one, as shown in the following theorem. Theorem 3..3 (Second-Order Accuracy). Let us assume that the RK (3.5) and SI- RK (3.5) methods are applied to equation (3.4). If the RK method (3.5) is at least second-order accurate, then the corresponding SI-RK method (3.5) with the same set of coefficients α i,k, β i,k 0 is second-order provided the coefficient C m is calculated by (3.4). Proof. First, we use the Taylor expansion of (3.3) with respect to t to obtain u n+ SI-RK = u(m) SI-RK C m( t) g (m) (m) SI-RK (f SI-RK + g(m) SI-RK u(m) SI-RK ) + O(( t)3 ), which using the consistency condition (3.) can be rewritten as u n+ SI-RK = u(m) SI-RK C m( t) g n SI-RK(f n SI-RK + g n SI-RKu n SI-RK) + O(( t) 3 ). It then follows from Lemma 3.. that u n+ SI-RK = un+ RK + O(( t)3 ).

57 44 Finally, our accuracy assumption on the RK method (3.5) implies that its truncation error is u n+ RK u(tn+ ) = O(( t) l ), l 3, which, in turn, implies that the truncation error of the SI-RK method (3.5) is u n+ SI-RK u(tn+ ) = O(( t) 3 ). We have thus proved that the SI-RK method is second-order accurate. Next, we prove that the SI-RK methods are A(α)-stable with α = π/4 and have stiff decay for the equation u = g(u, t)u. Theorem 3..4 (A(α)-Stability and Stiff Decay). Let us assume that the SI-RK methods (3.5) are applied to equation (3.4) with f(u, t) 0 and g(u, t) λ, where λ C is a constant with Re λ < 0. Then, the resulting methods, which can be written as u n+ = R(z)u n, z = λ t, satisfy the following two requirements: R(z), z C such that Re z Im z ( A(α)-stability with α = π ) 4 (3.3) and R(z) 0 as Re z, (3.4) provided α i,k 0 and β i,k 0 for all i, k. Proof. We first write the function R(z) using the following recursive relationship for

58 45 R (i) (z) := u (i) /u n : R (0) (z) =, (3.5) i ( ) R R (i) (z) (z) = α i,k, β i,k z i =,..., m, (3.6) k=0 R(z) = R(m) (z) + C m z. (3.7) It then follows from (3.6), (3.7) and positivity of β i,k that i R (i) (z) α i,k R (k) (z) max 0 k i R(k) (z), i =,..., m, k=0 which together with (3.5) implies that R (m) (z) R (0) (z) =. (3.8) The A(α)-stability with α = π/4 is then obtained from the inequality + C m z, which is clearly true as long as Re z Im z and C m 0 (note that the latter is ensured by (3.4) and positivity of α i,k 0 for all i, k). In fact, C m > 0 since C m may be equal to zero only if α i,k β i,k 0, which contradicts the consistency requirement (3.8). Therefore, (3.7) together with (3.8) implies (3.4), which completes the proof of the theorem. We next prove the steady state preserving property of the proposed SI-RK methods. Theorem 3..5 (Steady State Preserving Property). Let us assume that the SI-RK methods (3.5) with β i,k 0 i, k are applied to equation (3.4). Then, if the computed

59 46 solution is at a steady state at time t n, that is, u n = û such that f(û, t) g(û, t)û t, it will remain at the same steady state, namely, u n+ = û. Proof. We first prove by induction that u (m) = û. Indeed, if u (k) = û for all k i, then i u (i) = k=0 i ( ) u (k) + β i,k tf (k) α i,k = β i,k tg (k) = α i,k û = û, k=0 i k=0 (û ) βi,k tg(û, t (k) )û α i,k β i,k tg(û, t (k) ) where the last equality is obtained using the consistency requirement (3.7). We then substitute u (m) = û into the correction step of the SI-RK methods (3.5) to end up with u n+ = u(m) C m ( t) f (m) g (m) + C m ( tg (m) ) = û + C m( tg(û, t (m) )) û + C m ( tg(û, t (m) )) = û. At the end of this section, we prove the sign preserving property of the proposed SI-RK methods. Theorem 3..6 (Sign Preserving Property). Let us assume that the SI-RK methods (3.5) with β i,k 0 i, k are applied to equation (3.4) with the initial condition u 0 and function f satisfying {u 0 0, f 0} or {u 0 0, f 0}.

60 47 Then, sgn(u n ) sgn(u 0 ) (3.9) for all n provided α i,k 0 and β i,k 0 for all i, k. Proof. First, assume that u 0 0. It will be enough to show that u > 0 ((3.9) will then follow by induction). The positivity of u immediately follows from (3.5) by taking into account that f 0, g 0, α i,k 0, β i,k 0 and C m 0. The case of u 0 < 0 is completely analogous and thus the proof of the theorem is complete. 3.3 Absolute Stability of Two SSP-Based SI-RK Methods In this section, we study the absolute stability of two SI-RK methods, which are particular cases of the general SI-RK methods (3.5). The first SI-RK method, based on the second-order SSP-RK solver [5, 6] also known as the Heun method [59], reads u () = un + tf n tg n, u () = un + u() + tf () tg (), u n+ = u() ( t) f () g () + ( tg () ), (3.30)

61 48 and the second one, based on the third-order SSP-RK method, can be written as u () = un + tf n tg n, u () = 3 4 un + 4 u() + tf () tg (), u (3) = 3 un + 3 u() + tf () tg (), u n+ = u(3) ( t) f (3) g (3) + ( tg (3) ). (3.3) In what follows, the SI-RK method (3.30) will be referred to as the SI-RK method, while the SI-RK method (3.3) will be referred to as the SI-RK3. To analyze the absolute stability, we consider the following test problem y = λ y + λ y, λ C, Re(λ ) 0, λ R, λ 0, (3.3) where λ y and λ y are the nonstiff and stiff parts, respectively. We denote z := λ t and z := λ t. Applying the schemes (3.30) and (3.3) to (3.3), that is, substituting f(u, t) = λ u and g(u, t) = λ into (3.30) and (3.3), results in u () = + z z u n, u () = un + + z z u (), (3.33) u n+ = z z + z u (),

62 49 and respectively. u () = + z z u n, u () = 3 4 un z z u (), u (3) = 3 un z z u (), u n+ = z z + z u (3), (3.34) It should be observed that if g 0, the underlying ODE (3.4) is nonstiff and thus λ = 0 in the test problem (3.3). In this case, the SI-RK and SI-RK3 methods reduce to the second-order and third-order SSP-RK methods, respectively, and their stability regions are known, see, e.g., [5, 6]. Let us denote these stability regions by D SSP and D SSP3, respectively, and the corresponding time step restrictions by t t SSP and t t SSP3, given parameter λ. Theorem 3.3. (Absolute Stability of the SI-RK Method). The region of absolute stability of the SI-RK method (3.30) contains D SSP, that is, for any z 0, the solution of (3.33) satisfies u n+ u n provided t t SSP. Proof. First, we note that the stability functions for the SI-RK and the corresponding second-order SSP-RK methods are R SI-RK(z, z ) = z z + z [ + ( ) ] + z z and R SSP (z ) = + ( + z ), respectively. To prove the theorem, it will be enough to show that both + ( ) + z z (3.35)

63 50 and z z + z (3.36) for all z, z such that R SSP (z ) and z 0. It follows from the definition of R SSP (z ) that + z for all z S, where S = {z C z DSSP }. Note that S is a convex region that encloses the origin. Hence, for any z S and any z 0, we have all z D. z z S, which implies that (3.35) for We now turn to the proof of (3.36), which is obviously true for z = 0. We therefore consider z < 0, for which (3.36) is equivalent to z + z z + z. For a fixed z < 0, this inequality is satisfied in a disk with radius z + z centered at z = z. Denoting z := x + iy, we can write this domain as { } C(z ) := z z z + z { = x + iy ( y z + ) ) } (x z, z < 0. z (3.37) We thus need to show that D SSP C := z <0 C(z ). Finding the intersection of C(z ) s is equivalent to minimizing the set in (3.37) over z < 0, which results in C = = { x + iy y min z <0 [ ( z + ) ) ]} (x z { x + yi [ y + 3x /3 x, x ]}, 0. z Therefore the boundary of C consists of the curve y = +3x /3 x, x 0 and a part of the y-axis from (0, ) to (0, ). The boundaries of D SSP and C

64 5 are shown in Figure 3., which clearly indicates that, D SSP C, and thus the proof of the theorem is complete Figure 3.: Stability region D SSP contained in C. Remark Our numerical experiments clearly indicate that a similar result holds for the SI-RK3 method as well. However, no rigorous proof of that fact is available and therefore we formulate it as a conjecture and provide supporting numerical evidences. Conjecture 3.3. (Absolute Stability of the SI-RK3 Method). The region of absolute stability of the SI-RK3 method (3.3) contains D SSP3, that is, for any z 0, the solution of (3.34) satisfies u n+ u n provided t t SSP3. Discussion: The stability functions for the SI-RK3 and the corresponding thirdorder SSP-RK methods are R SI-RK3(z, z ) = z z + z [ 3 + ( ) + z + ( ) ] 3 + z z 6 z and R SSP3 (z ) = 3 + ( + z ) + 6 ( + z ) 3, respectively. The statement of the conjecture would be true if one could show that R SI-RK3(z, z ), z such that R SSP3 (z ) and z 0. (3.38)

