Structure preserving finite volume methods for the shallow water equations

Transcription

1 Structure preserving finite volume methos for the shallow water equations by Ulrik Skre Fjorholm THESIS for the egree of MASTER OF SCIENCE Master i Anvent matematikk og mekanikk) Department of Mathematics Faculty of Mathematics an Natural Sciences University of Oslo May 9

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3 Preface I wish to express my eepest gratitue to my supervisor Sihartha Mishra for all his help an support through the past year. This thesis is a prouct of our cooperation. I am also grateful to all my friens in reaing room B6 an B66 for making my time here at Blinern as enjoyable as it has been. Thanks in particular to Nils Henrik Risebro, Torstein Nilssen, John Christian Ottem, Nikolay Qviller an Robin Bjørnetun for proof-reaing the thesis. Last, but not least, I thank Ingeborg for all the support an interest she has shown me. Blinern, May 9. 3

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5 Contents Preface 3 Chapter. Introuction 7.. The shallow water equations 9.. Conservation laws.3. Finite volume methos 4 Chapter. Energy preserving an energy stable schemes 9.. Introuction 9.. Entropy preserving schemes.3. Energy preservation.4. Numerical iffusion 7.5. Two-imensional energy preserving an stable schemes Two-imensional ERoe Conclusion 39 Chapter 3. Well-balance schemes Introuction A well-balance energy preserving scheme A well-balance energy stable scheme A well-balance, secon-orer accurate scheme Generalization to two spatial imensions Conclusion 49 Chapter 4. Vorticity preservation Introuction Vorticity projection for the wave equation Vorticity projection for the shallow water system Conclusion 6 Bibliography 67 5

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7 CHAPTER Introuction The most wiely use moel for flui flows uner the influence of gravity is the shallow water system.) h t + hu) x + hv) y =, hu) t + hu + ) gh + huv) y =, x hv) t + huv) x + hv + ) gh =, where h is the height of the water column, g is the gravitational constant an u an v are velocity in the x- an y-irections, respectively. This system of equations has applications in weather simulations, tial waves, river an irrigation flows, tsunami preiction an more [, 5]. The system is an example of a hyperbolic conservation law; others inclue the Buckley-Leverett equation of reservoir flow, the magnetohyroynamic equations of plasma physics an the Euler equations of gas ynamics. In most cases these are nonlinear an highly nontrivial to solve analytically, an as such, numerical methos must be applie to obtain approximate solutions. Due to the nonlinear nature of conservation laws, stability results of numerical methos are har to obtain. Much work was one to this en in the 98 s by Osher, Harten, Lax an Tamor, among others. Of the most funamental concepts evelope was that of entropy preservation an entropy stability. By asserting that schemes satisfy certain entropy inequalities, one obtains stability estimates an convergence towars the correct solution. In the case of the shallow water equations, the relevant entropy is the energy of the solution. The main part of this thesis focuses on energy preserving an energy stable schemes for the shallow water equations. We evelop a novel energy preserving, secon-orer accurate scheme that is very simple to implement, is computationally cheap an is stable compare to other existing energy preserving schemes []. To allow for a correct issipation of energy in the vicinity of shocks, a novel numerical iffusion operator of the Roe type [3, 3] is esigne. The energy preserving scheme, together with this iffusion operator, gives an energy stable scheme for the shallow water system. We apply a stanar reconstruction proceure to obtain a secon-orer accurate scheme. y In the presence of a varying bottom topography, appropriate source terms must be ae to the shallow water system. This gives the non-homogeneous set of equations h t + hu) x + hv) y =, hu) t + hu + ).) gh + huv) y = ghb x, x hv) t + huv) x + hv + ) gh = ghb y, y 7

8 8. INTRODUCTION where b = bx, y) gives the elevation of the bottom from some absolute level. An important point of stuy for non-homogeneous conservation laws such as the above are steay states, solutions that are constant in time. The most natural steay state for.) is the lake at rest, where there is no flow at all. More general, moving steay states also exist. As many interesting flows are merely perturbations of a steay state, the ability of numerical methos for correctly computing steay states is essential. Such schemes are calle well-balance, since the early works of LeRoux an coworkers [6]. With a compatible iscretization of the source term in.), we foun that both our energy preserving an energy stable schemes are well-balance []. In aition to well-balance energy preserving an energy stable schemes, we consier a metho for controlling vorticity errors in numerical approximations of the shallow water equations. The new metho shows promise in problems where there is a strong interaction between the flow in the x an y irections [3]. A similar approach has been consiere earlier in the case of the Euler equations by Ismail an Roe []. This thesis is organize as follows. In the rest of the present chapter we erive the shallow water equations from basic principles. We review some theory on hyperbolic conservation laws, an in Section.3 we give a short introuction to finite volume methos. Chapter etails the construction of our new energy preserving an energy stable schemes. These are compare with other schemes in a number of numerical experiments. We also erive a secon-orer accurate version of the energy stable scheme. In Chapter 3, we investigate the well-balancing properties of the new schemes. In Chapter 4, we erive a scheme for the correct preiction of flows with strong vortical forces. All numerical methos presente in this thesis have been implemente in C++ using the linear algebra library Blitz++. The source coe can be ownloae from

9 .. THE SHALLOW WATER EQUATIONS 9.. The shallow water equations We moel the flow of a flui for which the epth is relatively small compare to the length scales, as is illustrate in Figure.b). If the variation in, say, the y-irection is small, then the problem is essentially one-imensional, as in Figure.a). First, we erive the one-imensional shallow water equations, an then provie the generalization to two spatial imensions..8 u v u.6 z.4 h z.5 h x.5 y W.5 x a) One-imensional b) Two-imensional Figure.. Intersection of flui profile. Since the height of the flui is small, we may assume that the velocity of the flui is constant in the vertical irection, ux, z, t) ux, t). Furthermore, we assume that the ensity ρ of the flui is constant. The mass of the flui over an interval [x, x ] is given by x hx,t) x ρ zx = ρhx, t) x. x If there is no creation or estruction of mass, then this quantity can only change ue to flui flow through the bounary of the interval. Denoting the rate of flow, the flux, over a point x by fx, t), then this means that.3) t x x x ρhx, t) x = fx, t) fx, t). Since the flui is transporte, or avecte, along the velocity fiel, the flux over a point x is given by the mass ensity times the velocity. Thus, fx, t) = hx,t) ρ z ux, t) = ρhx, t)ux, t). If h an u are ifferentiable functions of x an t, then.3) can be written as x x ρh t x, t) x = x x ρhu) x x, t) x. We use the notational convention h t = h t, h x = h x, etc.) Since x an x were arbitrarily chosen, this implies that.4) h t + hu) x =, which we refer to as conservation of mass. We say that h is a conserve variable. Momentum is also avecte by the velocity u, so the flux of momentum ue to avection is ρhu) u = ρhu.

10 . INTRODUCTION In aition to avection, momentum is affecte by the pressure p in the flui. Therefore the flux of momentum past a point x is fx, t) = ρhx, t)ux, t) + px, t). By the assumption of hyrostatic balance, the pressure at a point x, z) is simply the force of gravity exerte upon the water above that point. Thus, the total amount of pressure above x is px, t) = hx,t) ρg hx, t) z ) z = ρghx, t). Assuming ifferentiability as before, we get conservation of momentum ) hu) t + gh + hu =. Combining this with.4), we obtain the one-imensional shallow water equations.5) h t + hu) x =, ) hu) t + gh + hu =. We will always assume that h is positive, as negative height is meaningless. Actually, it may be shown that if h is nonnegative at the initial time, then it will stay nonnegative at all times. The erivation of.5) can be generalize to the two-imensional setup seen in Figure.b). We only give the erivation of conservation of mass. Let W R be any open, connecte set. Then the rate of change of mass over W exactly balances the flux through the bounary W. Thus, enoting the flux over a point x R by fx, t) = fx, t), gx, t)), we have t W ρhx, t) x = W x fx, t) ν Sx), where ν = ν x, ν y ) is the unit normal vector on W. Integrating by parts, this ientity is equivalent to ρhx, t) x = f x x, t) + g y x, t) Sx). t W Since W was arbitrary, this implies that W ρh t + f x + g y =. As for the one-imensional setup, mass is avecte with the flui, so writing u = u, v) for the velocity fiel, we have f = ρhu. Thus, h t + hu) x + hv) y =. Applying a similar argument to the momentum in each irection, we obtain the two-imensional shallow water system.)... Conservation laws The flux function of the one-imensional shallow water equations.5) epens on x an t only through h an u. Hence, the system can be written more compactly as a conservation law [9], which in general is of the form.6) U t + fu) x = for x R an t >. The solution U = Ux, t) : R [, ) R n is referre to as the conserve variables. In the case of the shallow water equations.5), we have U = [h, hu] an [ ] hu fu) = gh + hu.

