Well-balanced central finite volume methods for the Ripa system

Transcription

1 Well-balanced central finite volume methods for the Ripa system R. Touma a C. Klingenberg b a Lebanese American University, Beirut, Lebanon b Würzburg University, Würzburg, Germany This paper is dedicated to the memory of Professor Paul Arminjon Abstract We propose a new well-balanced central finite volume scheme for the Ripa system both in one and two space dimensions. The Ripa system is a nonhomogeneous hyperbolic system with a non zero source term that is obtained from the shallow water equations system by incorporating horizontal temperature gradients. The proposed numerical scheme is a second-order accurate finite volume method that evolves a non-oscillatory numerical solution on a single grid, avoids the process of solving Riemann problems arising at the cell interfaces, and follows a well-balanced discretization that ensures the steady state requirement by discretizing the geometrical source term according to the discretization of the flux terms. Furthermore the proposed scheme mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The proposed scheme is then applied and classical one and two-dimensional Ripa problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method. Key words: Well-balanced central schemes; Ripa system; surface gradient method; non-oscillatory and gradients limiting. Introduction In this paper we consider a system of one- and two- dimensional of shallow water equations with horizontal temperature gradients called the Ripa system, see 8], 7] and 8]. We shall derive a second order accurate numerical method for this system that is able to maintain stationary states. This well-balanced property we found R.Touma: Applied Numerical Mathematics May 5

2 to be essential even when computing non-stationary solutions. Without the well-balanced property the numerical solutions tend to become unstable also in cases where the underlying pde solution is stable; spurious oscillations and non-physical waves appear. The discretization of the Ripa system has been studied in two recent previous papers. Chertok, Kurganov and Liu 6] build a central scheme coupled with an interface tracking method. In ], the authors design a finite volume method that utilizes a new relaxation Riemann solver which is able to well balance the discretization. Our proposed method is an alternative to these two approaches by using the surface gradient method introduced in 4]. Here the water surface level is chosen as the basis for data reconstruction. This is coupled with an unstaggered central scheme UCS previously developed in ]. The unstaggered central schemes evolve a piecewise linear numerical solution on a single grid and follow the central finite volume schemes developed by 4], ], ], ], and ] to avoid the resolution of the Riemann problems arising at the cell interfaces thanks to a layer of staggered dual cells used at an intermediate step. A projection follows and generates the numerical solution at the centre of the original cells. The Ripa system is a variant of the shallow water equations system in which the temperature of the flowing liquid is represented. We note that several numerical methods for the shallow water equations were developed in the recent literature and interested reader is referred to ], ], 4], 5], ], 6], ], 7], 5], and ]. The Ripa system was first introduced in 8], 7], and 8] to model ocean currents. Written in its conservative form the two dimensional Ripa system is u t + F u x + Gu y = Su, x, with h hu hu hu + g u =, F u = h θ, hv huv hθ uhθ hv huv ghθz Gu = hv + g, and Su, x = x, h θ ghθz y vhθ where h denotes the water height, u, v is the velocity field, θ is the temperature of the liquid, g is the gravitational constant, and Z = Zx, y denotes

3 the bottom topography function. The Ripa system is hyperbolic system with real eigenvalues and linearly independent eigenvectors. The Jacobian matrices F/ u and G/ u are F u = ghθ u u gh uv v u uθ θ u and G u = uv v u ghθ v v gh vθ θ v The eigenvalues of the Jacobian matrix F/ u are λ = u ghθ, λ = λ = u, and λ 4 = u + ghθ, while the eigenvalues of the Jacobian matrix G/ u are µ = v ghθ, µ = µ = v, and µ 4 = v + ghθ. The eigenvalues are needed to dynamically calculate the time steps of the numerical scheme and to ensure its stability. Note that when u = ± ghθ the Ripa system features a resonance phenomenon the Jacobian matrix does not have a complete eigensystem, and thus the solution of the Riemann problems becomes very difficult to calculate. For this reason one would like to use numerical methods that avoids the resolution of the Riemann problems such as central schemes and others. Finally, we note that the solution of the Ripa system may be composed of shocks, contacts and rarefaction waves. For additional information on the structure of the solution of the Ripa system, one is referred to 6]. On the other hand, the Ripa system features the following two different steady states u = v =, θ = constant, h + Z = constant, 4 and u = v =, Z = constant, h θ = constant. 5 The goal of this paper is to develop a new well-balanced unstaggered central scheme WB-UCS for accurately solving one and two-dimensional Ripa systems while maintaining the steady states requirement at the discrete level. Here we note that under appropriate assumptions on the regularity of the solutions of the Ripa system, we can rewrite the Ripa system as a hyperbolic system in terms of the conservative variables h, hu, hv and h θ instead of hθ

4 in as follows. h hu hv hu hu + g + h θ huv + hv huv hv + g h θ h θ uhθ vhθ t x y ghθz = x, 6 ghθz y h θu x h θv y As a consequence, any consistent finite volume scheme can easily satisfy the steady state 5; more precisely if the initial data are such that u = v =, Z = constant and h θ = constant, then the numerical solution will satisfy the requirements in 5 at any later time. In this work we mainly focus on the steady state 4 and we develop a wellbalanced scheme that satisfies the constraints in 4 at the discrete level both in one space dimension Theorems and and two dimensions Theorems and 4. As for the steady state 5, we are still exploring/investigating appropriate numerical schemes that ensure the constraints in 5 at the discrete level. Ideally one would like to construct a numerical method that fulfills both equilibrium states 4 and 5 simultaneously. This indeed invites future research on this topic. The rest of the paper is structured as follows: in section, the central finite volume scheme is developed coupling it with the surface gradient method. In section, one- and two-dimensional experiments are presented, showing excellent results. In the appendix proofs of our theorems are given. Well-balanced central schemes for the Ripa system In this section, we develop a new well-balanced central finite volume scheme for the Ripa system - in one and two space dimensions. We follow a classical central finite volume construction for hyperbolic systems of conservation laws with a source term. In order to satisfy the steady state requirement 4, we extend the surface gradient method to the case of the Ripa system and show that the steady state conditions in 4 are exactly maintained at the discrete level. 4