65 5 In order to verify (3.38), we first observe that R SI-RK3(z, 0) = R SSP3 (z ) and first consider the case z 3. It follows from the definition of R SSP3 (z ) that 3 + z + 6 z3 for all z S, where S = {z C z DSSP3 }. Note that S is a region that encloses the origin O and contains the segments Oz for all z S. Hence, for any z S and any z 0, we have z z S, which implies that 3 + ( +z z ) + 6 ( +z z ) 3. Notice that D SSP3 is contained in B 3 (O), which is the disk of radius 3 centered at the origin, and therefore we have z z +3 z +z + z. Hence, we obtain that R SI-RK3(z, z ) in the case z 3. To study the case z ( 3, 0), we introduce a polynomial P (x, y) := R SSP3 (x + iy) and a rational function Q(x, y, z ) := R SI-RK3(x + iy, z ). For fixed values of z, the curves given by P (x, y) = 0 and Q(x, y, z ) = 0 are boundaries of the domains D SSP3 and D SI-RK3(z ), respectively (we denote by D SI-RK3(z ) the stability domain for the SI-RK3 method for a fixed value of z ). To determine whether D SSP3 D SI-RK3(z ), we only need to show that D SSP3 is enclosed by D SI-RK3(z ) for all z ( 3, 0). To this end, we consider P (x, y) and Q(x, y, z ) as polynomials of a single variable x and compute their resultant K(y, z ) := res(p, Q) = K(y, z ) (z ) 36 (z + ), where K(y, z ) is a specific function, whose explicit expression is quite complicated and not instructive and we thus omit it for the sake of brevity. Instead, we use a graphic software to visualize K. In Figure 3. (left), we plot log 0 ( K(y, z ) + ), which clearly indicates that K(y, z ) > 0 for all (y, z ) [.4,.4] ( 3, 0) (note that we take the y-bounds to be [.4,.4] in order to include the entire D SSP3 into the studied domain in the (y, z )-plane). This implies that D SSP3 and D SI-RK3(z ) have no intersections when z ( 3, 0). To cite an example, we take t = and illustrate

66 53 in Figure 3. (right) that D SSP3 D SI-RK3( ). Since K(y, z ) is continuous, we conclude that D SSP3 D SI-RK3(z ) for all z ( 3, 0) z y Figure 3.: Surface plot of log 0 ( K(y, z ) + ) (left); the stability domains D SSP3 and D SI-RK3( ) (right). To further verify the relationship between the stability functions R SI-RK3 and R SSP3, we numerically evaluate max z,z R SI-RK3(z, z ) under a stronger restriction on R SSP3 (z ) : R SSP3 (z ) α <. (3.39) The obtained results, presented in Figure 3.3, indicate even stronger dependence of R SI-RK3 and R SSP3, namely, R SI-RK3(z, z ) α, z such that R SSP3 (z ) α and z 0. Remark Since the statement of Conjecture 3.3. has not been proved, we suggest the following strategy, which should guarantee the stability of the SI-RK3 method in practice. The time step t should be chosen such that (3.39) is satisfied with α being slightly smaller than, say, α = 0.98, which will ensure that R SI-RK3(z, z ) as desired for the stability.

67 54 R SI-RK3 max z,z α Figure 3.3: Conjecture 3.3.: max R SI-RK3(z, z ) over z such that R SSP3 (z ) α and z 0 as a function of α. Remark The results in Figure 3.3 are obtained in the most straightforward manner by sampling the variable domains with very fine meshes and then computing the discrete maxima. We use a mesh grid with space 0.0 to sample z in the region D SSP3. To sample z [0, ), we employ 000 sample points (z ) i = ( ξ i ), where ξ i = i/000, i = 0,..., 999. Note that the points ξ i are uniformly spaced in [0, ). 3.4 Numerical Examples In this section, we test the second-order SI-RK3 method (3.3) on several examples including both scalar ODEs ( 3.4.) and systems of ODEs arising from semidiscretizations of PDEs ( 3.4.). We compare the results with the ones obtained using the second-order IMEX-SSP3(3,3,) method described in [49, Table 5]. The obtained results clearly demonstrate that the new SI-RK3 method outperforms the IMEX-SSP3(3,3,) when a large time step and/or coarse grid are used.

68 Scalar ODEs We consider the following scalar ODE: u = k u u, (3.40) where k is a positive real number. Equation (3.40) has one equilibrium point u = / k. Example 3. Accuracy Test. In this example, we apply the SI-RK3 scheme to equation (3.40) with k = 00 subject to the initial condition u(0) = 0.. We compute the solution until the final time T = 0. using uniform time stepping with t = /00, /00, /400, /800, /600 and show the absolute value of the error and the corresponding convergence rates in Table 3.. As expected, the experimental order of convergence is. t Error Rate /00.99E-04 /00 6.0E /400.64E / E /600.0E Table 3.: Example 3.: The errors and convergence rates of the second-order SI-RK3 method. Example 3. Steady State Preserving Test. In this example, we take k = 0000, which corresponds to the equilibrium point u = 0.0. We consider three different initial values: (a) u(0) = 0.9u, (b) u(0) = u, (c) u(0) =.u,

69 56 and solve equation (3.40) using both the SI-RK3 and IMEX-SSP3(3,3,) methods. The numerical results computed with t = /00, /00, /400, /800 and /600 are plotted in Figure 3.4. As one can see, when u is initially at the equilibrium (case (b)), the SI-RK3 method preserves the steady state exactly as expected (see Theorem 3..5), while the difference between the numerical steady states obtained by the IMEX-SSP3(3,3,) method and the exact steady state is of order O(( t) ). Similarly, in cases (a) and (c), the SI-RK3 method accurately captures and preserves the exact equilibrium, while the IMEX-SSP3(3,3,) method does not. Example 3.3 Sign Preserving Test. As in Example 3., we choose k = 0000 and solve equation 3.40 subject to the initial condition u(0) =, for which the exact solution must remain positive for all t. Once again, we implement both the SI-RK3 and IMEX-SSP3(3,3,) methods with t = /00, /00, /400, /800, /600 and plot the obtained results in Figure 3.5. As one can see, the SI-RK3 method preserves the positive sign of the computed solution (as proved in Theorem 3..6), while the IMEX-SSP3(3,3,) produces negative values of u Systems of ODEs Arising from Semi-Discretizations of PDEs In this section, we consider the Saint-Venant system [60] of shallow water equations with the friction term in classical Manning formulation [6 64]. In the onedimensional case, the system reads: h t + q x = 0, q t + (hv + g ) h x = ghb x g n h 7/3 q q, (3.4) where h(x, t) denotes the water depth, v(x, t) is the velocity, q(x, t) := h(x, t)v(x, t) is the discharge, and g is the gravitational constant. The first term on the RHS of

70 57 0. x 0 3 SI RK3 0. x 0 3 IMEX SSP3(3,3,) u(t) t=/00 t=/400 9 t=/800 Equilibrium t SI RK t=/00 t=/ t=/800 Equilibrium u(t) t=/00 t=/400 9 t=/800 Equilibrium t IMEX SSP3(3,3,) t=/00 t=/ t=/800 Equilibrium u(t) 0.0 u(t) t SI RK t=/00 t=/400 t=/800 Equilibrium t IMEX SSP3(3,3,) t=/00 t=/400 t=/800 Equilibrium u(t) u(t) t t Figure 3.4: Example 3.: Convergence toward the equilibrium for the SI-RK3 (left column) and IMEX- SSP3(3,3,) (right column) methods. (3.4) is the geometric source with B(x) representing the bottom topography function. The second term on the RHS of (3.4) models the bottom friction with n being the Manning coefficient. Solving the system (3.4) numerically is a challenging task due to the following reasons. First, the system admits several physically relevant steady states and many practically important solutions are in fact small perturbations of these steady states. A good numerical method should be well-balanced in the sense that it should be

71 SI RK3 t=/00 t=/400 t=/800 Equilibrium. 0.8 IMEX SSP3(3,3,) t=/00 t=/400 t=/800 Equilibrium u(t) u(t) t t Figure 3.5: Example 3.3: Solutions computed by the SI-RK3 (left) and IMEX-SSP3(3,3,) (right) methods. able to exactly preserve discrete versions of the relevant steady states since otherwise the numerical error can become larger than the waves to be captured. Second, the water depth h may be very small and even zero (in the islands and shore areas) and therefore it is crucial to design a numerical method that is capable of preserving the positivity of the computed water depth. Finally, when h is small in certain parts of the domain, the friction term in (3.4) becomes stiff and thus the use of an implicit or semi-implicit discretization of this term is required for designing an efficient numerical method. We solve the system (3.4) using the second-order semi-discrete central-upwind scheme from 3, see also [3, 3]. For simplicity, we consider the semi-discretization framework, in which the computational domain is divided into N uniform cells and thus the system of PDEs (3.4) reduces to an ODE system consisting of N coupled equations. As it was shown in [3], the overall method is both well-balanced and positivity preserving provided the system of ODEs is integrated using an explicit SSP ODE solver. To design a method, which is also efficient in the stiff regime, one can use the proposed SI-RK methods (3.5), which, as we have proved in 3., are both steady state and sign preserving. We consider a nonsmooth periodic flow over the slanted surface. In this setting, physically relevant steady states satisfy h Const, q Const, B x Const. In the