11 .. CONSERVATION LAWS Integrating the conservation law.6) over an interval x, x ), we see that the total amount of quantity over the interval can only change ue to the flux through the enpoints: x.7) Ux, t) x = fux, t)) fux, t)). t x Thus, conservation laws moel quantities in this case mass an momentum that are preserve over time. Any change in the quantity in a omain is cause solely by the flux through the bounary. The conservation law.6) can be written in the quasi-linear form.8) U t + f U)U x =, where f U) enotes the Jacobian of f. We say that the conservation law is hyperbolic if f U) has n real istinct eigenvalues [9]. The shallow water system is hyperbolic; inee, f U) has eigenvalues λ = u gh an λ = u + gh, which are real an istinct since h is positive. Other examples of hyperbolic conservation laws inclue the wave equation, the magneto-hyroynamics equations, the Einstein equation an the Euler equations [6]. The property of hyperbolicity implies a finite spee of propagation: Information travels at a finite spee given by the eigenvalues of f U). This is a funamental property of these PDE. As a consequence of this property, the task of solving an initial value problem can be tackle by iviing the spatial omain into smaller omains, solving the resulting inepenent problems an then patching these together. This may be contraste with parabolic equations such as the heat equation, where information travels at an infinite spee. Hence, any local change in initial ata implies an immeiate global change in the solution. The two-imensional shallow water system.) can be written in the similar form.9) U t + fu) x + gu) y =, for U = [h, hu, hv]. In one spatial imension, the conition of hyperbolicity of a conservation law is simply that f U) has real an istinct eigenvalues. For multiimensional conservation laws such as.9), we require the same for f U) an g U), but in aition we require that all nontrivial linear combinations of f U) an g U) have real an istinct eigenvalues. It is a straight-forwar calculation to show that.) is hyperbolic in this sense [6]. Specifically, f U) an g U) have eigenvalues.) λ = u gh, λ = u, λ 3 = u + gh, µ = v gh, µ = v, µ 3 = v + gh. We will first evelop the theory for the one-imensional conservation law an then generalize it to the multi-imensional case. Of the most interesting properties of hyperbolic conservation laws is the appearance of iscontinuities or shocks in even the simplest of problems. To emonstrate this, we will solve an initial value problem for the canonical scalar, hyperbolic conservation law calle Burgers equation [6] ) u.) u t + = x

12 t. INTRODUCTION x a) A selection of characteristics..5.5 t= t= 4 4 x b) The solution at t =. Figure.. The solution of Burgers equation with initial ata u x) = sinx). with initial ata ux, ) = u x). Carrying out ifferentiation of fu) in.), we obtain the equivalent PDE.) u t + uu x =. We solve this equation by the metho of characteristics [9]. Let u solve.), an for each x R, let the characteristic xt) solve { x t) = f uxt), t)) = uxt), t) for t >.3) x) = x. Then tuxt), t) =, so u is constant along characteristics. Hence information is transporte along the characteristics with spee equal to uxt), t) = u x ), the eigenvalue of f u x )). The solution of.3) is therefore xt) = tu x ) + x. Solving this equation for x then gives a solution ux, t) = u x x, t)) of.). The main assumption of this erivation is that there is a unique characteristic through each given point x, t) in the x-t-plane. This is an incorrect assumption; inee, solving xt) = xt) for two characteristics x an x gives a solution t = min u x x). Hence, if u x ) < for some x, then there will be an intersection of characteristics. At the point of intersection the solution is multi-value; it is both equal to u x)) an u x)). This is illustrate in the following example. We solve Burgers equation with initial ata u x ) = sinx ). A selection of characteristics is plotte in Figure.a). Along each characteristic xt), the solution is constant, equal to u at the point x). At time t =, there is an intersection of characteristics, an at this point, the solution is both equal to an. The problem of multi-valueness is resolve by the formation of a shock a iscontinuity in the solution. This is illustrate in Figure.b), where the solution of Burgers equation at t = is shown. To allow for iscontinuous solutions of a ifferential equation, one consiers the equation in a weak sense.

13 .. CONSERVATION LAWS 3 Definition.. A function U L loc R [, )) is a weak solution of.6) if Uϕ t + fu)ϕ x xt + ϕx, )Ux, ) x = for all ϕ C c R R [, )). While this formulation allows the presence of iscontinuities, uniqueness of solutions may not be possible without aing further amissibility criteria.... Existence an uniqueness. The mechanism that counters the formation of large graients in nature is viscosity. Viscosity is taken into account in the more funamental equation.4) U ε t + fu ε ) x = εu ε xx for an ε > [4]. Uner certain assumptions, this equation has a unique smooth solution. Letting ε one may obtain, using a compactness argument, a solution of.6). A solution that is the limit of such functions is calle a vanishing viscosity solution an is consiere to be the physically relevant solution of the conservation law [4]. It may be har to verify that a given weak solution is the vanishing viscosity solution. However, it is known that viscosity is closely relate to the concept of entropy, a measure of isorer in an isolate system. By the secon law of thermoynamics, entropy is nonecreasing in time; by a ifference in sign, mathematical entropy is nonincreasing in time. In the context of conservation laws, entropy shoul be preserve in smooth regions of the solution an be issipate ecrease) aroun shocks. Thus, for a smooth solution, the entropy shoul be a conserve variable. We formalize these ieas as follows. Let some function E = EU) : R n R be given as a measure of the entropy present in a solution U of the conservation law. Uner what conitions oes E satisfy its own conservation law.5) EU) t + QU) x = for some entropy flux function Q? By left-multiplying the equivalent form of the conservation law.8) by E U), we obtain EU) t + E U) f U)U x =. E U) enotes the transpose of the column vector E U), so left-multiplying by E U) is the same as taking the Eucliean inner prouct with E U).) Hence,.5) may be satisfie if an only if there exists a Q such that Q U) = E U) f U). Definition.. An entropy pair for the conservation law.6) is a pair E, Q) consisting of a convex function E : R n R an a function Q : R n R such that Q U) = E U) f U) for all U R n. Recall that convexity of a function E : R n R means that the matrix E U), the Hessian of E, is a positive matrix. The conition of convexity will soon be justifie. If the conservation law is scalar, then any convex ifferentiable function E : R R gives rise to an entropy pair; just let QU) = U R E s)f s) s. For instance, for the Burgers equation.), the entropy Eu) = u gives an entropy flux QU) = u s s = u3 3. For systems, the process of fining entropy pairs is more ifficult, although many conservation laws are equippe with rather