5 . One-dimensional well-balanced central scheme for the Ripa system We consider first the one-dimensional Ripa system written in its conservative form t u + x fu = Su, x, t >, x Ω R 7 ux, = u x with ux, t = h, hu, hθ, fu = hu, hu + gh, hθ, and Su, x =, ghθdz/dx,. Here hx, t denotes the water height, ux, t is the velocity in the x direction, g is the gravitational constant, and zx denotes the bottom topography function. The computational domain Ω is first partitioned using the control cells C i = x i /, x i+/ ] centered at the nodes x i. Without any loss of generality we assume that the numerical solution u n i is known at time t n and is defined at cell-centers x i. A piecewise linear reconstruction of the solution values is then performed and the exact solution ux, t to system 7 is approximated at time t n on the cells C i by ux, t n L i x, t n = u n i + x x i δ n i x C i, 8 where δi n approximates ux i, t n / x to a first order of accuracy. To avoid the process of solving the Riemann problems at the cell interfaces, the numerical solution to system 7 will be first calculated at time t n+ = t n + t on the dual cells D i+/ = x i, x i+ ]. Then a back projection step will be applied to retrieve u n+ i. We integrate system 7 over the domain Ri+/ n = D i+/ t n, t n+ ] and then we apply Green s theorem to the left-hand side; taking into account that ux, t Lx, t, we obtain u n+ i+ = u n i+ t n+ ] t n+ f ux i+, t dt f ux i, t dt t n t n + t n+ xi+ Su, xdxdt. 9 t n x i Here u n i+/ is the projected solution at time tn on the dual cells D i+/ and is obtained with the aid of a Taylor series expansion as follows u n i+ = u n i + u n i+ + 8 δ n i δ n i+ The flux integrals in equation 9 are approximated with second-order of accuracy using the midpoint quadrature rule leading to u n+ i+ = u n i+/ t fu n+ i+ fu n+ ] i + t n+ t n xi+ x i S ux, t, x dx dt, 5

6 where the required predicted values at the intermediate time t n+/ are obtained using a first-order Taylor expansion in time and the balance law 7 as follows u n+ i = u n i + t f i + Si n. The term Si n discretizes the source term in system 7 and should be carefully discretized in order to ensure well-balancing as we will be see later in this section. Next we discretize the integral of the source term using centered differences and the midpoint quadrature rule, we obtain: t n+ xi+ t n S ux, t, x dxdt Su n+ i, u n+ x i = t i+ g hn+ i +h n+ i+ θ n+ i +θ n+ i+ z i+ z i. The numerical solution at time t n+ is obtained by substituting equation in equation, and the computed values will be defined as the centers x i+/ of the dual cells D i+/ ; a back-projection step recovers the numerical solution on the original cells as follows u n+ i = u n+ i + u n+ + i+ 8 u n+ i u n+. 4 i+ To ensure the constant-temperature lake at rest steady state 4 property of the Ripa system one should pay special attention to the way the predicted values are computed at time t n+/ in equation, as well as to the back and forth projection steps in equations and 4. Note that in the steady state case h + Z = constant, u =, and θ = constant the Ripa system reduces to: h g t + x h θ = ghθz x, 5 hθ and therefore specific discretizations of the momentum s flux component and its corresponding source term component are required in order to ensure wellbalancing in the steady state case. In this work we discretize the source term in equation as follows: S n i = S n i,l + S n i,r + S n i,c 6 6

7 with S n i,l = σ i σ i 6 σ i S n i,r = σ i gh n i θ n i Θ z i z i + σ i σ i, S n i,c = σ i σ i + σ i 6 gh n i θ n i Θ z i+ z i, gh n i θ n i Θ z i+ z i. 7 The parameter σ i appearing in the discretized source term in equation 7 is a sensor function that forces the discretization of dz/dx in Su, x to follow the discretization of h/ x and is defined as follows: if h i = Θ hn i hn i, if h i = Θ hn i+ hn i, σ i = if h i =, if h i = hn i+ hn i. where Θ is the parameter of the MC-Θ limiter ]. This leads us to a first main result. Theorem In the context of the one-dimensional Ripa system 7, if at a given time t n, the numerical solution is such that h n i + z i =constant, u n i =, and θi n = constant, for all i, then the equations, 6, and 7 lead to u n+ i = u n i while equations and lead to u n+ = u n. i+ i+ PROOF. Highlights of the proof of theorem are given in the appendix. Remark Theorem states that the updated numerical solution n n+ i+/ obtained at time t n+ shares the same properties of the forward projected solution u n i+/ obtained at time t n on the cells D i+/, but its projection u n+ i doesn t necessarily satisfy the steady state requirement on the cells C i unless a special care of the 7

8 forward projection step and the backward projection step is taken into account. In general, the central scheme fails to satisfy the steady state requirement when it is used to solve steady state Ripa systems problems. This is due to the fact that the riverbed function and the water level function are both linear on the original cells C i, but not on the staggered cells D i+/. An additional treatment of the scheme is needed to remedy this situation. In this work we extend the surface gradient method, previously adapted to the shallow water equations, to the case of the Ripa systems and calculate the numerical derivative of the water height hx, t component in terms of the water level function Hx, t = hx, t + Zx. We also note that the hθ component of the solution should also be projected forward and backward using the water level component by following the surface gradient method as is described below. We first assume that the bottom topography function is defined at the cell interfaces i.e., z i+ is given at x i+. The cell centered bottom elevation function values are then obtained using the equation z i = z i+ + z i, and the linear interpolants zx of the bottom topography function on the cells C i are given by: zx = z i + z i+ z i x x i, x C i. The linearization of the water height hx inside each cell will be made indirectly by first linearizing the water level Hx = hx + zx, and then by using the relation hx = Hx zx. The linearization Hx = H i +H ix x i on the cells C i is obtained by using a limiting procedure of the numerical derivatives of H i = h i +z i ; the linearization of the water height hx is finally obtained by calculating h i as follows: h n i = H n i z i+/ z i /. 8 Similarly, in the forward projection step the linearization of the hθ component is made indirectly in terms of H using equation 8 as follows: h n i θ n i = Hi n z i+/ z i / θi n + h n i θi n. 9 Likewise, for the back projection step of the computed numerical solution u n+ i+/ back onto the original cells C i, we proceed again using the surface gradient method. We define the water level H i+/ on the staggered nodes as follows: H n+ i+/ = hn+ i+/ + z i+/ where z i+/ = z i+/ zi+/ z i + z i+ / is the corrected bottom value due to the fact that the bottom function zx is not linear inside the staggered 8