72 59 numerical examples below, we implement the SI-RK3 method (3.3), which exactly preserves the above steady states and the sign of the velocity. We again compare the performance of the SI-RK3 method with the IMEX-SSP3(3,3,) method, which is also efficient, but is neither steady state or sign preserving. These features play an important role in the ability of the numerical scheme to capture the correct numerical solution of the shallow water system even when it is still far from the steady state as demonstrated below. We numerically solve the system (3.4) with B x 0., n = 0.09 and subject to the following initial conditions: 0.0, x < 50, h(x, 0) = 0.0, x > 50, 0, x < 50, q(x, 0) = 0.04, x > 50. (3.4) We restrict the computational domain to [0, 00], which is divided into N = 00 uniform cells, and impose the periodic boundary conditions. Example 3.4 Time Steps Restricted by the CFL Condition. In this example, we conduct numerical experiments in which the time step t is adaptively determined based on the CFL condition with the CFL number 0.3. We first implement the proposed SI-RK3 method and plot the solution (h and v) computed at time t = 000 in Figures 3.6(a) and (b). As one can see, the results obtained on course (N = 00) and fine (N = 000) grids are in good agreement (obviously, the shock is sharper resolved on a fine grid). In addition, in Figure 3.6(c), we plot the value of the velocity v at x = 50 as a function of time. The figure clearly demonstrates that both the velocity v and the speed of the shock are captured quite accurately even with N = 00. Also note that the time steps shown in Figure 3.6(d) slightly increase in time, since they are adaptively selected using the CFL condition.

73 60 (a) N = 00 N = (b) N = 00 N = 000 h(x, 000) v(x, 000) x x (c) 0.8 (d) v(50, t) 0. 0 N = 00 N = t t(t) N = t Figure 3.6: Example 3.4: Solution obtained using the SI-RK3 method: the water depth (a) and velocity (b) at time t = 000; the velocity (c) at x = 50 as a function of time. The size of time steps (d) as a function of time. The solution computed on a course grid (N = 00) using the IMEX-SSP3(3,3,) time integration is shown in Figure 3.7 and compared with the reference solution obtained by the same IMEX-SSP3(3,3,) method with N = 000. As one can see, the results show considerable disagreements between the fine- and coarse-grid solutions: In Figure 3.7(a), a phase error in h can be noticed. Such a delay in shock propagation is due to the large numerical error in velocity magnitude, see Figure 3.7(b). In Figure 3.7(c), one can also notice that the error in velocity magnitude arises initially and gradually increases in time (here again we show the values of the computed v at x = 50 as a function of time). Notice that at large times, v(50, t) often admits negative values, which are unphysical and trigger oscillations that develop in both space and time as can be clearly seen in Figures 3.7(b) (d). Example 3.5 Fixed Time Step Restriction. We have demonstrated in Example 3. that the SI-RK3 method converges to the exact equilibrium as long as the stability restriction is satisfied while the IMEX-SSP3(3,3,) method delivers different

74 6 (a) N = 00 N = (b) N = 00 N = 000 h(x, 000) v(x, 000) x x (c) 0.8 (d) v(50, t) 0. 0 N = 00 N = t t(t) N = t Figure 3.7: Example 3.4: Solution obtained using IMEX-SSP3(3,3,) method: the water depth (a) and velocity (b) at time t = 000; the velocity (c) at x = 50 as a function of time. The size of time steps (d) as a function of time. numerical equilibria depending on the time step adopted within the stability region. When both ODE solvers are applied to the system of ODEs arising from the centralupwind semi-discretization of the shallow water system, the computed water depth h and velocity v are affected by the choice of t in a similar manner. To illustrate this, we use a combination of the CFL condition and fixed time restriction: t = min{ t CFL, t max }, and implement the SI-RK3 and IMEX-SSP3(3,3,) with t max = 0.3, 0.5 and 0.0. In Figures 3.8(a) (c), we show h(x, 000), v(x, 000) and v(50, t) computed using the SI-RK3 time integration method. As one can observe, the results obtained with different max s are visually indistinguishable. In addition, a satisfactory accuracy of the obtained solution is confirmed by comparing it with the reference ones computed by the same SI-RK3 method with N = 000 and t determined by the CFL condition. The t profile depicted in Figure 3.8(d) indicates that t is effectively controlled by

75 6 t max and is thus constant for almost all t. (a) 0.4 (b) h(x, 000) v(x, 000) x (c) t max = 0.0, N = 00 t max = 0.5, N = 00 t max = 0.3, N = 00 N = x 0.8 (d) v(50, t) 0. 0 t(t) t t Figure 3.8: Example 3.5: Solutions obtained using the SI-RK3 method with fixed time step restrictions: the water depth (a) and velocity (b) at time t = 000; the velocity (c) at x = 50 as a function of time. The size of time steps (d) as a function of time. The results obtained using the IMEX-SSP3(3,3,) time integration are shown in Figure 3.9 and are free of oscillations in both time and space due the fix time step employed. However, a great disagreement can be observed when a large value of t max is used. When the time step decreases, one can still notice a decreasing error (both in velocity magnitude and shock speed), which indicates that the dominating error is introduced by the time integration method. Of course, once a very small time step ( t max = 0.0) or a very fine mesh is used, the results obtained become comparable with those obtained by the SI-RK3 method.

76 63 (a) 0.4 (b) h(x, 000) v(x, 000) x (c) t max = 0.0, N = 00 t max = 0.5, N = 00 t max = 0.3, N = 00 N = x 0.8 (d) v(50, t) 0. 0 t(t) t t Figure 3.9: Example 3.5: Solutions obtained using the IMEX-SSP3(3,3,) method with fixed time step restrictions: the water depth (a) and velocity (b) at time t = 000; the velocity (c) at x = 50 as a function of time. The size of time steps (d) as a function of time.

77 64 Chapter 4 Well-Balanced Central Upwind Scheme for the Shallow Water System with Friction Terms The contents of this chapter have been submitted to and accepted by International Journal for Numerical Methods in Fluids. Shallow water models are widely used to describe and study free-surface water flow. While in some practical applications the bottom friction does not have much influence on the solutions, there are still many applications, where the bottom friction is important. In particular, the friction terms will play a significant role when the depth of the water is very small. In this chapter, we study shallow water equations with friction terms and develop a semi-discrete second-order central-upwind scheme that is capable of exactly preserving physically relevant steady states and maintaining the positivity of the water depth. The presence of the friction terms increases the level of complexity in numerical simulations as the underlying semi-discrete system becomes stiff when the water depth is small. We therefore implement an efficient semiimplicit Runge-Kutta time integration method that sustains the well-balanced and

78 65 sign preserving properties of the semi-discrete scheme. We test the designed method on a number of one- and two-dimensional examples that demonstrate robustness and high resolution of the proposed numerical approach. The data in the last numerical example correspond to the laboratory experiments reported in [77], designed to mimic the rain water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, we develop a special technique, which helps to achieve a remarkable agreement between the numerical and experimental results. 4. Introduction Shallow water equations are a set of hyperbolic partial differential equations derived by a vertical integration of Navier-Stokes equations. They are widely used in atmospheric sciences, oceano-graphy, coastal engineering and many other fields. In shallow water flow models, the horizontal length scale is considered to be much larger than the vertical one. As a result, the vertical effect can be neglected leading to a considerable simplification in the momentum equation, in which the vertical pressure gradients are replaced by the hydrostatic pressure. The simplest, yet commonly used, shallow water model is the Saint-Venant system [60], which in the two-dimensional (-D) case reads: h t + (hu) x + (hv) y = R(x, y, t), (hu) t + (hu + g ) h + (huv) y = ghb x, x (hv) t + (huv) x + (hv + g ) h = ghb y. y (4.) Here, h(x, y, t) is the water depth, u(x, y, t) and v(x, y, t) are the x- and y-components of the average velocity, R(x, y, t) is the water source term, B(x, y) is a function describing the bottom topography, g is the gravity constant. Solving the shallow water system numerically is a challenging task due to several

79 66 reasons. First, many physically relevant solutions of (4.) are small perturbations of steady states, characterized by a delicate balance between the flux and source terms. If the method does not accurately respect this balance, the numerical errors (which cannot be made too small on practically relevant grids) may lead to oscillations, in which the magnitude of artificial waves may be larger than the magnitude of the solution itself. The second major difficulty is related to the computation of solutions when the water depth h is very small or even zero. In such a case, small numerical oscillations may lead to appearance of negative values of h, which in turn would make it impossible to evaluate the eigenvalues of the system (4.), which are u ± gh and v ± gh. A good numerical method for the system (4.) should thus be well-balanced (in the sense that it must exactly preserve physically relevant steady states) and positivity preserving (in the sense that the computed values of h must be positive). In the past two decades, many well-balanced scheme have been developed (see, e.g., [3,4 6,3, 4, 6 8, 65 7]). Some of them preserve only lake at rest steady states, that is, u v 0, h + B constant, [3,4 6,3,4,8,65 68], other can preserve a nonflat steady-state solution as well, [6, 7, 70, 7]. There are also well-balanced schemes that preserve the positivity of h (see, e.g., [3, 4 6, 3, 8, 65, 66, 68]). In this chapter, we focus on studying the effects of the bottom friction terms in the shallow water model and thus we consider the following modified version of (4.): h t + (hu) x + (hv) y = R(x, y, t), (hu) t + (hu + g ) h + (huv) y = ghb x τ x x ρ, (hv) t + (huv) x + (hv + g ) h = ghb y τ y y ρ, (4.) where τ x and τ y are the two components of the bottom friction and ρ is the water

80 67 density. The friction terms are computed by using the following formulae: τ x ρ = ghix, τ y ρ = ghiy, (4.3) where the I x and I y are the components of the bottom friction slope. There are many ways to model friction terms, see, e.g. [7,73]. In this chapter, we focus on the classical Manning formulation (see, e.g., [6 64]): I x = n h 4/3 u u + v, I y = n h 4/3 v u + v, (4.4) where n is the Manning coefficient. Notice that if h 0, the friction term (4.4) becomes a stiff damping term, which increases the level of complexity in the development of efficient numerical methods for the system (4.). The system (4.) still admits lake at rest steady states. However, we are interested in simulating drainage of the rain water in urban areas. In such situations, the simplest yet physically relevant quasi one-dimensional (-D) steady-state solutions correspond to the case when the water flows over a slanted infinitely long surface with a constant slope as illustrated in Figure 4. (left). Such steady states (both -D and -D) are discussed in 4.. h(x) B(x) x Figure 4.: The bottom setting of numerical examples. The figure on the left corresponds to the quasi -D steady state. The figure on the right illustrates the case of urban draining with obstacles like houses. The slope of the bottom and the height of the houses are out of scale.