14 4. INTRODUCTION natural entropies. In the linear case fu) = AU for a symmetric matrix A, one entropy pair is the energy EU) = U U, with entropy flux QU) = U AU. For the shallow water equations, the entropy of interest is the total energy. This is given by EU) = hu + gh ), an is the sum of kinetic an potential energy. It is reaily verifie that.6) Eh, hu) = hu + gh ) an Qh, hu) = hu3 + guh is an entropy pair for the shallow water equations.5). The calculations carrie out to obtain the entropy equality.5) are not vali when iscontinuities are present in the solution. To obtain a similar result for weak solutions, we must realize the solution as the limit of solutions of the viscous problem.4). Assume that an entropy pair E, Q) of the conservation law is given. By multiplying the equation.4) by E U ε ) on both sies, we get EU ε ) t + QU ε ) x = εe U ε ) Uxx ε = ε EU ε ) xx Ux) ε E U ε )Ux ε ) εeu ε ) xx, the inequality following from the positivity of E U ε ). In the limit ε, the right-han sie vanishes formally), an we obtain the entropy inequality.7) EU) t + QU) x. This is to be interprete in the sense of istributions: EU)ϕ t + QU)ϕ x xt + Ex, )ϕx, ) x R for all ϕ C c R [, )) with ϕ. Definition.3. We say that a weak solution U of the conservation law satisfies the entropy conition for an entropy pair E, Q) if.7) hols. In omains where the solution is smooth, the entropy equality.5) must necessarily hol. The entropy inequality.7) is only relevant near iscontinuities in U or its erivatives. The concept of entropy has proven important for existence an uniqueness of conservation laws. In [3], Kruzkov showe that for a scalar equation, even in several imensions, there is a unique weak solution that satisfies the entropy conition for a specific class of entropy pairs. A more general result for systems of conservation laws in one imension was prove by Lax [4]. He showe that a solution satisfies the entropy conition for a strictly convex entropy function whenever it is the physically correct solution. More importantly, the entropy conition is a sufficient conition whenever the initial ata is sufficiently small..3. Finite volume methos As the task of solving a conservation law explicitly is tractable only in the simplest of cases, we must employ numerical methos when solving more realistic problems. Of the available approaches, the finite volume methos are the most successful [6]. These methos rely on partitioning the computational omain into a finite set of control volumes, over which the conserve variables are average. This average form allows the presence of iscontinuities in the solution contrary to finite ifference schemes, which rely on a pointwise approximation of the PDE. The omain is ivie into intervals I i = [ ) x i /, x i+ / for i Z R

15 .3. FINITE VOLUME METHODS 5 U i I i U i+ U i I i I i+ x i x i / x i x i+/ x i+ Figure.3. Division of R into control sets I i. of length i = x i+ / x i /, an Ux, t) is approximate by a piecewise constant function U i t) = Ux, t) x i I i see Figure.3). By.7), U i satisfies.8) t U it) = fuxi+ /, t)) fux i /, t)) ). Integrating this equation from t to t + t for some time step t >, we fin an expression for U i at the next time step, U i t + t) = U i t) t+ t t fuxi+ /, s)) fux i /, s)) ) s. However, this expression relies explicitly on the exact solution U, an so must be approximate appropriately. The following algorithm is ue to Gounov [5]. At the initial time t =, we average the initial ata over each gri cell I i, obtaining a piecewise constant approximation U i ). At each cell interface x i+ / we solve the Riemann problem {.9) U RP U i if x < x i+ / x, ) = U i+ if x > x i+ /. This is a well-efine initial value problem that may be solve exactly. We obtain an approximate solution at the next time.) U i t) = U i ) t f U RP x i+ /, s) ) f U RP x i /, s) )) s. The process escribe above is repeate iteratively until t has reache some en time t max. The solution of the Riemann problem consists of a set of at most n waves emanating from x = x i+ /. The spee of propagation of each of these waves in the x-t-plane is boune by the maximum eigenvalue of f U i ) an f U i+ ). To keep waves from ifferent Riemann problems from interacting with each other, we select t so that the CFL conition c t,

16 6. INTRODUCTION where c = max λ k Ux, )), k x R is satisfie. The quantity c t is enote as the CFL number in the remainer; the CFL conition requires the CFL number to be less than. In general, it is har to solve the Riemann problem exactly. Instea, one consiers so-calle approximate Riemann solvers. In the formula.), we replace fu RP x i+ /, s)) by a numerical flux function F i+ / = F U i, U i+ ). The semiiscrete form of the scheme is then.) t U i = ) Fi+ / F i /. Note that this upating formula only epens on U, t k ) in the cells I i, I i an I i+ ; thus, the scheme is a 3-point scheme. More general p+)-point schemes exist [4], an will inee be consiere later when we increase the orer of accuracy of the scheme. The specific form of the semi-iscrete formula.) is not inciental. A finite volume metho that may be written in this form is calle conservative [6]. Conservation implies that the flow into a gri cell I i+ from the left exactly balances the flow out of the cell I i to the right; they are both equal to F i+ /. Thus, a iscrete conservation of the quantity U is obtaine. Inee, by summing.) over i = k,..., l, we obtain ) l U i = F k+ / F l /, t i=k a iscrete version of.7). There lies great freeom in choosing the numerical flux F i+ /, but to ensure that the scheme approximates the correct equation, a certain consistency criterion is impose. F is terme consistent with the conservation law.6) if F U, U) = fu) for all U R n. Thus, when the solution in two neighboring gri cells are equal, the flux through the cell interface shoul exactly equal the flux given by the conservation law. By integrating.) over t t k, t k + t k ) an applying any numerical metho of integration, we fin a solution U i t k+ ) at the next time step. This integral may be iscretize with a number of methos. Perhaps the simplest one is the first-orer accurate forwar Euler metho. Writing Ui k = U i t k ) an LUi k ) for the right-han sie of.), this metho is given by U k+ i = U k i + t k LU k i ). In this thesis we also use the secon- an thir-orer accurate, non-oscillatory Runge-Kutta methos [7].) an.3) U i = U k i + t k LU k i ), U i = U i + t k LU i ), U k+ i = U k i + U i ) U i = U k i + t k LU k i ), U i = 3 4 U k i + 4 U i + t 4 LU i ), U k+ i = 3 U i k + 3 U i + t LUi ). 3

17 .3. FINITE VOLUME METHODS Two-imensional conservation laws. The numerical metho outline above may be generalize to conservation laws in two an more) spatial imensions.9). The computational omain is ivie into gri cells I i,j = [ [ ) x i /, x i+ /) yj /, y j+ / with x i+ / x i / an y j+ / y j / y. The solution U is approximate by the cell average U i,j = Ux, y, t) xy, y I i,j an the numerical metho is written in the form.4) t U i,j = Fi+ /,j F i /,j) y Gi,j+ / G i,j /) for numerical fluxes F i+ /,j = F U i,j, U i+,j ) an G i,j+ / = GU i,j, U i,j+ ). These are approximate Riemann solvers of the problems U t + fu) { x = U i,j if x < x i+ / Ux, ) = U i+,j if x > x i+ / an U t + gu) y { = U i,j if y < y j+ / Uy, ) = U i,j+ if y > y j+ /, respectively. F an G are sai to be consistent with the conservation law.9) if we have F U, U) = fu) an GU, U) = gu) for all U R n..3.. Two stanar schemes. Two well-known finite volume fluxes are the Rusanov an the Roe fluxes. We inclue them here for reference. See also [6]. Rusanov: The Rusanov or Local Lax-Frierichs) flux is.5) F Rus i+ / = fu i) + fu i+ )) c i+ /U i+ U i ), where c i+ / = max k { λ ku i ), λ k U i+ ) } is an estimate of the local wave spee. Roe: The Roe flux relies on a linearization of the conservation law aroun a point Ũ. It is given by.6) F Roe i+ / = fu i) + fu i+ )) R Λ R U i+ U i ), where R is the matrix of eigenvectors of f Ũ) an ) Λ = iag λ Ũ),..., λnũ). The state Ũ suggeste in Roe s paper [3] calle the Roe average ensures that isolate single shocks are resolve exactly. For the shallow water equations, this state is given by Ũ = h, hũ) with.7) h = h i + h i+ an ũ = hi u i + h i+ u i+ hi + h i+.

18 8. INTRODUCTION.3.3. Further notation. For a gri function u i we efine the ifference an average operators [u] = u i+ / i+ u i an u i+ / = u i + u i+. The following ientities are reaily verifie:.8) [uv ] i+ / = u i+ / [v ] i+ / + v i+ / [u] i+ /, u i+ / u i / = u i+ / [u] i+ / + u i / [u] i /. For a two-imensional gri function u i,j we write [u] i+ /,j = u i+,j u i,j, [u] i,j+ / = u i,j+ u i,j, u i+ /,j = u i,j + u i+,j, u i,j+ / = u i,j + u i,j+.