9 cell D i+/. Next, we obtain the limited discrete derivatives H n+ i+/ from the staggered n+ water level values H i+/ and obtain hn+ i+/ as follows: h n+ i+/ n+ = H i+/ z i+ z i. Similarly for the back-projection of the hθ component in u n+ i+/, the linearization is performed in terms of H n+ i+/ and using equation as follows h n+ i+/ θn+ i+/ n+ = H i+/ z i+ z i θ n+ i+/ + hn+ i+/ θ i+/. Theorem In the context of the one-dimensional Ripa system 7 and the central scheme along with the prediction step, the projection steps, 8, 9, and the back projection steps 4,,, and if at a given time t n the numerical solution u n i satisfies the lake at rest with a constant temperature steady state 4, i.e. h n i + z i = constant, u n i =, and θi n = constant for all i, then the numerical solution generated using the developed central scheme exactly satisfies the steady state 4 at the discrete level and u n+ i = u n i holds for all i and all n =,,. PROOF. Highlights of the proof of theorem are given in the appendix of the paper. Remark We finally note that the proposed well-balanced numerical scheme follows the same stability condition as the original NT scheme 4], since the numerical integration of the source term as well as the forward and backward projection steps of h and hθ components according to the surface gradient method don t add any restriction on the stability requirement of the numerical base scheme. The numerical results presented in section 4 are performed with a CFL number of magnitude.485, and the time step t is calculated dynamically in terms of the eigenvalues of the one-dimensional Ripa system λ i, i =,, and the spacial step according to the formula max λ i t.485 for i {,, }. For further information regarding the stability of central schemes, one is referred to 4], ].. Two-dimensional well-balanced central scheme for the Ripa system In this section we extend the one-dimensional well-balanced unstaggered central scheme to the case of the two-dimensional Ripa system with a variable 9

10 bottom topography function. We consider the two-dimensional Ripa system t u + x fu + y gu = Su, x where u, fu, gu, and Su, x are defined in system. We assume that the computational domain Ω R is uniformly discretized using the Cartesian cells C i,j = x i /, x i+/ ] y j /, y j+/ ] centered at the nodes x i, y j, and we define the dual cells D i+/,j+/ to be the rectangles x i, x i+ ] y j, y j+ ]. We assume further that the initial conditon is defined at the centers of the cells C ij. Without any loss of generality we assume that the numerical solution to the two-dimensional Ripa system is known at time t n and is also defined at the center of the cells C ij. We start by constructing the linear interpolants L i,j x, y, t n = u n i,j + x x i δ n i + y y j σ n j ux, y, t n, x, y C i,j, 4 that approximate the solution to system on the cells C ij, where δi n, σj n denotes a limited numerical gradient of the numerical solution u n i,j. We then integrate equation on the rectangular box Ri+/,j+/ n = D i+/,j+/ t n, t n+ ] and we apply Green s theorem to the integral to the left-hand side to obtain t u + x fu + y gu dv = Su, x, y dv. 5 R n i+/,j+/ R n i+/,j+/ Just like the one-dimensional case, the staggering process will avoid solving Riemann problems, but the obtained solution at time t n+ will be defined on the dual cells D i+/,j+/. A back-projection step will be necessary to retrieve the solution values at centers of original cells C i,j. Gradients limiting avoids spurious oscillations and Euler s midpoint quadrature rule ensures second-order of accuracy provided the integral of the source term is properly approximated. Furthermore, in order to ensure well-balancing and the steadystate requirement at the discrete level, the source term should be discretized according to the discretization of the flux terms. Taking into account that the numerical solution is piecewise linear over the dual cells D i+/,j+/ and is defined at their centers x i+/, y j+/, equation 5 reduces to u n+ i+/,j+/ = un i+/,j+/ t t y f n+/ i+,j g n+/ i,j+ f n+/ i,j + f n+/ i+,j+ f n+/ i,j+ ] g n+/ i,j + g n+/ i+,j+ g n+/ i+,j ] + t y Sun+/ i,j, u n+/ i+,j, u n+/ i,j+, u n+/ i+,j+ 6

11 where u n i+/,j+/ is the projected solution at time tn on the staggered dual cells D i+/,j+/, and is evaluated using a Taylor expansion in space as follows u n i+/,j+/ = u n 4 i,j + u n i+,j + u n i,j+ + ui+,j+ n + 6 δ i,j + δ i,j+ δ i+,j δ i+,j+ + 6 σ i,j σ i,j+ + σ i+,j σ i+,j+. 7 The flux integral with respect to time is approximated to second order of accuracy using the midpoint quadrature rule and the term f n+/ i,j = fu n+/ i,j is obtained using a prediction step at an intermediate time t n+/. The solution at time t n+ on the cells C ij of the original grid is then obtained using a back projection step as follows: u n+ i,j = u n+ + u n+ + u n+ + u n+ 4 i,j i+,j i,j+ i+,j+ + 6 δi,j + δ i,j+ δ i+,j δ i+,j+ + 6 σi,j + σ i,j+ σ i+,j σ i+,j+ 8 where δ, σ i+/,j+/ denotes a limited numerical gradient of the numerical solution obtained at time t n+ at the point x i+/, y j+/. For a further detailed description of the unstaggered central scheme for homogeneous hyperbolic systems one is referred to 9]. The term t Su n+/ i,j, u n+/ i+,j, u n+/ i,j+, u n+/ i+,j+ in equation 6 is used to approximate the spatial integral of the source term over the domain Ri+/,j+/ n with a second-order of accuracy. In the case of the Ripa system, the spatial integral of the source term is discretized using centered differences and the midpoint quadrature rule as follows:

12 S u n+ i,j, u n+ i+,j, u n+ i,j+, u n+ i+,j+ = y + y g hθn+ i+,j + hθ n+ i,j g hθn+ i,j+ + hθ n+ i,j g hθn+ i+,j+ + hθ n+ i,j+ g hθn+ i+,j+ + hθ n+ i+,j z i+,j z i,j z i,j+ z i,j y z i+,j+ z i,j+. 9 z i+,j+ z i+,j y The predicted values u n+/ ij at the intermediate time step t n+/ in equations 6 and 9 are estimated using a first-order Taylor expansion in time and the RIPA system as follows: u n+ i,j = u n i,j + t f i,j x gy i,j y + Sn i,j, where f i,j x is a limited numerical partial derivative of the flux function fu with respect to the x variable, and is obtained using the Jacobian matrix f/ u. The term Sij n in the predictor step discretizes the source term in equation at time t n on the cells C i,j as follows Si,j n ghθ n z i,j = x ghθ n z i,j y ij ij S = S =. S S 4 = The terms S and S in equation are used to discretize the gradient of the bottom topography function according to the discretization of the water