81 68 A well-balanced Roe-type numerical scheme that exactly preserves steady states shown in Figure 4. was proposed in [74]. However, to maintain the positivity of the water depth h, the scheme in [74] requires one to use very small time steps and thus may not be robust in certain settings. Another Godunov-type scheme for the -D version of (4.) was proposed in [75]. Though this method does not suffer from restrictive time stepping, it is capable of preserving lake at rest steady states only. In this chapter, we develop a central-upwind scheme for the system (4.), which is well-balanced, positivity preserving and efficient. Central-upwind schemes have been proposed for general hyperbolic system of conversation law in [35 38] and extended to the shallow water equations and related models in [3, 3, 76]. These schemes belong to the family of Godunov-type central schemes that are Riemann-problem-solverfree, robust and highly accurate. The extension of central-upwind schemes to the shallow water systems with friction terms is very natural and it is described in both the -D ( 4.3.) and -D ( 4.3.) cases (the -D scheme presented here is restricted to Cartesian meshes). The efficiency of the proposed central-upwind scheme hinges on the use of an efficient second-order semi-implicit ODE solver we have recently developed in [0] and briefly describe in The designed scheme is tested in a number of numerical experiments including those with realistic urban bottom topography structures, schematically shown in Figure 4. (right). The results presented in 4.4 demonstrate the superb performance of the proposed numerical method. The data in Example 4.8 correspond to the laboratory experiments reported in [77], designed to mimic the rain water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, the proposed central-upwind scheme has been modified to accurately handle such situations as follows. First, the houses are removed from the computational domain, which becomes a punctured domain with many internal solid wall boundary pieces. Second, the rain water falling

82 69 over the houses is redistributed to the areas near the edges of the houses. This helps to achieve a remarkable agreement between the numerical and experimental results. 4. Steady-State Solutions In this section, we discuss steady-state solutions of the shallow water system (4.). We begin with the simplest -D case, in which the system (4.) reduces to h t + (hu) x = R(x, t), (hu) t + (hu + g ) h x = ghb x g n h /3 u u. (4.5) In the situation when the water source is zero (R 0), the steady-state solution satisfies the time-independent system: (hu) x = 0, (hu + g ) h x = ghb x g n h /3 u u. (4.6) In general, this system is not solvable, but one can obtain a particular nontrivial (u 0) steady state in the form hu constant, h constant, B x constant. (4.7) This solution corresponds to the situation when the water flows over a slanted infinitely long surface with a constant slope. Indeed, if we assume that B x C, where C > 0 is a constant, and denote hu q 0, then the second equation of (4.6) can be rewritten as ( ) q 0 h + gh h x = ghc g n u u, (4.8) h/3

83 70 and one obtains h h 0 = ( ) n q0 3/0, hu q 0, B x C, (4.9) C where h 0 is the so-called normal depth. A simple analysis of the ODE (4.8) shows that this steady state is expected to be stable in the supercritical case, that is, when h 0 is below the critical depth h c : h 0 < h c := ( q0 ) /3. (4.0) g The structure of -D steady states is substantially more complicated. However, the quasi -D steady-state solutions: h constant, hu constant, hv 0, B x constant, B y 0, (4.) or h constant, hu 0, hv constant, B x 0, B y constant. (4.) are still physically relevant to the situation depicted in Figure 4.. In the next sections, we design both -D and -D central-upwind schemes that exactly preserve the above steady states (4.9) and (4.), (4.), respectively. 4.3 Numerical Method In this section, we present a well-balanced and positivity preserving semi-discrete central-upwind scheme for the shallow water equations with friction terms in both -D and -D. The scheme is derived along the lines of [3] and therefore here we only describe its main components following the key ideas from [3]. In particular,

84 7 the well-balanced property of the scheme will be ensured by a special finite-volumetype quadrature used for discretizing the geometric source term on the right-hand side (RHS) of the system. We also introduce a new variable for the water surface w := h + B. As it has been shown in [3], working with w (rather than with h) is important for preserving lake at rest steady states at the discrete level. Even though in this work, we focus on different types of steady states, described in 4., our goal is to design a numerical method, which preserves both the lake at rest and steady states (4.9), (4.) and (4.). The positivity preserving property is achieved by (i) replacing the bottom topography function B with its continuous piecewise linear (or bilinear in the -D case) approximation (done exactly the same way as in [3]) and (ii) a special positivity preserving correction of the piecewise linear reconstruction for the water surface w (which is different from the one proposed in [3]). It should be observed that a successful implementation of the central-upwind scheme would be impossible without the use of an accurate and efficient time integration method that maintains the aforementioned important features of the semidiscrete scheme. In what follows, we first overview the -D and -D central-upwind schemes in 4.3. and 4.3., respectively, and then, in 4.3.3, describe a new semiimplicit Runge-Kutta ODE solver we have recently developed in [0] One-Dimensional Central-Upwind Scheme We start with a description of a well-balanced positivity preserving central-upwind scheme for the -D shallow water equations (4.5). We first rewrite the system (4.5) in an equivalent form in terms of w := h + B and q := hu: w t + q x = R(x, t), [ q q t + w B + g ] (w B) x n (4.3) = g(w B)B x g (w B) q q. 7/3

85 7 We use the notations U := w, F (U, B) := q q w B + g q, (w B) and R(x, t) 0 S(U, R, B) :=, M(U, B) := n g(w B)B x g (w B) 7/3 q q, so that the system of balance laws (4.3) takes the following vector form: U t + F (U, B) x = S(U, R, B) + M(U, B). (4.4) For simplicity, we introduce a uniform grid x α := α x, where x is a small spatial scale and the corresponding finite volume cells C j := [x j, x j+ ], and assume that at certain time level t, the solution is realized in terms of its cell averages, U j (t) = x C j U(x, t) dx, which are evolved in time according to the semi-discrete central-upwind scheme (see [3, 3, 37]): d dt U j(t) = H j+ (t) H j (t) +S j (t) +M j (t) (4.5) x with the central-upwind numerical fluxes, H j±, given by H j+ = a + F (U, B j+ j+ j+ ) a F (U +, B j+ j+ j+ ) a + a + j+ j+ a + a j+ j+ a + j+ a j+ [ ] U + U. j+ j+ (4.6) Note that all of the indexed quantities in (4.6) depend on time, but from now on we suppress the time-dependence of indexed quantities in order to shorten the notation. In (4.6), U ± j+ are the right and left point values of the piecewise linear recon-

86 73 struction Ũ j (x) = U j + (U x ) j (x x j ), x C j, j, (4.7) at x = x j+ : U + j+ := U j+ x (U x) j+, U j+ := U j + x (U x) j. (4.8) To ensure the second-order accuracy and a non-oscillatory nature of the reconstruction, the numerical derivatives (U x ) j are to be computed using a nonlinear limiter, for example, the generalized minmod limiter: (U x ) j = minmod ( θ U j+ U j x, U j+ U j, θ U j U j x x ), where the minmod function is defined by minmod(z, z,...) := min(z, z,...), if z i > 0 i, max(z, z,...), if z i < 0 i, 0, otherwise, and the parameter θ [, ] controls the amount of numerical dissipation: The larger the θ the smaller the numerical dissipation. The use of a limiter does not, however, guarantee the positivity of h ± j+ := w ± j+ B j+, where B j+ := B(x j+ + 0) + B(x j+ 0). (4.9) Hence, to ensure the positivity of h ±, we first replace the bottom topography func- j+ tion B(x) with its continuous piecewise linear approximation, B(x) = B j + (B j+ B j ) x x j, x C j, j, (4.0) x