19 CHAPTER Energy preserving an energy stable schemes.. Introuction The two most funamental questions for any numerical approximation are: Does the scheme converge as the gri is refine, an if it oes, is the limit the correct solution? In our framework, those questions are: Does the solutions obtaine by a finite volume metho.) converge to a weak solution of the conservation law.6), an furthermore, is that limit the physically relevant solution? The first question was answere by Lax an Wenroff [4]. Theorem. Lax-Wenroff). If a sequence {U k) } k of solutions compute by a consistent an conservative scheme with gri sizes t k) an k) converges bounely a.e. as t k), k), then the limit is a weak solution. Thus consistency an conservation is essential to ensure that the limit is a solution of the conservation law. Note that this theorem oes not aress the question of whether the scheme will at all converge, only that if it oes, then the limit is a weak solution. The secon question is answere in the form of a iscrete version of the entropy conition. Given an entropy pair E, Q), we look for numerical methos whose solutions satisfy a iscrete form of the entropy inequality.7),.) t EU it)) + Qi+ / Q i /) for all i Z an t > for a numerical entropy flux Q i+ / = Q U i t), U i+ t) ) that is consistent with Q. The numerical entropy flux Q is assume to be continuous as a function Q : R n R n R. The following theorem whose iea an proof are very similar to Lax an Wenroff s shows that the iscrete entropy conition is sufficient for the limit function to be the physically correct solution. Theorem. Osher [3]). Let {U k) i t)} k be a sequence of functions compute by a consistent an conservative scheme that converges bounely a.e. to a function U. Assume that there is an entropy flux Q, consistent with Q, such that.) is satisfie for every U k). Then U is a weak solution of.6) that satisfies the entropy conition for E, Q). Proof. The fact that the limit is a weak solution follows from the Lax- Wenroff theorem. Let k N; we write U i = U k) i. Let ϕ Cc R [, )) with ϕ an enote ϕ i = ϕ i t) = ϕx i, t). By multiplying.) by ϕ i, summing over i Z an integrating over t [, ), we obtain t EU i) ϕ i + ) Qi+ / Q i /) ϕ i t. i= Integrating an summing by parts, this becomes EU i )) ϕ i ) + i= i= EU i ) t ϕ i + Q i+ / 9 ) ϕ i+ ϕ i t.

20 . ENERGY PRESERVING AND ENERGY STABLE SCHEMES We may view U i an ϕ i as step functions on R [, ), letting Ũx) = U i an ϕx) = ϕ i for x I i. Then the above can be written as E Ũx, ) ) ϕx, ) x + E Ũx, t) ) ϕx, t) tx R R t.) ) ϕx + ) ϕx) + Q Ũx), Ũx + ) tx. R Both Ũ an ϕ converge bounely a.e. to U an ϕ, respectively, so by Lebesgue s ominate convergence theorem, the first two terms converge to EUx, ))ϕx, ) x + EUx, t))ϕ t x, t) tx R R as k. Since Q is continuous an consistent with Q, the thir term converges to QUx, t))ϕ x x, t) tx. Thus the left-han sie of.) converges to EU)ϕ t + QU)ϕ x tx + R thus proving the inequality. R R EUx, ))ϕx, ) x, This result gives a motivation for looking for schemes that satisfy the iscrete entropy inequality.). However, the theorem is not a constructive one; it gives no hint to how such a scheme may be esigne. This problem was tackle by Tamor in 984 for scalar equations an was generalize to systems in 987 [34, 35]. Given an entropy pair E, Q), a finite volume scheme was terme entropy preserving if it satisfies the iscrete entropy equality.3) t EU it)) + Qi+ / Q i /) = for a numerical entropy flux Q consistent with Q. By summing over i = k,..., l we see that ) l EU i t)) = t Q k+ / Q l / i=k for all choices of enpoints k, l. Hence, if the entropy flux through the bounary is zero, then the right-han sie vanishes, an so the total amount of entropy in the interval [x k, x l ] is preserve over time. A scheme was terme entropy stable if it satisfies the iscrete entropy inequality.). It was foun in [34, 35] that a scheme is entropy stable precisely when it contains more numerical iffusion than an entropy preserving scheme. Moreover, an explicit conition was foun that ensures that a scheme is entropy preserving. Thus, by esigning an entropy preserving scheme, one may etermine entropy stability in a scheme by comparison. These notions will be mae more precise in the next few sections. In Section.3 we esign a novel entropy preserving scheme for the shallow water system.5) an stuy it through a series of numerical experiments. In Section.4 we a numerical iffusion to obtain an entropy stable scheme. In Section.5 we generalize these ieas to the full two-imensional shallow water system.).

21 .. ENTROPY PRESERVING SCHEMES.. Entropy preserving schemes A solution of the conservation law.6) shoul satisfy the entropy inequality.7). Specifically, this implies the stability estimate EU) x t R on the solution. By requiring that numerical approximations satisfy the iscrete entropy inequality.), we ensure that the solutions o not converge towars entropy violating weak solutions. What is more, we get a stability estimate t i EU i ) in the approximate solutions. Recall that the entropy equality.5) was obtaine by multiplying.6) by E U) on both sies. Doing this to the semi-iscrete scheme.), we obtain.4) ) t EU it)) = E U i t)) ) F i+ / F i /. If the right-han sie were equal to Qi+ / Q i /) for some Qi+ / = Q ) U i, U i+ that is consistent with Q, then.4) woul inee be an entropy equality, an the scheme woul be entropy preserving. Given the importance of the quantity E U), it is terme the vector of entropy variables an is enote by V = E U) [34]. As E is strictly convex, we have E U) = U V >, an so the mapping V = V U) from conserve variables to entropy variables is ifferentiable an injective. As a consequence, we may work with conserve variables an entropy variables interchangeably. Given U i, we write V i = E U i ). We wish to fin a more irect criterion on F for the ientity Vi ) Fi+ / F i / = Qi+ / Q i / to hol. The next theorem gives precisely this. The entropy potential is efine as ψu) = V U) fu) QU). Theorem.3 Tamor [35]). Assume that a consistent numerical flux F i+ / satisfies.5) [[V ] i+ / F i+ / = [ψ ] i+ /. satisfy the is- Then solutions compute by the scheme with numerical flux F i+ / crete entropy equality.3) with numerical entropy flux.6) Qi+ / = V i+ /F i+ / ψ i+ /. Hence, Q is consistent with Q. Proof. Taking the inner prouct of.) with V i = E U i ), we get t EU i) = V i F i+ / Vi ) F i / aing an subtracting) = V i+ /F i+ / ) [V ] i+ F / i+ / V i /F i / + ) ) [V ] i F / i /

22 . ENERGY PRESERVING AND ENERGY STABLE SCHEMES by.5) an the efinition of Q) = = Q i+ / + ψ i+ / [ψ ] i+ / Q i / ψ i / [ψ ] i / Qi+ / Q i /). For consistency of Q, we see that if U i = U i+, then Q i+ / = V i+ /F i+ / ψ i+ / ) = Vi f i Vi f i Q i ) = Q i. In the scalar case n =, the equation.5) has the unique solution F i+ / = [ψ ] i+ /. [V ] i+ / As mentione earlier, Burgers equation has an entropy pair Eu) = u /, Qu) = u 3 /3. The corresponing entropy variables an potential are V u) = u an ψu) = u 3 /6, so the entropy preserving flux for Burgers equation is F i+ / = 6 [u 3 ] i+ / [u] i+ / = u i + u iu i+ + u i+. 6 The more general case of a system of conservation laws.6) is harer, but some existence results exist. The flux originally propose in [35] is in the form of a path integral,.7) F i+ / = fv ξ)) ξ, where V ξ) = V i + ξ [V ] i+. It is a straight-forwar calculation that F is consistent an satisfies.5); see [35]. To see how this numerical flux is applie, we / consier the conservation law with flux fu) = AU for a symmetric matrix A. As previously mentione, an entropy pair for this conservation law is EU) = U U, QU) = U AU, with entropy variables V = E U) = U. Inserting into.7) an evaluating the integral gives the entropy preserving flux F i+ / = fu i) + fu i+ )). However, for more general, nonlinear flux functions, the integral.7) might be har to calculate explicitly..3. Energy preservation Our aim is to esign energy stable schemes for the shallow water equations. These are schemes whose solutions satisfy the iscrete entropy inequality t EU it)) + Qi+ / Q i /) with E = hu + gh ) the energy of the shallow water system an Q any numerical entropy flux function that is consistent with Q = hu3 + guh. The first step towars esigning energy stable schemes will be to esign an energy preserving scheme a scheme satisfying the iscrete entropy equality.3) for the energy. Preservation of energy in the shallow water equations has been stuie extensively by, among others, Arakawa et. al [,, 3], an has been eeme important especially in long-term meteorological simulations. We go on to present three energy