13 height gradient, and are obtained with the aid of sensor functions as follows: ghθ n i,jθ z i,j z i,j, if σ =, if σ =, S = ghθ n i,jθ z i+,j z i,j, if σ =, ghθ n z i+,j z i,j i,j, if σ =. ghθ n i,jθ z i,j z i,j, if σ =, y if σ =, and S = ghθ n i,jθ z i,j+ z i,j, if σ =, y ghθ n z i,j+ z i,j i,j, if σ =. y The parameters σ and σ in equation are two sensor parameters that direct the discretization of z and z according to the discretizations of x ij y ij x h ij and y h ij, respectively. They are defined as follows:, if x h ij Θ h n i,j hi,j n /,, if x h ij, σ =, if x h ij Θ h n i+,j hi,j n /,, if x h ij h n i+,j h n i,j/., if y h ij Θ h n i,j hi,j n / y,, if y h, and σ = ij, if y h Θ h n ij i,j+ hi,j n / y,, if y h h n ij i,j+ h n i,j / y. The parameter Θ appearing in the formulae for S and S is the MC-Θ limiter parameter. Theorem In the context of the two-dimensional Ripa system -, the central finite volume method 6, and the directed discretization formulas of the source term 9,,, and, if the steady state 4 is satisfied at the discrete level at time t n i.e., h n i,j + z i,j = constant, u n i,j = vi,j n =, and θi,j n = constant, then the updated numerical solution verifies the equations u n+/ i,j = u n i,j and u n+ i+/,j+/ = un i+/,j+/ for all i, j. PROOF. The proof of theorem can be constructed in a similar way the

14 the proof of theorem ; highlights are presented in the appendix section. Remark Theorem states that if the numerical solution at time t n corresponds to the case of an equilibrium state, then the updated solution at time t n+ is such that u n+ i+/,j+/ = un i+/,j+/, but the equality un+ i,j = u n i,j doesn t necessarily holds. The forward and backward projection steps 7 and 8 need to be adjusted according to the surface gradient method in order to ensure the steady state requirement. In the steady state case 5 the two-dimensional Ripa system becomes h t hθ + x g h θ + ghθz y g = x, 4 h θ ghθz y and one can see that the first and fourth components h and hθ of the projected solutions need to be carefully computed. As in the one-dimensional case, we assume that the bottom function is defined at the corners x i+, y j+ of the original cells C i,j and the we set the bottom topography function value at the cell centers to be z i,j = z i,j + z i,j+ + z i+,j + z i+,j+ 4 5 In the forward projection step 7 we linearize the water height h in the first and fourth components of the numerical solution u n i,j in terms of the water level H n i,j = h n i,j +z i,j by first constructing the linear interpolants for the water levels on each control cell C ij as follows Hx, y, t n = H n i,j + x x i H x n i,j + y y j H y n i,j, x, y C i,j, 6 where H x n i,j, H y n i,j denotes a limited numerical gradient of the water level function H n i,j. The numerical gradient of the water height function hx, y, t n can now be calculated using the equations h x n i,j = H x n i,j z x i,j and h y n i,j = H y n i,j z y i,j. 7 The gradient of the bottom topography function is discretized using centered differences as follows: zi+/,j / + z i+/,j+/ z x i,j = z i /,j / + z i /,j+/ / 8 4

15 zi /,j+/ + z i+/,j+/ z y i,j = z i /,j / + z i+/,j / / y 9 Equations 8 and 9 are used in the forward projection step 7 of the first and fourth components of u n i,j only. As for the back projection step 8, we follow a similar procedure and linearize the water height in terms of the water level function. As in the onedimensional case, we first correct the water bed function at the center of the cells D i+/,j+/ as follows z i+,j+ z i+,j+ = z i+,j+ z i+/,j+/ + z i+/,j / + z i /,j+/ + z i+/,j+/ 4 Next, we calculate the water level values as follows: H n+ i+,j+ H n+ i+,j+. 4 using corrected waterbed function = h n+ + z i+,j+ i+,j+. 4 The projection step of the water height at time t n+ back onto the original grid can be performed now using the limited gradient components of h n+ i+/,j+/ computed by discretizing the spatial partial derivatives in h x = H x z x and h y = H y z y as follows: h x n+ = H i+,j+ x n+ i+,j+ z i+,j+ +z i+,j+ z i,j+ +z i+,j+ 4 h y n+ = H i+,j+ y n+ i+,j+ z i+,j+ +z i+,j+ z i+,j +z i+,j+, y where the discrete derivatives H x n+ i+,j+ the staggered values H n+ i+,j+ and H y n+ i+,j+ using a gradients limiting procedure. are obtained from Theorem 4 In the context of the two-dimensional shallow water equations, and the proposed two-dimensional finite volume method 6 along with the forward and backward projection steps 7 and 8 obtained with the aid of the surface gradient method 7 and 4, if at a given time t n, the numerical solution satisfies the steady state 4 at the discrete level i.e., h n i,j + z i,j = constant, u n i,j = vi,j n =, and θi,j n = constant, then we can show that the updated numerical solution u n+ i,j satisfies the steady state at the discrete level and the equation u n+ i,j = u n i,j holds for all i, j. The detailed proof of theorem 4 is given in the appendix section. 5

16 Numerical experiments. One-dimensional numerical experiments: The one and two-dimensional well-balanced schemes developed in this paper are now applied and used to solve classical Ripa problems... One-dimensional Riemann problem over a flat bottom Our first numerical experiment is a classical Riemann problem with two constant states across the point x = center of the computational domain Ω =, ]. The initial conditions are as follows 5,, if x <, h, u, θ =,, 5 if x >. The computational domain Ω is discretized using gridpoints and the numerical solution is calculated at the final time t =.. Figure shows the water height, the temperature, and the pressure obtained at the final time using the well-balanced scheme dotted curve. The reference solution solid curve is obtained using gridpoints. The obtained results are in perfect match and in a good agreement with those appearing in 6] and 9]. Figure h h u θ P Figure.. D dam break problem: Water height h, momentum hu, temperature θ, and pressure P obtained at time t = using the UCS scheme dotted line and the WB-UCS scheme solid line. shows the profile of hθ obtained using the unstaggered central scheme with 6