87 74 for which the following property is satisfied: B j := B(x j ) = B(x) dx = x C j B j+ + B j. (4.) Remark Notice that for the slanted bottom topography (B x constant), B(x) B(x). Also notice that equation (4.9) reduces to B j+ is continuous at x = x j+. = B(x j+ ) if B Next, the reconstruction for w near (almost) dry areas has to be corrected since the use of (4.8) may lead to negative values of h ± j+ as it was shown in [3]. We therefore propose the following correction procedure: In the cells, where the original reconstruction (4.8) produces negative values of h, we make the slope of h to be equal to the slope of B. Namely, we proceed as follows: if w j+ < B j+ or w + j < B j, then take (w x ) j = (B x ) j = w j+ = h j + B j+, w + j = h j + B j, where h j := w j B j. This correction (unlike the correction procedure in [3] or its more sophisticated modification recently proposed in [5]) will not only guarantee the positivity of h ±, but will also be able to exactly reconstruct the steady-state j+ solution (4.9). It should also be pointed out that when the solution is expected to have (almost) dry areas, say, when the computational domain contains islands and/or coastal areas, the values of h could be very small or even zero. This may not allow us to (accurately) compute the values of the velocity u, which may become artificially large. In such cases, when in some cells the point values h ± j± are smaller than an a-priori chosen positive number ε, that is, h ± j± < ε, the piecewise linear reconstruction of q in (4.7) should be replaced with a piecewise linear reconstruction of u in the entire computational domain. Namely, the velocity at the cell centers is first computed by

88 75 the desingularization formula u j = h j q j h j + max ( h j, ε ), (4.) and then the point values of the velocity at the cell interfaces x = x j+ from u + j+ := u j+ x (u x) j+, u j+ := u j + x (u x) j, are obtained where the numerical derivative (u x ) j are evaluated using the same nonlinear limiter as in (4.7). For consistency, the values of the discharge at cell interfaces are recomputed using q ± j± = h ± j± u ±. j± Remark We note that one may use other strategies to compute the desingularized velocity: u = hq h4 + max (h 4, ε 4 ), or q, if h < ε, u = h 0, otherwise, see, e.g., the discussion in [3]. Our numerical experiments demonstrate that the proposed method is not sensitive to the selection of the desingularization procedure used. Next, equipped with the values of h ± j+ and u ±, we complete the construction of j+ the central-upwind flux (4.6) by estimating the one-sided local speeds of propagation as follows: a + j+ a j+ { = max u + + j+ = min gh +, u + j+ j+ gh j+ }, 0, }. { u + gh +, u gh, 0 j+ j+ j+ j+ (4.3) The final step in the derivation of the semi-discrete scheme is the discretization

89 76 of the source terms: S j (t) x S(U, R, B) dx, M j (t) C j x C j M(U, B) dx. We calculate the first component of S j using the midpoint rule: S () j = R(x j, t), while approximate the geometric source in S () j using a special quadrature derived in [3, 3], which guarantees the well-balancedness of the resulting scheme: B j+ B j j = gh j. (4.4) x S () The second component of M j is computed using the desingularization procedure (4.): ( ) 7/3 M () j = g n h j h j + max ( h j, ε ) q j q j. (4.5) We remark that the semi-discrete scheme (4.5), (4.6) is a system of timedependent ODEs which should be solved using a high-order (at least second order accurate) and efficient method as we discuss in below Two-Dimensional Central-Upwind Scheme In this section, we describe the central-upwind scheme for the -D shallow water system (4.). As in the -D case, we rewrite the system (4.) in terms of the new unknown vector U = (w, q := hu, p := hv) T : U t + F (U, B) x + G(U, B) y = S(U, R, B) + M(U, B), (4.6)

90 77 where the fluxes and the source terms are: F (U, B) = G(U, B) = ( q, ( p, q w B + g (w B), ) T qp, w B ) T, qp w B, p w B + g (w B) S(U, R, B) = (R, g(w B)B x, g(w B)B y ), ) T M(U, B) = (0, g n h q q + p, g n 7/3 h p q + p. 7/3 We denote by C j,k the computational cells C j,k = [x j, x j+ ] [y k, y k+ ], where x α := α x and y β := β y, where x and y are small spatial scales, and write a central-upwind semi-discretization of (4.6) as the system of ODEs: d H x dt U j+ j,k =,k Hx j,k x H y j,k+ H y j,k y +S j,k +M j,k, (4.7) for the time evolution of the cell averages, U j,k (t) = x y C j,k U(x, y, t) dxdy. As in the -D case, we follow [3, 38] and obtain the central-upwind numerical fluxes in the form: H x j+,k = a + j+,kf (U E j,k, B j+,k) a j+,kf (U W j+,k, B j+,k)) a + j+,k a j+,k H y j,k+ a + j+,ka j+,k + a + [U j+,k a j+,k W Uj,k], E j+,k =b + G(U j,k+ j,k N, B j,k+ )) b G(U j,k+ j,k+ S, B j,k+ )) b + b j,k+ j,k+ (4.8) + b + b j,k+ j,k+ b + j,k+ b j,k+ [U S j,k+ U N j,k].

91 78 Here, B j±,k and B j,k± are the values of the piecewise bilinear approximation of B: B(x, y) = (B j+,k+ B j+,k B j,k+ + B j,k + B j,k + (B j+,k B j,k ) x x j x ) (x x j x y )(y y k ) + (B j,k+ B j,k ) y y k, (x, y) C j,k, y (4.9) with B j+,k+ = B ( x j+, y k+ [3] for the case of discontinuous B. ) if the function B is continuous at (xj+, y k+ ); see Similarly to the -D case, the piecewise bilinear approximant B satisfies the following property: B j,k := B(x j, y k ) = B(x, y) dx dy = x y C j,k where B j+,k := B(x j+, y k ) and B j,k+ 4 (B j+,k + B j,k + B j,k+ := B(x j, y k+ ). + B j,k ), (4.30) Remark For the slanted bottom topography satisfying either B x constant, B y 0 or B x 0, B y constant, the approximant (4.9) is exact, that is, B(x, y) B(x, y). The values U E,W,N,S j,k in (4.8) are the point values of the piecewise linear reconstruction Ũ(x, y) = U j,k + (U x ) j,k (x x j ) + (U y ) j,k (y y k ), (x, y) C j,k, (4.3)

92 79 at (x j+, y k ), (x j, y k ), (x j, y k+ U E j,k := Ũ(x j+ U W j,k := Ũ(x j ), (x j, y k ), respectively. Namely, we have: 0, y k ) = U j,k + x (U x) j,k, + 0, y k ) = U j,k x (U x) j,k, Uj,k N := Ũ(x j, y k+ 0) = U j,k + y (U y) j,k, Uj,k S := Ũ(x j, y k + 0) = U j,k y (U y) j,k. As in the -D case, the numerical derivatives (U x ) j,k and (U y ) j,k are to be computed using a nonlinear limiter, say, the generalized minmod limiter (for details see [3]). To preserve the positivity of water height h, we follow the -D approach presented in 4.3. and correct the reconstructed values of w as follows: if w E j,k < B j+,k or ww j,k < B j,k, then take (w x) j,k = (B x ) j,k = w E j,k = B j+,k +h j,k, w W j,k = B j,k +h j,k; if w N j,k < B j,k+ or wj,k S < B j,k, then take (w y ) j,k = (B y ) j,k = wj,k N = B j,k+ +h j,k, wj,k S = B j,k +h j,k, where h j,k := w j,k B j,k. Once again, we observe that the obtained point values of h may be very small or even zero. Similarly to the -D case, when the solution contains (almost) dry areas, that is, if h j+,k < ε or h j,k+ < ε somewhere in the computational domain, we reconstruct the velocities u and v instead of the discharges q and p. To this end, we first compute the velocities at the cell centers: u j,k = h j,k q j,k h j,k + max ( h j,k p j,k h j,k, ε ), v j,k = h j,k + max ( h j,k, ε ), (4.3) and evaluate the point values at the cell interfaces using the piecewise linear recon-

93 80 structions: u E j,k := u j,k + x (u x) j,k, u W j,k := u j,k x (u x) j,k, u N j,k := u j,k + y (u y) j,k, u S j,k := u j,k y (u y) j,k v E j,k := v j,k + x (v x) j,k, v W j,k : v j,k x (v x) j,k, v N j,k := v j,k + y (v y) j,k, v S j,k := v j,k y (v y) j,k, where the numerical derivatives (u x ) j,k, (v x ) j,k, (u y ) j,k and (v y ) j,k are computed with the help of the same nonlinear limiter used in (4.3). The obtained values are then used to recompute the corresponding point values of q and p: q E(W,N,S) j,k = h E(W,N,S) j,k u E(W,N,S) j,k, p E(W,N,S) j,k = h E(W,N,S) j,k v E(W,N,S) j,k. The local one-sided speeds of propagation a ± j+,k and b± j,k+ in (4.8) can be estimated as follows: a + = j+,k max{ue j,k + gh E j,k, uw j+,k + gh W j+,k, 0}, a = j+,k min{ue j,k gh E j,k, uw j+,k gh W j+,k, 0}, b + = max{v N j,k+ j,k + gh N j.k, vs j,k+ + gh S j,k+, 0}, = min{vj,k N gh N j.k, vs j,k+ gh S j,k+, 0}. b j,k+ (4.33) Finally, a well-balanced discretization of the source term is obtained using the