23 .3. ENERGY PRESERVATION 3 preserving schemes: The average flux.7), the PEC scheme presente in [36], an a novel, much simpler flux..3.. The AEC scheme. The integral.7) can be evaluate explicitly [] to obtain the energy preserving flux F i+ / = [ F ) i+, F ), / i+ /] with.8) F ) i+ = h iu i / 3 + h i+u i+ + h iu i+ + h i+u i u3 i 4 u3 i+ 4 + u iu i+ + u i+u i 4 4 F ) i+ = / h iu i + h i+u i+ + 6 h iu i+ + 6 h i+u i + 4 h iu i u i+ + 4 h i+u i u i+ + 7g 4 h i + 7g 4 h i+ g h ih i u4 i + 96 u4 i+ 48 u i u i+. We enote this as the AEC Average Energy Conservative) scheme. It is reaily shown that the AEC flux is consistent with the shallow water system..3.. The PEC scheme. An explicit solution of.5) was foun by Tamor in [36]. In this flux, the path integral in.7) is replace by a piecewise linear path along orthogonal irections in R n. Let {l k, r k } n k= be an orthogonal eigensystem in R n. Define ) V [] = V i an V [k] = V [k ] + [V ] i+ l / k r k for k =,..., n. Note that V [n] = V i+. Let.9) F [k] = ψ V [k]) ψ V [k ]) [V ] i+ l l k for k =,..., n. / k The entropy preserving flux is given by.) F i+ / = n F [k]. k= To see that this flux satisfies.5), multiply by [V ] i+ / n [V ] i+ F / i+ / = k= to get ψ V [k]) ψ V [k ]) = [ψ ]. i+ / In the flux.), we follow Tamor an Zhong [37] an choose the system of eigenvectors as that of the Jacobian of the flux function f, evaluate at the Roe average.7). We enote the resulting scheme as the PEC Pathwise Energy Conservative) scheme The EEC scheme. The AEC an PEC schemes have several isavantages. Both fluxes contain complex expressions that require a large number of floating point operations at each flux evaluation. Moreover, the PEC has some numerical stability issues, most notably that a ivision by zero may occur in the expression for F [k].9). The thir scheme that we will consier avois these problems by applying.5) in a more irect manner. To separate this flux from the others, we enote it by F. The entropy variables an entropy potential of the energy of the shallow water equations are [ ].) V = E gh u U) = an ψ = u guh.

24 4. ENERGY PRESERVING AND ENERGY STABLE SCHEMES An energy preserving scheme with numerical flux F i+ / = [ F ) i+ /, F ) i+ /] must satisfy.5), which in this case is ) [gh] F i+ [u /] / i+ F ) / i+ + [[u] / i+ F ) / i+ = / g [uh ]. i+ / By the first ientity in.8), this can be written as g [h] i+ F ) / i+ u / i+ / [u] i+ F ) / i+ + [u] / i+ F ) / i+ / = g [h] i+ / h i+ /u i+ / + gh i+ / [u] i+ /. By equating jumps in the same variable we get the set of equations F ) i+ / = h i+ /u i+ /, F ) i+ u ) / i+ / F i+ = / gh i+ /, which have solution [ ] h i+ /u i+ /.) Fi+ / = gh i+ / + h i+ /u. i+ / The resulting finite volume scheme is then.3) t U i = Fi+ / F i /), which we enote as the EEC Explicit Energy Conserving) scheme. Theorem.4. The EEC scheme.3) is consistent with.5) an is seconorer accurate. Furthermore, it is energy preserving, i.e. it satisfies the energy quality.3) with numerical entropy flux.4) Qi+ / = gh i+ / which is consistent with Q. u i h i+ + u i+ h i + h i+ /u i+ /u i u i+, Proof. Consistency of the EEC scheme follows immeiately from the efinition. A simple truncation error analysis shows that the scheme is of secon orer. The scheme satisfies the energy preservation criterion.5) by construction, an so satisfies the iscrete energy equality with Q as in.6), which in this case is.4) Numerical experiment. We test the energy preserving properties of the AEC, PEC an EEC schemes. To measure the change in energy over time, we will consier the relative change in energy over time, Et) E) L, E) L where Et) = i EU it)). By the iscussion in the beginning of this chapter, Et) shoul be constant as long as the entropy flux through the bounary is zero. The energy flux Q is zero exactly when u =. Therefore we stop the simulation before any waves hit the bounary. We consier a stanar am-break problem, { if x <.5) hx, ) = ux, )..5 if x > The correct solution consists of a left-going rarefaction wave an a right-going shock. We compute with the energy preserving schemes on a mesh of gri points up to time t =.4 using the secon-orer accurate Runge-Kutta RK) metho.) for

25 .3. ENERGY PRESERVATION 5 temporal iscretization. Figure.a)-c) show the compute solutions. There is little iscernible ifference between the solutions, an all three schemes resolve the shock an the rarefaction wave correctly. However, in the wake of the shock there are unphysical oscillations with perio of the orer of. This is to be expecte from an energy preserving scheme. With the lack of any iffusive mechanism, nonlinear ispersive effects reistribute energy into higher wave numbers. As the highest wave number on a iscrete mesh is, the oscillations will have perio precisely of orer. This is verifie in Figure., where we repeat the experiment with the EEC scheme on a sequence of gri sizes. In all four solutions the perio of the oscillations are of the orer of. Note that the amplitue of the oscillations in all experiment are of the orer of the initial jump in h. To get ri of the oscillations, numerical issipation must be ae to allow a iffusion of energy a) Height with EEC c) Height with AEC x 4 EEC PEC AEC b) Height with PEC ) Relative energy vs. time. Figure.. Height h at t =.4 compute with the three energy preserving schemes. In Figure.) we see the relative energy of the solution as a function of time. There is a certain increase in energy over time, although we have prove that energy shoul be preserve. We claim that this increase is solely ue to the error in the iscretization.) of the time integral. To emonstrate this, we repeat the experiment with the EEC scheme using a smaller CFL number; hence, t is lowere accoringly. In aition, we compute with the Runge-Kutta RK3) metho.3), which is more accurate than RK. The effect can be seen in Figure.3. There is a rastic ecrease in the amount of energy iffusion for both methos. Furthermore, the RK3 metho is less iffusive than the RK metho. We remark that lowering the CFL number or changing the metho of integration has no visible effect on the solution variables. This shows just how small the scales we are ealing with are.

26 6. ENERGY PRESERVING AND ENERGY STABLE SCHEMES a) points b) 4 points c) 8 points ) 6 points Figure.. Height h at time t =.4 compute with the EEC scheme Numerical experiment: Computational cost. Next, we compare the computational efficiency of the schemes. As the error in energy ecreases with the CFL number, it is interesting to see how the CFL number must be chosen to get a certain boun on this error. For each scheme we fin the CFL number that ensures that the energy issipation error is less than 3, 4 an 5. The runtime of each scheme is shown in Table.. Clearly the EEC scheme is the fastest of the three, with runtimes almost three times lower than the PEC scheme. The AEC scheme is not far behin, but for some reason, we were unable to reuce errors to 5 with this scheme. Energy error EEC PEC AEC Table.. Normalize runtimes for the three energy preserving schemes on the one-imensional am break problem with three ifferent levels of error in energy. In conclusion, we have esigne a novel scheme that preserves the energy of the shallow water system exactly moulo temporal iscretization errors. The scheme is very simple to implement, an as it requires only a few floating point operations, is computationally very efficient. However, the presence of a shock resulte in the prouction of unphysical oscillations. This will be resolve in the next section.