17 dashed curve and without dotted curve well-balancing on gridpoints. The reference solution solid curve is obtained on gridpoints. The obtained results are in good agreement in this case and the UCS scheme is capable of generating the correct waves mainly because the source term of the RIPA system vanishes in this test case flat bottom topography. 5 4 UCS WB UCS Reference hθ Figure.. D dam break problem: Water height at time t = obtained using the UCS scheme dashed line and the WB-UCS scheme dotted line; the solid line is the reference line obtained on gridpoints... Perturbation of a lake at rest problem Our next one-dimensional experiment is a small perturbation of a lake at rest problem as considered in 6]. The non-flat bottom topography function is given by.85{cosπx +.9] + }, if x.8, Zx =.5{cosπx.4] + }, if. x.5,, otherwise. The initial water level is h + z = 6, the initial velocity and temperature are v = and θ = 4, respectively. We first validate the well-balanced numerical scheme and compute the steady state solution until t =. Figure top left shows the waterbed and the water level at rest at the final time. Figure top right shows that the steady state remains satisfied at the discrete level at the final time. Next we perturb the lake at rest and take hx, = hx, + χ.5,.4] x, with χ.5,.4] x = if x.5,.4] and χ.5,.4] x = elsewhere in the computational domain. As time evolves, the initial perturbation splits into two pulses moving in opposite directions, 7

18 8 x 5 H = h + z 6 4 h u h θ P 5 4 Figure.. Lake at rest problem: steady state computed at time t =. and leave the computational domain from its endpoints. Figure 4 top left shows the water height at time t =.4. H = h + z h u h θ P 5 4 Figure. 4. Perturbation of a lake at rest problem: Numerical solution at the final time t =.4 showing the propagation of pulses... Dam break problem over a rectangular bump Here we extend the shallow water equations problem on a discontinuous waterbed presented in 7] to the case of the Ripa system. The waterbed features 8

19 of a rectangular bump defined as follows: 8, if x < 75, zx =, otherwise. The initial conditions for h, u, θ are as follows zx,,, if x, h, u, θ = 5 zx,, 5, otherwise. The numerical solution is computed at the final time t = using the wellbalanced scheme and the obtained numerical results are reported in figure 5 where we plot H = h + z, hu, and hθ obtained on gridpoints dotted curve and gridpoints reference solid curve H=h+z h u θ 8 hθ Figure. 5. D dam break problem over a rectangular bump: Water height obtained at time t = using the well-balanced scheme on grid points dotted curve and grid points solid curve. Figure 6 shows the profile of the pressure P = hθ / at the final time obtained on, 4, and gridpoints. When the surface gradient method is not applied in the forward and backward projection steps, the numerical scheme generates spurious oscillations and non-physical waves due to the non wellbalanced effects. Figure 7 shows the water level function obtained with and without the surface gradient method solid and dashed curves, respectively; non-physical waves start in the neighborhood of waterbed jumps and propagate in the computational domain. The non-physical waves are invisible when the numerical solution is computed using the well-balanced scheme. 9

20 8 points 4 points points 6 4 Pressure Figure. 6. D dam break problem over a rectangular bump: Profile of the pressure P = h θ/ at the final time. 5 Waterbed Without well balancing Initial water level Well balanced scheme 5 h Oscillations due to the non well balanced scheme Figure. 7. D dam break problem over a rectangular bump: Water level obtained with and without surface gradient method for the forward and backward projection steps

21 ..4 Dam break problem over a flat bottom Our final one-dimensional problem is a dam break problem over a flat bottom topography. The initial conditions feature two constant states defined as follows:,,, if x.5, h, u, θ =,,.5, otherwise. The computational domain Ω =, ] is discretized using, 4, and gridpoints and the numerical solution is calculated at the final time t =. using the well-balanced central scheme. Figure 8 shows the water height at the final time obtained using, 4, and gridpoints. We note hθ points.55 4 points points Figure. 8. D dam break problem over a flat waterbed: Water height at time t =. obtained using the UCS scheme dotted line and the WB-UCS scheme solid line. that when the waterbed is flat, the source term in the Ripa system vanishes and the resulting scheme is a homogeneous hyperbolic systems that can be easily solved using finite volume methods. This test case was solved using both the proposed well-balanced numerical scheme and the original unstaggered central scheme ]. Figure 9 shows the profiles of h, hu, hθ, and the pressure P = h θ/ obtained using the numerical base scheme with solid curve and without dotted curve well-balancing at time t =.. A good agreement between the obtained numerical results is observed in figure 9, thus confirming the consistency of the well-balanced scheme.

22 h h u θ P Figure. 9. D dam break problem over a flat waterbed: Profile of the water height h, momentum hu, temperature θ, and pressure P obtained at time t =. using the WB-UCS scheme on grid points dotted curve and gridpoints solid curve.. Two-dimensional numerical experiments: Now we validate the two-dimensional well-balanced central schemes developed in this paper and we solve some classical Ripa problems... Rectangular dam break problem First, we consider the classical two-dimensional rectangular dam break problem over a flat bottom topography. The initial conditions feature two constant states defined as follows:,,,, if.5 x.5, h, u, v, θ =,,,.5, otherwise. The computational domain Ω =, ] is discretized using gridpoints and the numerical solution is calculated at the final time t =. using the well-balanced central scheme. Figure shows the profile of the obtained water height at the final time, while figure shows cross sections of the water height along the x axis obtained at the final time on 5,, and gridpoints and compared to the solution of the corresponding one-dimensional problem obtained on gridpoints. Figure shows a good agreement between the obtained numerical results. The structure of the obtain solution features two shock waves and two contact waves propagating away from the y axis and

23 .9.8 hθ Figure.. Rectangular dam break problem: Water height at time t =. obtained using the UCS scheme dotted line and the WB-UCS scheme solid line. two rarefaction waves propagating towards the y axis..8.6 h.4. 5 gridpoints gridpoints gridpoints Reference Figure.. Rectangular dam break problem: Profile of the water height h obtained on 5,, and gridpoints and compared to the solution of the corresponding one-dimensional problem obtained on gridpoints.