94 8 same desingularization process as in (4.3) and is thus given by S () j,k = R(x j, y k, t), S () j,k = gh B j+,k B j,k j,k, x S (3) j,k = gh B j,k+ B j,k j,k, y M () j,k = 0, M () j,k = gn q j,k +p j,k M (3) j,k = gn q j,k +p j,k ( ) 7/3 h j,k h j,k + max ( h j,kε ) q j,k, ( ) 7/3 h j,k h j,k + max ( h j,kε ) p j,k. As in the -D case, in order to obtain a fully discrete scheme, the system (4.7), (4.8) should be integrated by a stable and efficient ODE solver of at least second order of accuracy. We discuss the details of time integration in the next section Time Integration Method As it was outlined in the previous sections, both the -D and -D semi-discrete centralupwind schemes are the systems of time dependent ODEs which should be solved by an accurate, stable and efficient method. A family of explicit strong stability preserving Runge-Kutta methods (SSP-RK) has been widely used in numerical simulations of various shallow water systems, see, e.g., [5, 6]. The presence of the stiff friction term in both (4.5) and (4.7) can lead, however, to a great loss in accuracy and efficiency of the ODE solver. In shallow water applications that included dry and/or almost dry areas, the explicit treatment of the friction terms imposes a severe time step restriction which is several order of magnitude smaller than a typical time step used for the corresponding friction-free version of the studied system. An attractive alternative to explicit methods is implicit-explicit (IMEX) SSP

95 8 Runge-Kutta solvers, which treat the stiff part of the underlying ODE system implicitly and thus typically have the stability domains based on the nonstiff term only, see, e.g., [45 50]. However, a straightforward implementation of these methods may break the discrete balance between the fluxes, geometric source and the friction terms maintained by the derived semi-discrete central-upwind scheme and the resulting fully discrete method will not be able to preserve the relevant steady states and the positivity of the computed water depth. To overcome this difficulty, we have recently developed a family of second-order semi-implicit time integration methods for systems of ODEs with stiff damping term [0]. In these methods, only a portion of the stiff term is implicitly treated and therefore the evolution equation is very easy to solve and implement compared to fully implicit or IMEX methods. The important feature of the ODE solvers we introduced in [0] resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only. The new semi-implicit methods are based on the modification of explicit SSP-RK methods and are proven to have a formal second order of accuracy, A(α)-stability and stiff decay. We now briefly describe the application of the second-order semi-implicit ODE solver from [0] to the ODE system (4.5), (4.6) (the implementation of the ODE solver to the system (4.7), (4.8) is similar and we thus omit it for the sake of brevity). } We first introduce the grid function of the numerical solution U := {U j. We then denote the discretization of the sum of fluxes, geometric and water source terms by F[U] j := H j+ H j +S j, (4.34) x

96 83 and introduce the discrete friction coefficient ( ) 7/3 G(U j ) := gn h j h j + max ( h j, ε ) q j, (4.35) so that the discretization of the friction term (4.5) can be written as M () j = G(U j )q j. Using these notations, the ODE system (4.5) can be rewritten as d dt w j = F () [U] j, d dt q j = F () [U] j + G(U j )q j. (4.36) We now implement the SI-RK3 method to the system (4.36) (the SI-RK3 method is a second-order semi-implicit Runge-Kutta method based on the third-order SSP- RK method; for details, see [0, Section 3]). The resulting fully discrete scheme can be written as w I j = w j (t) + tf () [U(t)] j, w II j = 3 4 w j(t) + ) (w Ij + tf () [U I ] j, q II j 4 w III j = 3 w j(t) + ( ) w II j + tf () [U II ] j, q III j 3 q I j = q j(t) + tf () [U(t)] j, (4.37a) t G(U(t) j ) = 3 4 q j(t) + 4 qi j + tf () [U I ] j, t G(U I j) = 3 q j(t) + 3 qii j (4.37b) + tf () [U II ] j, t G(U II j ) (4.37c) w j (t + t) = w III j, q j (t + t) = qiii j ( t) F () [U III ] j G(U III j ), + ( t G(U III j )) (4.37d) where U I = (w I, q I ) T, U II = (w II, q II ) T, and U III = (w III, q III ) T. In the following theorem, we prove that the fully discrete scheme (4.37) is well-

97 84 balanced. Theorem The fully discrete central-upwind scheme (4.37) is a well-balanced in the sense that it preserves steady-state solutions satisfying h h 0 = ( ) n q0 3/0, q q 0, B x C, R 0 (4.38) C as long as h 0 ε, where ε is a desingularization parameter used in (4.). Proof. We first note that for the steady states (4.38), the numerical derivatives of U are given by U x ( C, 0) T, and the numerical flux reduces to H j+ = ( q 0, q 0 + g ) T h 0 h 0. Therefore, H j+ H j also equal to zero, that is, 0. The sums of the components of the source terms are S () j +M () j = = 0, S () j +M () j = gh 0 C g n h 7/3 0 q 0 q 0 = 0. (4.39) The second equation can be proven true when h 0 ε according to the definition of h 0. Then, using the notations (4.34) and (4.35), we obtain that (4.39) is equivalent to F () [U] j = 0, F () [U] j + G(U j )q j = 0, which after being substituted into (4.37) implies h n+ j = h n j = h 0 and q n+ j = q n j = q 0. We therefore have proved that the steady state (4.38) is preserved. Remark Notice that the -D version of Theorem can be proved in a similar manner. Remark In [0], we have proved that the time step restriction for the SI-RK3 method is determined by the nonstiff (explicitly treated) term only and no extra time

98 85 restrictions due to the stiffness of the friction term is required. This implies that for the ODE system (4.36), arising from the central-upwind semi-discretization of the -D shallow water system, the size of the time step is to be calculated based on the CFL condition, namely, we need to select t ( t) := x a, { } a = max a +, a, (4.40) j j+ j+ where a ± j+ are the local propagation speeds defined in (4.3). Remark We would like to emphasize that [3, Theorem.] directly applies to the first stage of the SI-RK3 method (4.37) and hence the time step restriction (4.40) guarantees the positivity of h I j := w I j B j for all j provided h j (t) 0 for all j. The positivity of h II j, h III j and h j (t + t) will then be ensured by the same theorem provided t ( t) I and t ( t) II, where ( t) I and ( t) II are computed using (4.40) applied to the intermediate solutions U I and U II, respectively. To satisfy all of the aforementioned time step restrictions, we implement the following adaptive strategy: (i) Given the solution U(t), set t := κ( t), where κ (0, ) and ( t) is given by (4.40); (ii) Use t to compute U I by (4.37a); (iii) Given the intermediate solution U I, compute ( t) I by (4.40); (iv) If ( t) I < t, set t := κ( t) I and go back to Step (ii); (v) Use t to compute U II by (4.37b); (vi) Given the intermediate solution U II, compute ( t) II by (4.40); (vii) If ( t) II < t, set t := κ( t) II and go back to Step (ii); (viii) Use t to compute U III and U(t + t) by (4.37c) and (4.37d), respectively. In all of the numerical examples below, we have used κ = 0.9. In the -D case, we have implemented a similar adaptive strategy to ensure the

99 86 positivity of h. However, the basic CFL condition is more restrictive (see [3, Theorem 3.]), and one has to choose { x t ( t) := min 4a, y } 4b }, a = max {a +j+ j,k,k, a j+,k { }, b = max b +, b, j,k j,k+ j,k+ where a ± j+,k and b± j,k+ are the local propagation speeds defined in (4.33). 4.4 Numerical Examples In this section, we test the designed well-balanced positivity preserving central-upwind scheme on several -D and -D problems including the ones with the data originating from the recently performed laboratory experiments reported in [77]. In all of the examples below, the gravitation constant g = 9.8 (in Example 4.4, we use g = 9.8), the minmod parameter θ =.3, and we select the desingularization parameter ε = 0 8 (in Example 4.4, we use ε = 0 4, 0 8 and 0 ) One-Dimensional Examples Example 4. Steady Flow Over a Slanted Surface. We begin by illustrating the well-balance property of the designed scheme, that is, we test the ability of the scheme to exactly preserve the steady-state solution (4.9), which is schematically shown in Figure 4. (left). To this end, we consider the system (4.5) on the interval [0,.5] with R 0 and subject to the constant initial data given by (4.9). We introduce a uniform grid with x = 0.05 and set zero-order extrapolation boundary conditions at both ends of the domain, that is, h 0 = h,h N+ = h N. We run five sets of experiments with different values of h 0, q 0, B x and n, shown in Table 4., in which the first four are taken from [77]. The solution is evolved until time t = 00. The right column of Table 4. clearly illustrates that our scheme preserves the studied steady-state solutions within machine accuracy.