27 .4. NUMERICAL DIFFUSION 7. x x 5 a) RK, CFL = x x 8 b) RK, CFL = c) RK3, CFL = ) RK3, CFL =.5 Figure.3. Relative energy of solutions compute with the EEC scheme on mesh points for two ifferent RK schemes at two ifferent CFL numbers..4. Numerical iffusion The EEC scheme was esigne to preserve energy. However, energy shoul be issipate at shocks. As we saw in the previous section, the scheme prouces a cascae of energy into higher wave numbers, which results in oscillations in the trail of the shock. To amen this, numerical iffusion must be ae to the scheme to obtain an energy stable scheme. To quantify this numerical iffusion, we write the numerical flux in the viscous form [35].6) F i+ / = fui ) + fu i+ ) ) P i+ / [V ] i+ /. The construction of the viscosity coefficient matrix P is given in the following lemma. Lemma.5. Assume that the numerical flux F i+ / = F U i, U i+ ) is Lipschitz in each parameter an is consistent with the conservation law. Then there exists a matrix P i+ / such that F i+ / can be written as.6). Proof..6) can be rewritten as.7) P i+ / [V ] = f i+ / i + f i+ F i+ / = F U i+, U i+ ) F U i, U i+ ) ) F U i, U i+ ) F U i, U i ) ) by the consistency of F. As previously remarke, the mapping from conserve variables U to entropy variables V is injective an ifferentiable, so we might as

28 8. ENERGY PRESERVING AND ENERGY STABLE SCHEMES well consier F as a function of V. Letting V ξ) = V i + ξ [V ] i+ for ξ [, ], we / have F V i, V i+ ) F V i, V i ) = = ξ F V i, V ξ))ξ F V i, V ξ))ξ [V ] i+ /, where F is the partial erivative of F with respect to its secon argument. Applying this to.7) then gives P i+ / = F V ξ), V i+ )ξ F V i, V ξ))ξ. The conition that will be impose to obtain entropy stability will be one of comparison; an entropy stable scheme will be one that contains more iffusion than an entropy preserving scheme. To formalize this, let F be a numerical flux satisfying the requirement of Theorem.3, so that the corresponing scheme is entropy preserving, an let F be any other numerical flux consistent with.6). We enote their numerical viscosity matrices by P an P, respectively, an we let D i+ / = P i+ / P i+ / be their ifference. Theorem.6 Tamor [35]). Assume that.8) [V ] i+ / D i+ / [V ] i+ / for all V i an V i+. Then the scheme with numerical flux F satisfies the entropy issipation estimate t EU i) + Qi+ / Q i /) = ).9) [V ] i+ 4 D / i+ / [V ] + [[V i+ / ] i D / i / [V ] i /, where Q i+ / = Q i+ / V i+ /D i+ / [V ] i+ / is consistent with Q an Q is as in.6). Proof. As in the proof of theorem.6, we left-multiply.) by Vi sies. Aing an subtracting F from F then gives t EU i) = = + V i F i+ / Vi F i / Vi F i+ / F i+ /) Vi F i / F ) i /) Qi+ / Q i /) ) Vi D i+ / [V ] V i+ / i D i / [V ] i. / The secon part of the right-han sie can be rewritten as ) on both V i ) D i+ / [V ] V i+ / i D i / [V ] i /

29 .4. NUMERICAL DIFFUSION 9 = ) V i+ /D i+ / [V ] V i+ / i /D i / [V ] i / ) [V ] i+ 4 D / i+ / [V ] + [V i+ / ] i D / i / [V ] i / ) V i+ /D i+ / [V ] V i+ / i /D i / [V ] i, / as [[V ] i± D / i± / [V ] i±. This gives the entropy issipation estimate. / Q is consistent with Q since, if U i = U i+, then [[V ] =, an so Q i+ / i+ / = Q i+ / + = Q i. Remark.7. A sufficient conition for.8) is that the symmetric part of the matrix D i+ / is positive for all V i, V i+. Since D i+ / in general epens nonlinearly on V i an V i+, this is only a necessary conition when we are in the scalar case, in which case.8) means that the number D i+ / is nonnegative. In general it is har to irectly esign entropy stable iffusion operators. Our approach will be to moify the iffusion operator of the Roe flux.6) to obtain an entropy stable iffusion operator. The Roe flux is selecte specifically for its simplicity an goo accuracy, an other fluxes might inee be chosen. We will replace the flux average term fu i)+fu i+ )) in.6) by the energy preserving EEC flux.3), an the iffusion operator R Λ R [U ] i+ by a term / of the form A i+ / [V ] for a positive matrix A i+ / i+ /. A similar approach has been consiere by Roe in [33] for the Euler equations. The construction of the matrix A i+ / relies on the following result, a special case of a more general result of Barth [5]. Lemma.8. Let U, U i an U i+ be given states. We efine the symmetric positive efinite change-of-variables matrix at U = [h, hu] as U V = [ ] u g u u. + gh i) Define the following scale version of the eigenvector matrix R of f U): R = [ ] g u gh u +. gh Then we have ii) We have RR = U V. [U ] i+ / = U V ) i+ / [V ] i+ /, where U V ) i+ / = [ ] ui+ / g u i+ / u i+ + gh / i+ / is U V evaluate at the arithmetic averages.) h = h i+ /, an u = u i+ /. Proof. The results follow simply by insertion. Now, given left an right states U i an U i+, we evaluate the Roe matrix R Λ R at the state given in lemma.8 ii), U i+ / = [h i+ /, h i+ /u i+ /]. We are free to evaluate the matrix wherever we want, although by using states other than the Roe average we will lose the property of exact resolution of single shocks. By inserting this state we get R Λ R [U ] i+ / = R Λ R U V ) i+ / [V ] i+ /

30 3. ENERGY PRESERVING AND ENERGY STABLE SCHEMES We efine the numerical flux = R Λ R RR [V ] i+ / = R Λ R [V ] i+ /..) F ERoe i+ / = F i+ / R Λ R [V ] i+ /, an we enote the corresponing scheme as the ERoe entropy stable Roe) scheme. The following theorem is a result of theorem.6. Theorem.9. The ERoe scheme is consistent with the shallow water system.5), it is first-orer accurate an is energy stable, i.e. it satisfies the energy issipation estimate t EU i) + = where, Qi+ / Q i /) ) [V ] i+ R Λ R ) / i+ / [V ] + [[V i+ / ] i R Λ R ) / i / [V ] i / Q i+ / = Q i+ / 4 V i+ /R Λ R [V ] i+ / is consistent with Q. Here, Q is the energy flux.4) of the EEC scheme. Proof. Comparing the numerical viscosity matrices of the energy preserving EEC scheme an the ERoe scheme, we fin that their ifference is D i+ / = P i+ / P i+ / = R Λ R, which is a positive symmetric matrix, an hence satisfies the stability criterion.8) for all V i an V i+. The energy issipation estimate then follows from theorem Numerical experiments. We test the energy stability of the ERoe scheme in the am break problem of the previous section. The results are compute an compare with the Rusanov scheme.5) an the Roe scheme.6), along with a reference solution compute with the Rusanov scheme on a fine mesh of 3 gri points. Plots of height at t =.4 can be seen in Figure.4a). While the shock an rarefaction wave are more smeare out than for the EEC scheme, we see that all unphysical oscillations have been cleare out. Thus, aing numerical iffusion to the EEC scheme has ha the esire effect Reference ERoe Roe Rusanov x 3 Reference ERoe.5 Rusanov Roe a) Height at t = b) Energy vs. time. Figure.4. Solutions compute with the Roe, Rusanov an ERoe schemes for the one-imensional am break problem with mesh points.