24 .. Circular dam break problem Next, we consider the classical two-dimensional circular dam break problem over a flat bottom topography. The initial conditions feature two constant states defined as follows:,,,, if x + y.5, h, u, v, θ =,,,.5, otherwise. The computational domain Ω =, ] is discretized using, gridpoints and the numerical solution is calculated at the final time t =. using the well-balanced central scheme. Figure shows the profile of the water height obtained using the proposed numerical scheme. We see a shock and a contact wave propagating radially towards the exterior of the domain, and a circular rarefaction wave is propagating towards the center of the domain. Figures h Figure.. Circular dam break problem: Profile of the water height h obtained time t =. on gridpoints. and 4 show scatter plots of the cross sections of the water height h along the diagonal line y = x and the y-axis x=, obtained on 5,, and gridpoints... Perturbation of a steady state on an irregular waterbed For our next numerical experiment, we consider a small perturbation of a steady state problem, a variant of the problem presented in 6]. The initial 4

25 gridpoints gridpoints gridpoints.7 Water height Diagonal axis Figure.. Circular dam break problem: Scatter plot of the cross sections of the water height h at time t =.5 along the diagonal line y = x obtained on 5,, and gridpoints..8.6 Water height.4. 5 gridpoints gridpoints gridpoints Line x= Figure. 4. Circular dam break problem: Scatter plot of the cross sections of the water height h at time t =.5 along the line x = obtained on 5,, and gridpoints. 5

26 data features the steady state h + z, u, v, θx, y, =,,, 4/ for all x, y, ]. The waterbed topography is defined as follows.5 exp x y +.5 ], if x zx, y =.6 exp x.5 + y.5 ], if x > The steady state solution is perturbed by introducing the variation of the water height hx, y, = hx, y, +. for all x, y inside the annulus. x.9. The well-balanced numerical scheme is first validated and the numerical solution of the steady state without perturbation is computed until time t =.8 on a gridpoints. The obtained water height is shown in figure 5 along with the waterbed function. Figure 6 shows the profile of the Figure. 5. Perturbation of a steady state problem: Profile of the water height at a steady state at time t =.8 obtained on gridpoints obtained using the WB-UCS scheme. water height obtained at time t =.5 and shows the propagation of the water perturbation across the computational domain. Figure 7 shows some contour lines of the water height at time t =.5 obtained using the wellbalanced scheme; these results are in good agreement with corresponding ones presented in 6]. Figure 8 shows s scatter plot of the cross sections of hθ along the diagonal line y = x obtained at the final time t =.5 on 5,, and using the WB-UCS scheme. 6

27 Figure. 6. Perturbation of a steady state problem: Profile of the perturbed water height at time t =.5 obtained on gridpoints obtained using the WB-UCS scheme..5.5 h Figure. 7. Perturbation of a steady state problem: Contour lines of the water height at time t =.5 on gridpoints obtained using the WB-UCS scheme. 7

28 hθ gridpoints gridpoints gridpoints. Diagonal line Figure. 8. Perturbation of a steady state problem: Scatter plot of the cross sections of hθ along the diagonal line y = xs at time t =.5 obtained on 5,, and gridpoints using the WB-UCS scheme. 4 Conclusion In this paper we have developed a new well-balanced finite volume method for the Ripa system within the framework of one and two-dimensional unstaggered central schemes. The proposed scheme has the advantages of being an unstaggered central scheme, second-order accurate, and it evolves the numerical solution on a single Cartesian grid. To satisfy the well-balanced constraint in the case of the Ripa system we discretize the source term according to the discretization of the flux divergence term with the aid of sensor functions. Furthermore, to ensure that the equilibrium solution is exactly satisfied at the discrete level, we carefully adapt the surface gradient method to the case of the Ripa system by linearizing of the water height function, on both the original and staggered grids, according to the linearization of the water level function. The temperature component is linearized in a similar fashion. The proposed scheme is then validated and classical problems from the recent literature are successfully solved. The numerical simulations we carried out deal with dam-breaks and hydraulic jumps with variable bed topography and varying temperature gradients. The numerical results obtained for both one 8

29 and two-dimensional Ripa system problems are in perfect agreement with corresponding results appearing in the literature, thus confirming the efficiency and potential of the proposed numerical scheme to accurately solve Ripa system problems. Appendix We now present the proofs of theorems -4. Proof of theorem First we show the predicted solution as time t n+/ is invariant in time i.e., is equal to the solution at time t n u n+/ i = u n i. Recall that u n+/ i is computed using ataylor series expansion in time as follows u n+ i = u n i + t f i + Si n, and the source term Si n is discretized according to the discretization of h n i. If h i is discretized using the backward difference, i.e., h i = Θ hn i h n i, then the sensor function becomes σ i = and we obtain Si,L n = gh n i θi n Θ z i z i, Sn i,r =, and S n i,c = Therefore, Si n = Si,L n + Si,C n + Si,R n = gh n i θi n Θ z i z i and in the context of the steady state 4, fu = gh θ gh n i θi n h n i because θ i n is constant in the steady state case. Therefore the prediction step becomes, so f n i = 9

30 u n+ i = u n i + t gh n i θi n h n i + gh n i θi n Θ z i z i = u n i + t gh n i θi n h n i + Θ z i z i ] = u n i + t gh n i θi n Θh i + z i θh i + z i ] But since h n i +z i = H = constant for all i then Θh n i +z i Θh n i +z i = leading to u n+ i = u n i A similar proof can be done for the values of the sensor function σ i. Next we show that u n+ i+ = u n. Recall that i+ u n+ i+ = u n i+ t fu n+ i+ fu n+ i + Sun+ i, u n+ i+ where the source term is discretized using the formula Su n+ i, u n+ i+ = t g hθn+ i +hθ n+ i+ z i+ z i In the steady state case 4 we have θi n = constant and u n i = zero velocity, and from the first part of theorem just established above, we have h u n+ i = u n i, thus we obtain u = and fu = gh θ, therefore the hθ discretized flux comes