100 87 Test h 0 q 0 n B x Froude Number h h / Table 4.: Example 4.: Errors in computing the steady-state solution (4.9) for different sets of data. Example 4. Small Perturbation of Steady Flow Over a Slanted Surface. In this example, we take the same initial data from Example 4., but introduce a small perturbation to the initial water surface, namely, 0.h 0, x.5, h(x, 0) = h 0 + 0, otherwise, q(x, 0) q 0. (4.4) We first consider a supercritical case (Test in Table 4.) and present several time snapshots of the computed solution in Figure 4.. As one can see, the perturbation first changes its shape and propagates to the right, eventually leaving the domain. At large times, the computed solution converges to the steady state, see Figure 4. (right) t=0 w B x t= w B x t=00 w B x.5.5 Figure 4.: Example 4.: Evolution of the solution in the supercritical case (Test in Table 4.). We then proceed with a subcritical case (Test 4 in Table 4.) and again replace the initial data in Example 4. with (4.4). The evolution of the perturbed solution is shown in Figure 4.3. In this case, the shape of the propagating perturbation is

101 88 different from the one in the supercritical case, but at large times the computed solution still converges to the steady state t=0 w B t=0.5 w B t=00 w B x x x Figure 4.3: Example 4.: Evolution of the solution in the subcritical case (Test 4 in Table 4.). Finally, we consider Test 5 from Table 4., in which the magnitude of the slope B x is much larger than in the other tests. In this case, the perturbation propagates much faster than in the previous two tests (see Figure 4.4), but the scheme still performs very well and the numerical steady state is achieved at large times..5 t=0 w B.5 t=0.05 w B.5 t=00 w B x x x Figure 4.4: Example 4.: Evolution of the solution with large bottom slope (Test 5 in Table 4.). Example 4.3 Rainfall-Runoff Over a Slanted Surface. In this example, which is a -D modification of Example T3 from [74], we test the stability and accuracy of proposed numerical scheme in the scenario with very shallow water, bed friction and large bottom slope. The surface is assumed to be almost dry at time t = 0, that is, we set h(x, 0) 0, q(x, 0) 0, (4.4)

102 89 and the bottom is set to be a slanted surface. The rain of a constant intensity starts falling at time t = 0 and stops at time t = 00. This is modeled by taking 0 4, 0 t 00, R(x, t) = 0, otherwise. The water drains through the right boundary, at which we set h N+ := 0, q N+ := 0, where N is a total number of cells inside the computational domain [0,.5] (we take N = 00 in this example). On the left side of the domain we use solid wall boundary conditions. We consider six sets of data with different values of friction coefficient n and slope B x (see Table 4.) and run the simulations until time t = 50. Test n B x Table 4.: Example 4.3: Friction coefficients and slopes. In Figure 4.5, we plot the first component of the numerical flux at the right edge of the computational domain (H () N+/ ) as a function of time. Notice that this is an approximation of the outlet discharge, which is a measurable quantity in experimental settings. In all six cases, the results obtained by the designed central-upwind scheme are in a good agreement with the results reported in [74]. We would like to point out that in [74], the simulations were performed in the -D domain of the width 0., so that all of the discharge values in Figure 4.5 are to be multiplied by the factor of 0. in order to be compared with those in [74]. As one can clearly see from Figure 4.5, the reported results are oscillation-free confirming the robustness of the designed

103 90 central-upwind scheme. Test x Test x Test 3 x Test 4 x Test 5 x Test 6 x Figure 4.5: Example 4.3: Outlet discharge as a function of time. While comparing with the results in [74], the discharge values are to be multiplied by 0. which is the width of the -D computational domain in [74]. Remark It should be observed that if an upwind numerical scheme is used instead of the central-upwind one, the ghost cell values (4.4.) may be unacceptable. We refer the reader to [74], where supercritical boundary conditions have been

104 9 implemented using a different ghost cell values: h N+ := ( q ) /3 N, q g N+ := q N. In all of our computations, both types of boundary conditions produce similar results. However, the supercritical boundary conditions sometime cause small oscillations. Example 4.4 Convergence to a Non-Trivial Equilibrium. In this example taken from [75], we simulate a subcritical flow over a nonflat bottom. As described in [75], the system (4.5) with R(x, t) 0, n = 0.03 and g = 9.8 admits the non-trivial steady-state solution with h st (x) = exp [ 35 4 ( ) ] x 75, q st (x), 50 and the bottom topography function B satisfying ( ) B 4 (x) = 9.8 h 3 st(x) h st(x) h 0/3 st (x). We prescribe zero initial condition (h(x, 0) q(x, 0) 0) throughout the computational domain [0, 50] and implement the following boundary conditions by setting the ghost cell values to be h 0 := h h, q 0 :=, h N+ := h st (x N+ ), q N+ := q N. In addition, we fix the first component of inlet numerical flux by setting H () := for all times.

105 9 The solution of the studied initial-boundary value problem is expected to converge to the aforementioned steady state as t. We test our scheme on different grids varying the number of cells N from 50 to 600 and study the convergence rate by comparing the computed solutions with the steady-state one at time T max = 00. The solution (w and q) computed with N = 50 is plotted in Figure 4.6 together with the steady-state solution. As one can see, the proposed method captures the exact steady state quite accurately even on such a coarse grid (the obtained results are a little better than the one reported in [75]). When the mesh is refined, the secondorder convergence rate is observed in h, while the q component converges with almost third order, see Table 4.3, where we show both the L - and L -norms of the errors together with the experimental convergence rates..5 w N=50 Steady state B q N=50 Steady state Figure 4.6: Example 4.4: Solution (w(x, 00) and q(x, 00)) computed using N = 50 and the exact steady-state solution. h q N L -norm Rate L -norm Rate L -norm Rate L -norm Rate E-03.53E-04 7.E-04.66E E E E E E E E E E E E E E E E E Table 4.3: Example 4.4: The L - and L -errors and convergence rates for h and q.

106 93 Example 4.5 Oscillating Lake. We consider shallow water flow with wet/dry fronts in a frictional parabolic bottom. This example is a modification of the numerical example studied in [78, 79]: we use the same initial data and bottom topography function, but we replace the linear friction term used in [78, 79] with the Manning friction term. The main goal of this example is to study the impact of the small parameter ε in the desingularization procedure (4.) on the computed solution. We consider the bottom topography described by the function [ ( x ) ] B(x) = 0, 3000 and set the following initial conditions: w(x, 0) = max{k x + k, B(x)}, q(x, 0) 0, where k = and k = We use the Manning friction term with the coefficient n = 0.0. We run the simulation over the computation domain [ 5000, 5000], which is divided into N = 000 cells. We use three different values of the desingularization parameter ε = 0 4, 0 8 and 0 and compute the solutions until the final time t = The snapshots of the computed solutions at t = 300, 000 and 3000 are shown in Figure 4.7. As one can see, the solutions obtained with the different values of ε are almost indistinguishable. This clearly demonstrates that the proposed central-upwind scheme is not sensitive to the selection of ε. Example 4.6 Wet/Dry Front Propagation over a V-Shape Bottom Topography. In this example, we simulate a package of water released on the left part of a V-shape valley (see Figure 4.8). The water steams down the left part and gradually accelerates. After passing the lowest point of the bottom, it starts climbing up the

107 w(x, 300) ε=0 4 ε=0 8 ε=0 B q(x, 300) ε=0 4 ε=0 8 ε= u(x, 300) w(x, 000) ε=0 4 ε=0 8 ε=0 B w(x, 3000) ε=0 4 ε=0 8 ε=0 B q(x, 000) ε=0 4 ε=0 8 ε= q(x, 3000) ε=0 4 ε=0 8 ε= ε=0 4 0 ε=0 8 ε= u(x, 000) ε=0 4 ε=0 8 ε= u(x, 3000) ε=0 4 0 ε=0 8 ε= Figure 4.7: Example 4.5: Snapshots of w, q and u computed with ε = 0 4, 0 8 and 0. right slope, meanwhile gradually losing its speed. Before the water starts streaming down the right slope, the wet/dry front achieves its maximum height. Our goal is to study this front propagation quantitatively. We consider a V-shape bottom topography function, B(x) = 3 x, and take the friction coefficient n = 0.0. Initially, a package of water has a parabolic profile and zero velocity: h(x, 0) = max {0,.5(x 0.3)(x 0.7)}, q(x, 0) 0. We take a computational domain to be sufficiently large ([0, ]) so that the vibrating

108 95 water body never reaches the boundaries. We run the computation on different grids varying the number of cells N from 00 to 6400 and compute a reference solution using N = According to the reference solution, the wet/dry front arrives at its highest point on the right slope at time T max = In Figure 4.8, we plot the water profile computed with N = 400 and 6400 at t = 0 and t = T max. Also, in Figure 4.9, we zoom at the wet/dry front area and compare the solutions computed using N = 400, 600 and 6400 with the reference solution. Finally, in Table 4.4, we show the L -norm of the errors and compute the experimental convergence rates, which are slightly above. This rate is expected given the fact that the solution is not smooth Figure 4.8: Example 4.6: Water profiles at t = 0 (dashed line) and t = T max (solid line) computed with N = 400 (left) and 6400 (right) N=400 N=600 N=6400 N= Figure 4.9: Example 4.6: Zoom at the wet/dry fronts of the computed and reference (N = 5600) solutions.

109 96 h q N L -error Rate L -error Rate 00.40E E E E E E E E E E E E E E Table 4.4: Example 4.6: The L -errors and convergence rates for h and q Two-Dimensional Examples Example 4.7 Oscillating Lake. Similar to the -D Example 4.5, we consider shallow water flow with a wet/dry front in a frictional parabolic bowl. This example is a modification of the oscillating lake example from [80]: as in the -D case, we replace the linear friction term with the nonlinear (Manning) one. Our goal is to demonstrate the ability of the proposed central-upwind scheme to accurately capture wet-dry fronts. Following [80], we take the parabolic bottom topography, B(x, y) = (x + y ), and the following initial conditions: w(x, y, 0) = max{k x+k y+8.75, B(x, y)}, q(x, y, 0) 0, p(x, y, 0) = 5(w(x, y, 0) B(x, y)), where k = and k = We use the Manning friction term with the coefficient n = 0.0. We run the simulation until the final time T max = 00 in the computation domain [ 5000, 5000] [ 5000, 5000], which is divided into N N grid cells with N = 50, 00, 00, 400 and 800. The fine mesh (N = 800) solution is shown in Figure 4.0, where

110 97 we show the computed water surface (w), water depth (h), as well as the discharges (q and p) and velocities (u and v). As one can see, the wet/dry front is quite sharply captured by the central-upwind scheme and the velocities, which are computed using the desingularization procedure near the front, do not contain large spikes and thus the efficiency of the scheme is maintained. To verify the rates of convergence of the proposed method, we measure the difference between the solutions computed on two consecutive grids using the weighted L -differences, which are defined as follows: φ N ψ N := N N j= k= N φ N j,k ψj,k, N where φ N := {φ N j,k } and ψn := {ψj,k N } are two functions prescribed on the N N grid. To apply this formula to the solutions computed on two different grids, we project the fine grid solution onto the coarse grid using the conservative projection. Then, to measure the experimental convergence rates r for h, we use the ratio of the weighted L -differences: ( ) h N/ h N/4 r = log. h N h N/ The rates for q and p are computed similarly. The obtained results, reported in Table 4.5, indicate that the experimental convergence rate of the proposed central-upwind scheme is slightly smaller than. N h N h N/ Rate q N q N/ Rate p N p N/ Rate E-03.56E E E E E E E E E E E Table 4.5: Example 4.7: The weighted L -differences between the solutions computed on consecutive grids and the corresponding convergence rates.