31 .4. NUMERICAL DIFFUSION 3 In Figure.4b) we isplay the relative energy in the solution over time. The energy is clearly not preserve. Instea, a issipation of energy is taking place as it shoul, since there is a shock in the solution. Comparing the three schemes, the Rusanov scheme issipates slightly more energy than the other two. It is well-known that the Rusanov scheme is more issipative than many other schemes. We see that all three schemes issipate more energy than the reference solution oes. We can conclue that, in this experiment, there is too much numerical iffusion in all three schemes. However, fining the correct amount of iffusion can be very ifficult, an a smaller amount of iffusion may come at the cost of unwante oscillations. Numerical experiment: Large am break. In the previous experiment, the solutions compute with the three schemes were almost inistinguishable. We now consier an experiment that emphasizes the ifferences between the Roe an the ERoe scheme. The initial ata is given by { 5 if x <.) hx, ) = ux, ). if x > The correct solution consists of a left-going rarefaction wave an a right-going shock. We compute on a mesh of gri points up to t =.4, using a CFL number of.45, as before. A reference solution is compute with the Rusanov scheme on a mesh of 3 gri points. The result can be viewe in Figure Reference ERoe Roe Figure.5. Height at time t =.4 compute with the Roe an ERoe schemes for the large am break problem with mesh points. While the solutions are close away from the origin, there is a large iscrepancy at x =. Here, the Roe scheme immeiately prouces an unphysical steay shock, whereas the ERoe scheme computes the rarefaction wave as it shoul. It is wellknown that the Roe scheme is not entropy energy) stable, an that it may prouce steay shocks were there shoul be none see e.g. [36]). Numerical experiment: Near-zero heights. This experiment will illustrate how well the ERoe scheme preserves positivity. Recall that the height variable h is positive in exact solutions of the shallow water system, an a negative value of h woul be meaningless. preserve the positivity of h. The initial ata is given by.3) hx, ), ux, ) = Hence, we shoul require that approximate solutions { u if x < u if x >

32 3. ENERGY PRESERVING AND ENERGY STABLE SCHEMES for a u >. Depening on how large u is, the solution will create a gap in the water aroun the origin, with epths close to zero. We let u = 4, an we try to compute up to time t =. on the same mesh as before. The results are shown in Figure.6. The Roe scheme broke positivity Reference ERoe Reference Roe a) Height with ERoe at t =..5.5 b) Height with Roe at t =.56 Figure.6. Solutions compute with the Roe, an ERoe schemes for the expansion problem with mesh points. at aroun t =.56, an the simulation was halte. The ERoe scheme, on the other han, compute the solution correctly, albeit somewhat more iffuse than the reference solution. We o not claim that the ERoe scheme is positivity preserving. In fact, the scheme broke positivity when we increase u to 8. Still, this an the previous experiment show that the ERoe scheme is more stable than the Roe scheme in the presence of large shocks an close-to-zero heights. Numerical experiment: Computational efficiency. With the apparent avantages of the ERoe scheme over the Roe an Rusanov schemes, it is natural to ask whether these avantages come at the price of an increase in runtime. As a measure of accuracy we will see how close the compute height lies to the correct solution. We compute a reference solution of the am break problem of Section.4. with the Rusanov scheme on a mesh of 3 gri points, an for each of the three schemes, we fin the mesh at which the relative error in height in the L norm is less than,.5 an. per cent. The runtime of each scheme is presente in Table.. Clearly, the ERoe scheme gives the best performance, an issipates less energy than the other schemes. Surprisingly, the Rusanov scheme performs better than the Roe scheme, issipating less energy at the same runtime. Relative height error.5. Rusanov Roe ERoe Table.. Normalize run-times for the Rusanov, Roe an ERoe schemes on the one-imensional am break problem with three ifferent levels of relative error in height.4.. Increasing the orer of accuracy. While the ERoe scheme is energy stable an has several attractive features, it is only first-orer accurate. Consequently, sharp features such as shocks are iffuse out, as seen in the am break experiment in Section.4.. In this section we apply a stanar proceure to increase the orer of accuracy of the scheme to obtain a secon-orer accurate scheme

33 .4. NUMERICAL DIFFUSION 33 x i x i / x i x i+/ x i+ Figure.7. Original value U soli line) an reconstructe values Ũ ashe line) which we enote as the ERoe scheme. This scheme will resolve shocks much more accurately, at the cost of a slight increase in runtime. The metho that we will employ relies on a reconstruction of the solution at each time step. Recall that in the finite volume framework, we solve for the average U i = I i Ux, t) x over a gri cell I i. Thus, the approximate solution is constant in each gri cell. By applying a first-orer interpolation or reconstruction of U in the neighboring gri cells, we obtain a linear in each gri cell) function Ũx) = U i + σ i x x i ) for x I i, where σ i R n is the slope. Given a scheme t U i = F Ui, U i+ ) F U i, U i ) ), we replace U in the flux terms by the reconstructe variables: t U i = F Ui E, U W i+) F U E i, Ui W ) ), where an U W i = U i + σ i x i / x i ) = U i σ i Ui E = U i + σ i x i+ / x i ) = U i + σ i are the left an right states of the reconstructe variables in the gri cell I i. See Figure.7 for an illustration. This proceure formally increases the orer of accuracy of the scheme to two [6]. Higher orer schemes may be obtaine by using e.g. a quaratic or cubic reconstruction of U, but we will not consier this here. There are several choices available for selecting the slope σ i. Three caniates are the forwar, backwar an central ifferences U i+ U i, U i U i, U i+ U i. In each gri cell I i an for each component of U i, we will select the one of the three that gives the least oscillatory solution. In Figure.7, we see that the reconstructe value in gri cell I i increases the gap at the cell interface x i+ /. This may lea to unwante oscillations in the solution, particularly near shocks. To avoi oscillations,

34 34. ENERGY PRESERVING AND ENERGY STABLE SCHEMES we use a so-calle slope limiter in the construction of the slopes σ i. We employ the minmo limiter k) U.4) σ k) i+ i = mm U k) i U k) i U k) i U k) i+,, U ) k) i where.5) mmx, y, z) := { max{ x, y, z } if sign x) = sign y) = sign z) otherwise, an σ k) i is the kth component of σ i. See LeVeque [6] for more information on slope limiters. Instea of performing reconstruction in the conserve variables, we will reconstruct the energy variables. Given states {U i } at time t, we let V i = V U i ), an perform a linear reconstruction of V, obtaining left an right states Vi E an Vi W. Let Ui E = UVi E ) an Ui W = UVi W ). The secon-orer accurate ERoe scheme then has flux Fi+ ERoe = F U / i, U i+ ) R Λ R Vi+ W Vi E ), where R an Λ are evaluate at the average state h = he i + h W i+, u = ue i + u W i+ compare to.)). We o not use the reconstructe values in the EEC flux F, as it is alreay secon-orer accurate Reference ERoe ERoe.5 x a) Height Reference ERoe ERoe b) Energy vs. time Figure.8. Solutions compute with the first an secon-orer accurate versions of the ERoe scheme with mesh points Numerical experiment. To witness the gain in accuracy with the ERoe scheme, we repeat the am break problem of Section.4.. As is clearly seen in Figure.8a), the solution is now much less iffuse out. Both the shock an the rarefaction wave are more accurately reprouce, an the shock spans only a few gri cells. Figure.8b) isplays the energy over time. The amount of energy iffusion is more than halve, an the profile lies much closer to the reference solution.

35 .5. TWO-DIMENSIONAL ENERGY PRESERVING AND STABLE SCHEMES Two-imensional energy preserving an stable schemes Next, we generalize the schemes of the preceing sections to the two-imensional shallow water system.). As so much is similar to the one-imensional case, the exposition will be brief an without proofs. First, we introuce the two-imensional equivalent of an entropy pair. Definition.. A triple E, Q x, Q y ) is an entropy triple for the conservation law.9) if E : R n R is strictly convex an for all U R n. Q x ) U) = E U) f U) an Q y ) U) = E U) g U) It is reaily verifie that the energy.6) EU) = hu + hv + gh + ghb ) of the shallow water system gives rise to an entropy triple E, Q x, Q y ) with Q x = hu 3 + huv ) + guh an Q y U) = hu v + hv 3) + gvh. The energy variables are now.7) V = E U) = gh u u v an the energy potentials in each irection are +v, ψ x = guh an ψ y = gvh. Theorem.3, which gives a conition for schemes to be entropy preserving, is easily generalize to the two-imensional case. By the same argument as for the one-imensional EEC scheme, we fin that the two-imensional EEC scheme has numerical fluxes.8a) Fi+ /,j = an.8b) Gi,j+ / = h i+ /,j u i+ /,j h i+ /,ju i+ /,j) + g h ) i+ /,j h i+ /,j u i+ /,j v i+ /,j h i,j+ / v i,j+ / h i,j+ / u i,j+ / v i,j+ / h i,j+ /v i,j+ /) + g h ) i,j+ / Lemma.. The two-imensional EEC scheme is consistent with the twoimensional shallow water equations an is energy preserving, i.e. it satisfies the energy equality where.9a) an t EU i) + Qx i+ /,j Q x i /,j) + Qy y i,j+ Q ) y / i,j = / Qx = V i+ /,j F i+ /,j [ψ x ] i+ /,j.9b) Qy = V i,j+ / G i,j+ / [ψ y ] i,j+ /. Note that the two-imensional EEC scheme reuces to its one-imensional counterpart whenever there is no variation in the y-irection.,.