31 fu n+ i+ = leading to: ghn+ i+ θ n+ i+, and fu n+ i = ghn+ i θ n+ i u n+ = u n i+ i+ t + t ghn+ i+ θ n+ i+ g hθn+ i +hθ n+ i+ z i+ z i ghn+ i θ n+ i Performing basic algebra operations, and taking into account that θ n+/ θ n+/ i+ = θi n = constant, we obtain i = u n+ i+ = u n i+ t θn i ] ghn+ i+ + h n+ i h n+ i+ + z i+ h n+ i + z i But since u n+ i = u n i and h n i + z i = H = constant for all i, then h n i + z i h n i+ + z i+ =. Therefore, h n+ i + z i h n+ i+ + z i+ = h n i + z i h n i+ + z i+ =, leading to u n+ = u n which means that if the steady state requirement was satisfied at time t n on the dual cells, it will remain as such at i+ i+ time t n+. Proof of theorem To show that the steady state 4 is maintained at time t n+ provided it was maintained at time t n we will proceed component wise and show that u n+ i = u n i, for all i. Below we will present the proof for the hθ component and the prove can be easily generalized the the h component. We note that in the steady state case 4 the hu component doesn t change in time because u = and because of the well-balanced discretization. We use the surface gradient method for both the forward and backward pro-

32 jection step; for the forward projection we have hθ n i+/ = hθ n i + hθ n i+ + 8 hθ n i hθ n i+ ] 4 We discretize the derivatives of hθ using the product rule and with the aid of the water level function Hi n = h n i + z i as follows hθ n i = h n i θi n + h n i θi n = H n i z i+/ z i / ] θ n i + h n i θ n i. Substituting in equation 4 one obtains while taking into account that H n i = H = constant and θ n i is constant for all i hθ n i+/ = h n i + hi+ n z i+/ z i + z i+ ]] θ n i 44 Similarly, for the backward projection step we apply the surface gradient method and one obtains while taking into account that θ n+/ i+/ is constant n+/ and H i+/ = hn+/ i+/ + z i+/ is constant for all i hθ n+ i = h n+ i / + ] hn+ i+/ 8 z i z i + z i+ z i ] θi n 45 Substituting equation 44 in 45 we obtain hθ n+ i = hθ n i. The proof for h n+ i = h n i is performed in a similar way. Therefore we conclude that if the steady state 4 was satisfied at time t n, then it will remain as such at the next time t n+. Proof of theorem The proof of theorem follows the same strategy as the one of theorem. We assume that u n i,j satisfies the steady state 4 and we show first that u n+/ i,j = u n i,j and the u n+ i+/,j+/. When the steady state case 4 is satisfied at time t n at the discrete level, the Ripa system reduces to h + x gh θ ghθ z + y = x gh θ ghθ z y hθ t

33 The predicted solution at time t n+/ is calculated using a Taylor expansion in time and the balance law as follows u n+ i,j = u n i,j + t f x i,j g y i,j y + Sn i,j 46 where S n i,j is the discretized source term obtained with the aid of sensor functions. If the partial derivatives of the flux functions are approximated using backward differences, then the sensor functions take the values σ = and σ =. Using the product rule and taking into account that θ n i,j is constant, the numerical partial derivatives of h θ required in equation 46 become h n h θ x n i,j = hθ n i,j h n i,j i,jθ h n h θ y n i,j = hθ n i,j h n i,j i,jθ y The source term is then discretized according to f x and g y as follows Si,j n = S = S = ghθ n i,jθ z i,j z i,j S = ghθ n i,jθ z i,j z i,j y S 4 = where Θ is the parameter of the MC-Θ limiter., Substituting in 46 and performing elementary algebra operations, we obtain u n+ i,j = u n i,j + t = u n i,j + t ghθ n i,jθ h i,j h i,j + z i,j z i,j ghθ n i,jθ h i,j h i,j + z i,j z i,j y y ghθ n i,j Θ h i,j + z i,j h i,j + z i,j ] ghθ n i,j Θ h y i,j + z i,j h i,j + z i,j ]

34 And since h i,j +z i,j = H = constant for all x i, y j, then we obtain u n+ i,j Similar proofs can be done to show that u n+ i,j the sensor functions σ i and t j. = u n i,j. = u n i,j for the other values of Next we show to show that u n+ = u n i+,j+ i+,j+ The solution at time t n+ on the dual cells D i+/,j+/ is obtained using equation 6 as follows u n+ = u n i+,j+ i+,j+ t t y fu n+ i+,j + fu n+ i+,j+ fu n+ i,j fu n+ i,j+ gu n+ i,j+ + gu n+ i+,j+ gu n+ i,j gu n+ i+,j + t y S u n+ i,j, u n+ i+,j, u n+ i,j+, u n+ i+,j+ ] ] were the integral of the source term is approximated with second-order of accuracy as follows S u n+ i,j, u n+ i+,j, u n+ i,j+, u n+ i+,j+ = y g hθ i+,j+hθ i,j z i+,j z i,j g hθ i,j++hθ i,j z i,j+ z i,j + y g hθ i+,j++hθ i,j+ g hθ i+,j++hθ i+,j z i+,j+ z i,j+ z i+,j+ z i+,j y To show that u n+ = u n, we shall proceed component wise. Since i+,j+ i+,j+ u n i,j satisfies the steady state 4, then h u =, f = gh θ ghθ z, g =, and S = x gh θ ghθ z y hθ and we also have from part of theorem u n+ i,j = u n i,j The fluxes and the source term that correspond to the h component are 4

35 zeros, then the h component is updated in equation 47 as follows h n+ i+,j+ = hn i+,j+ The hu component is updated using equation 47 hu n+ = hu n i+,j+ i+,j+ g t h θ n+ 4 h θ n+ + h θ n+ h θ ] n+ i+,j i,j i+,j+ i,j+ + t g hθn+ i+,j + hθ n+ i,j z i+,j z i,j + t g hθn+ i+,j+ + hθ n+ i,j+ z i+,j+ z i,j+ 49 Taking into account that θ n i,j = constant, h n i,j + z i,j = H = constant, and u n+ i,j = u n i,j for all i, j, then 49 reduces to hu n+ i+,j+ Similarly we can show that hv n+ = hv n. i+,j+ i+,j+ = hu n. i+,j+ The fluxes and source term that correspond to the hθ component are zeros, then hθ is updated as follows hθ n+ i+,j+ = hθ n i+,j+ Thus we conclude that u n+ i+,j+ = u n i+,j+ for all i, j. Proof of theorem 4 Now we want to show that the equality u n+ i,j = u n i,j holds for all i, j whenever the steady state 4 is satisfied at the discrete level at time t n. We shall proceed componentwise; we note that the equalities hu n+ i,j = hu n i,j and hvi,j n+ = hvi,j n are immediate from theorem since we have u n i, j = vi,j n = for all i, j. We start by showing the equality h n+ i,j = h n i,j; we recall that for both the first and fourth components of u n i,j, the forward projection step 7 is performed using the surface gradient method as follows 5