111 98 h Figure 4.0: Example 4.7: Solution (w, h, q, u, p and v) computed using the fine grid. Example 4.8 Rainfall Runoff Over An Urban Area. We consider another rainfall-runoff example, which now occurs over a more complicated -D surface containing houses as outlined in Figure 4. (right). The setting corresponds to the laboratory experiment reported in [77]. The surface structure is shown in Figure 4.. The experimental setting was built to mimic an urban area within the laboratory simulator of size meters by.5 meters. To model urban buildings, several blocks were placed onto the surface according to different

112 99 geometries three of which are shown in Figure 4.. In these three configurations, the houses are aligned in either the x- (Configuration X0), y-direction (Configuration Y0) or both (Configuration S0). The bottom topography function B(x, y) is defined as (x, y), outside the house region, B(x, y) = H(x, y; x h, y h ), inside the house region centered at (x h, y h ). The precise data of the surface structure S(x, y) was provided by Dr. Luis Cea and the house-roof configuration H(x, y; x h, y h ) was computed according to the following formulae: 0.3 y y h, (x, y) [x h 0.5, x h + 0.5] [y h 0., y h + 0.], H(x, y; x h, y h ) = 0, otherwise, for houses aligned in x-direction and 0.3 x x h, (x, y) [x h 0., x h + 0.] [y h 0.5, y h + 0.5], H(x, y; x h, y h ) = 0, otherwise, for houses aligned in y-direction. Notice that across the walls of the houses the bottom topography is discontinuous and thus the bilinear interpolant (4.9) has very sharp gradients there. Similarly to the -D rainfall-runoff example, we set the following almost dry initial conditions: h(x, y, 0) 0, q(x, y, 0) 0, p(x, y, 0) 0. The rain of a constant intensity starts falling at time t = 0 and stops at t = T s, that

113 00.5 X0.5 Y0.5 S Figure 4.: Example 4.8: Three house configurations (X0,Y0 and S0) and contour plot of the urban area structure S. In each configuration, the houses are placed in the blank rectangles such that the house ridges coincide with the dotted lines. is, R(x, y, t) = 000, 0 t T s, 0, otherwise. In different experiments, different values of T s = 0, 40, 60 or have been used. The computational domain [, ] [0,.5] is divided into N x N y uniform cells (we have taken N x = 80 and N y = 00). The solid wall boundary conditions have been set at the left, right and top parts of the boundary, while absorbing boundary conditions have been implemented at the lower part ( x, y = 0). As in the -D case, we use the ghost cell technique and the ghost cell values are set to be h j,0 := 0, q j,0 := 0, p j,0 := 0, j =,..., N x. N x ( At this boundary, the total outlet discharge is computed using y H y (). j,/) In Figure 4., the outlet discharge is plotted as a function of time and compared with the experimental data provided to us by Dr. Luis Cea. Notice that in some cases (for example, for Configuration Y0 with T s = 0 or 40 and Configuration S0), the computed curves have lower peaks then the experimental ones. We believe that such a delay of outlet discharge can be explained by inability of the system (4.) (4.4) to accurately model the situation occurring near the house walls where the size of j=

114 0 the jumps in the bottom topography is about -3 orders of magnitude larger than the water depth. We therefore modify the model by removing the houses from the computational domain and placing the entire rain water, which would accumulate over the houses, near the house walls inside the computational domain. The details of a modified approach are given in Example 4.8a. 5 x X0, T 0 4 s = x Y0, T 0 4 s = x S0, T 0 4 s = x X0, T 0 4 s = x Y0, T 0 4 s = x S0, T 0 4 s = x X0, T 0 4 s = x Y0, T 0 4 s = x S0, T 0 4 s = Figure 4.: Example 4.8: Outlet discharge (the experimental data, dashed line, vs. results, solid line) as a function of time. the computed Example 4.8a Modified House Treatment. We consider the same setting as in Example 4.8, but make the following modifications. First, we remove the houses from the computational domain which becomes a punctured rectangle. A typical hole is depicted in Figure 4.3: The house walls

115 0 become the internal boundary, which is numerically treated using the solid wall ghost cell technique. Second, we redistribute the rain water falling onto the roof so that it is placed inside the modified computational domain. In the laboratory experiment, the water falling on the house blocks streams down from the long (lower) edges and finally joins the surface water flow, while in reality, the gutter system is commonly used and the rain water streams down from the rain pipes typically located at the house corners. Ghost Cells Houses Edge Ghost Cells Houses Edge Figure 4.3: Example 4.8a: Special treatment of a house region. The house edges are considered to be a part of an internal boundary. The rainfall on the roof is uniformly distributed into the shaded cells. There are two cases: the one with the gutter system installed (left) and the one without such a system (right). In the numerical experiments, we adopt two different strategies to mimic the above two draining situations. In both cases, the building-roof rainfall is uniformly distributed on certain cells near and outside the building edges. These are the shaded cells in Figure 4.3. The modified rain source can then be written as follows: 000 R(x, y, t) = ( + A ) h, in the shaded cells, A s 000, otherwise, which is, as before, switched on only for t [0, T s ]. Here, A h is the area of the house and A s is the area of the shaded region near that house. If the gutter system is

116 03 installed, then in all studied configurations (X0, Y0 and S0) with N x = 80 and N y = 00, we have A h A s = x y = 4. If the gutter system is not installed, then this ratio depends on a house orientation, but since in this numerical example we take x = y = /40, then A h A s = x = y = 4 for all of the configurations (X0, Y0 and S0). Again, we compare the outlet discharges obtained by our numerical scheme and the laboratory measurement. As we can see in Figure 4.4, our numerical results are now in much better agreement with the experimental results, especially for Configurations Y0 (with T s = 0 and 40) and S0. In other cases, the achieved resolution is also slightly higher though the improvement is not so essential. It should also be observed that in the case of a longer rain duration (T s = 60), the computed outlet discharge remains almost constant for t [30, 60] for Configurations X0 and Y0. In fact, if the rain source is not switched off (T s = ), the outlet discharge will converge to a steady state as shown in Figure 4.5. The snapshots of a water height at this steady state are shown in Figure 4.6. Finally, in Figure 4.7, we plot time snapshots of the water height obtained for Configurations X0, Y0 and S0 (we now use T s = 40). The graphs illustrate how the rain water drains and clarify the dependence of the water flow on the surface configuration. In the examples using Configuration X0, the draining stream takes advantage of the space along the vertical central line x = 0 and a big mainstream is formed. In the examples using Configurations Y0 and S0, there are houses on the central line x = 0 that block the flow there. Therefore, the mainstream of draining water has to bypass these houses and thus small lakes are developed behind two

117 04 5 x X0, T 0 4 s = x Y0, T 0 4 s = x S0, T 0 4 s = x X0, T 0 4 s = x Y0, T 0 4 s = x S0, T 0 4 s = x X0, T 0 4 s = x Y0, T 0 4 s = x S0, T 0 4 s = Figure 4.4: Example 4.8a: Outlet discharge (the experimental data, solid line, vs. the computed results, dashed line) as a function of time. 5 x X x Y x S Figure 4.5: Example 4.8a: Outlet discharge as a function of time: convergence to a steady state when the rain source is not switched off. of the houses, see Configurations Y0 and S0 in Figure 4.7. This explains why the total amount of water flew out of the domain may not be the same as in the measured data since in the experimental setting the lakes drain through the gaps between

118 05 Figure 4.6: Example 4.8a: Water height snapshots at the steady state achieved when the rain source is not switched off. the building blocks and the bottom surface. Remark We would like to point out that the aforementioned two different strategies of modifying the rain source term near the houses lead to practically the same results. In the reported numerical experiments, we have redistributed the rain falling onto the house into the cells along the longer house edges (see Figure 4.3, right) as discussed in the beginning of Example 4.8a. Remark Another point we would like to clarify is the necessity of including the bottom friction terms in the studied shallow water model. In Figure 4.8, we compare the measured data with the numerical result obtained by solving the frictionless system (4.). In this case, the computed outlet discharge changes dramatically even when the simplest Configuration X0 with T s = 0 is considered. 4.5 Conclusion We have studied the shallow water system with friction terms and developed a positivity preserving semi-discrete central-upwind scheme, which is well-balanced in the sense that capable exactly preserves special types of steady-state solutions: both the