36 h n i+,j y 6 + y 6 y 6 y 6 = 4 H n x i,j H n x i,j+ H n x i+,j H n x i+,j+ H y i,j H n y i+,j H n y i,j+ H n y i+,j+ h n i,j + h n i+,j + h n i,j+ + h n i+,j+ z i+,j +z i+,j+ z i,j +z i,j+ z i+,j+ +z i+,j+ z i,j+ +z i,j+ z i+,j +z i+,j+ z i+,j +z i+,j+ z i+,j+ +z i+,j+ z i+,j+ +z i+,j+ z i,j+ +z i+,j+ z i,j +z i+,j y z i+,j+ +z i+,j+ z i+,j +z i+,j y z i,j+ +z i+,j+ z i,j+ +z i+,j+ y z i+,j+ +z i+,j+ z i+,j+ +z i+,j+ y 5 Since at time t n the steady state constraint H n i,j = h n i,j + z i,j = constant is maintained, then all numerical partial derivatives of H n i,j in equation 5 are zero. Furthermore, equation 5 leads to z i+,j + z i+,j+ z i,j + z i,j+ = z i+,j + z i+,j+ z i,j z i+,j+ + z i+,j+ z i,j+ + z i,j+ = z i+,j+ + z i+,j+ z i,j+ z i+,j + z i+,j+ z i+,j + z i+,j+ = z i+,j z i+,j + z i+,j+ z i+,j+ + z i+,j+ z i+,j+ + z i+,j+ = z i+,j+ z i+,j+ + z i+,j+ 5 6

37 and similarly we have the following equations z i,j+ + z i+,j+ z i,j + z i+,j = z i,j+ + z i+,j+ z i,j z i+,j+ + z i+,j+ z i+,j + z i+,j = z i+,j+ + z i+,j+ z i+,j z i,j+ + z i+,j+ z i,j+ + z i+,j+ = z i,j+ z i,j+ + z i+,j+ z i+,j+ + z i+,j+ z i+,j+ + z i+,j+ = z i+,j+ z i+,j+ + z i+,j+ 5 Substituting the identities we obtain h n i+,j+ = 4 h n i,j + h n i+,j + h n i,j+ + h n i+,j zi+,j zi+,j+ zi+,j z i+,j + z i+,j+ z i,j + z i+,j+ z i,j+ zi+,j+ z i+,j+ zi,j+ zi+,j+ zi,j+ z i,j+ + z i+,j+ + z i+,j+ + z i+,j+ z i,j + z i+,j+ z i+,j + z i+,j+ zi+,j+ z i+,j+ + z i+,j+ which simplifies leading to 7

38 h n i+,j+ = 4 h n i,j + h n i,j+ + h n i+,j + h n i+,j+ + 4 z i,j + z i,j+ + z i+,j + z i+,j+ 8 4z i+,j+ + z i+,j+ + z i+,j +z i,j+ + z i+,j+ 5 Similarly, for the back-projection step, we follow the surface gradient method n+ and discretize the water height in terms of the water level H i+/,j+/ = h n+ i+/,j+/ + z i+/,j+/ as follows. h n+ i,j = y 6 + y 6 y 6 y 6 h n+ i,j + h n+ + h n+ + h n+ i+,j i,j+ i+,j+ H x n+ i,j H x n+ i,j+ H x n+ i+,j H x n+ i+,j+ H y n+ i,j H y n+ i+,j H y n+ i,j+ H y n+ i+,j+ z i,j +z i+,j z i,j +z i,j z i,j+ +z i+,j+ z i,j+ +z i,j+ z i+,j +z i+,j z i,j +z i+,j z i+,j+ +z i+,j+ z i,j+ +z i+,j+ z i,j +z i,j+ z i,j +z i,j y z i+,j +z i+,j+ z i+,j +z i+,j y z i,j+ +z i,j+ z i,j +z i,j+ y z i+,j+ +z i+,j+ z i+,j +z i+,j+ y 54 Note that when H i,j = h n i,j + z i,j = H = constant, then z i+,j+ = H remains constant and therefore H x n+ i+,j+ equation 54 becomes. H n+ i+,j+ = h n+ = H y n+ i+,j+ i+,j+ + = ; 8

39 h n+ i,j = H which reduces to z i,j + z i+,j + z i,j+ + z i+,j+ zi,j zi,j+ zi+,j zi+,j+ zi,j zi+,j zi,j+ + z i+,j z i,j + z i,j ] + z i+,j+ z i,j+ + z i,j+ ] + z i+,j z i,j + z i+,j ] + z i+,j+ z i,j+ + z i+,j+ ] + z i,j+ z i,j + z i,j ] + z i+,j+ z i+,j + z i+,j ] + z i,j+ z i,j + z i,j+ ] zi+,j+ + z i+,j+ z i+,j + z i+,j+ ] 55 h n+ i,j = H z i,j + z i+,j + z i,j+ + z i+,j+ ] zi,j zi+,j zi,j zi,j zi,j zi,j+ zi,j Taking into account that z i,j = 4 z i,j = 4 z i+,j = 4 zi,j zi,j + z i,j + z i,j+ + z i,j+ + z i+,j + z i+,j+ + z i+,j+ + z i+,j + z i,j+ + z i+,j+ + z i+,j + z i,j+ + z i+,j+ + z i,j + z i+,j + z i+,j + z i,j+ + z i+,j+ + z i+,j+ + z i,j+ + z i+,j + z i+,j+ zi,j + z i,j+ + z i+,j + z i+,j+ + z i+,j + z i,j+ + z i+,j+ + z i,j + z i,j+ + z i ],j+ zi+,j + z i+,j + z i+,j+ + z i+,j+ ] ] ] ] ] ] ] ] ] 9