36 36. ENERGY PRESERVING AND ENERGY STABLE SCHEMES.5.. Numerical experiments. We test the EEC scheme on a two-imensional equivalent of a am break problem. The initial ata is given by { if x.3) hx, y, ) = + y <.5 ux, y, ). otherwise The exact solution is an expaning shock wave with an inwars-going rarefaction wave. We compute on a uniform mesh of gri points in the omain x, y) [, ] [, ] up to time t =.. We use g = as before. As seen in Figure.9a), the shock an the rarefaction wave are compute correctly, but there are unphysical oscillations in the wake of the shock. As before, these are cause by a lack of numerical iffusion, an will be ampene when aing a iffusion operator in the next section. 6 x 4 CFL = CFL =. CFL =. CFL = a) Height at t =.. b) Energy for four ifferent CFL numbers. Figure.9. Solution of the cylinrical am break problem compute on a uniform mesh with the EEC scheme. Figure.9b) shows the relative change in energy over time for four ifferent choices of CFL numbers. Clearly, the increase in energy is ampene with smaller time steps, an in the limit t, the scheme preserves energy exactly. Numerical experiment: Computational cost. Next, we perform a test of the computational efficiency of the EEC an PEC schemes, similar to the one-imensional am break problem. We run the problem.3) on a 5 5 mesh an we fin the CFL numbers that give a relative energy error of 3, 4 an 5. The runtimes are isplaye in Table.3. Clearly, the EEC scheme gives the smallest energy error for the same runtime, with ifferences in runtime of the orer of 5 or 6. Relative energy error EEC PEC Table.3. Normalize run-times for the two energy preserving schemes on the two-imensional cylinrical am-break problem with three ifferent levels of error in energy..6. Two-imensional ERoe The generalization of the ERoe scheme to the two-imensional shallow water system is straight-forwar. If R x, Λ x, R y an Λ y are the eigenvector an eigenvalue

37 .6. TWO-DIMENSIONAL EROE 37 matrices of f U) an g U), respectively, then the numerical fluxes F ERoe an G ERoe will be of the form an F ERoe i+ /,j = F i+ /,j Rx Λ x R x ) [V ] i+ /,j G ERoe i,j+ / = G i,j+ / Ry Λ y R y ) [V ] i,j+ /. To obtain the specific states at which the eigenmatrices are evaluate, we apply the following lemma, a straight-forwar generalization of lemma.8. Lemma.. Let U, U i,j, U i+,j an U i,j+ be given states. We efine the symmetric positive efinite change-of-variables matrix at U = [h, hu, hv] as U V = V U = u v u u + gh uv. g v uv v + gh i) Define the following scale version of the eigenvector matrices R x an R y of f U) an g U), respectively: R x = u gh u + gh g, v gh v.3) R y = u gh u g v gh v +. gh Then we have R x R x ) = R y R y ) = U V. ii) We have [U ] i+ /,j = U V ) i+ /,j [V ] i+ /,j an [U ] = U i,j+ / V ) i,j+ / [V ] i,j+ / where U V ) i+ /,j an U V ) i,j+ / are U V, evaluate at the arithmetic averages.3a) h = h i+ /,j, u = u i+ /,j, v = v i+ /,j an.3b) h = h i,j+ /, u = u i,j+ /, v = v i,j+ / respectively. By the same technique as before, we en up with eigenvector matrices R x an R y.3), evaluate at the average states.3). We have the following energy stability result. Proposition.3. The two-imensional ERoe scheme is consistent with the shallow water system.). Furthermore, it is energy stable, i.e. it satisfies the energy issipation estimate where an t EU i,j) + Qx i+ /,j Q x i /,j) + y Qy i,j+ Q ) y / i,j, / Q x i+ /,j = Q x i+ /,j V i+ /,jr x Λ x R x ) [V ] i+ /,j Q y i,j+ / = Q y i,j+ / V i,j+ /R y Λ y R y ) [V ] i,j+ /

38 38. ENERGY PRESERVING AND ENERGY STABLE SCHEMES are consistent with Q x an Q y, respectively. Here, Q x an Q y are the energy fluxes.9) of the two-imensional EEC scheme..6.. Two-imensional ERoe. We perform a two-imensional reconstruction in the energy variables to obtain a two-imensional, secon-orer accurate version of the ERoe scheme. A stanar piecewise linear reconstruction gives reconstructe values V x, y) = V i,j + σ i,j x x i ) + γ i,j y y j ) for x, y) I i,j, with σ i,j an γ i,j etermine by a slope limiter, similar to the one-imensional ERoe scheme. The numerical fluxes of the two-imensional ERoe scheme are then Fi+ ERoe /,j = F U i,j, U i+,j ) Rx Λ x R x ) Vi+,j W Vi,j E ) an G ERoe i,j+ / = GU i,j, U i,j+ ) Ry Λ y R y ) V S i,j+ V N i,j ), an F an G are as in.8). Theorem.4. The two-imensional ERoe scheme is secon-orer accurate an is consistent with.)..6.. Numerical experiments. We repeat the cylinrical am break problem of Section.5.. As shown in Figure., the oscillations have been cleare out. The shock an rarefaction wave are slightly more iffuse out. The ERoe scheme resolves the expaning shock more accurately than the Roe an ERoe schemes. a) Roe b) ERoe c) ERoe Figure.. Approximate heights for the cylinrical am break problem at time t =. compute on a mesh with the Roe, ERoe an ERoe schemes.

39 .7. CONCLUSION 39 7 h = 9.5 u = v = h = u = v = FLOW Figure.. Setup of the physical am break. Numerical experiment: Physical am break. We test the two-imensional EEC, ERoe an ERoe schemes in a slightly more realistic problem that was first stuie in [] an was also consiere in [8, 38]. We compute in a basin of 4 4 meters with a am along the y-axis. At t =, a 8 meter long section of the am fails, an water flows through the breach. The initial ata is given by { if x > hx, y, ) = an u = v =. 9.5 if x < The setup is isplaye in Figure.. Along the bounary of the omain we impose Neumann bounary conitions, an along the am we set a reflective bounary. The acceleration ue to gravity is set to g = 9.8. We compute on a mesh of gri points up to t = 5, an show the solution compute by the EEC, ERoe an ERoe schemes in Figure.. The EEC scheme solves the resulting shock an rarefaction waves quite accurately, but with unphysical oscillations ue to a lack of energy issipation. The iffusion operator of the ERoe scheme corrects these, but at the cost of a less accurate resolution of the shock wave. The secon-orer accurate ERoe scheme gives a sharper resolution of the shock an the rarefaction waves..7. Conclusion We have esigne novel energy preserving an energy stable finite volume schemes for the shallow water system in both one an two space imensions. The energy preserving scheme is easy to implement, has a low computational cost an is secon orer accurate. The energy stable ERoe scheme clears out the oscillations of the EEC scheme an is more stable an accurate than comparable schemes, but is only first-orer accurate. A reconstruction in the energy variables gives a secon-orer accurate metho.

40 4. ENERGY PRESERVING AND ENERGY STABLE SCHEMES a) EEC b) ERoe c) ERoe Figure.. Height compute with the EEC, ERoe an ERoe schemes in the physical am break problem.

41 CHAPTER 3 Well-balance schemes 3.. Introuction Until now we have only consiere the case of a flat bottom topography. This is unrealistic in practice; when computing flows in rivers, near shores or across riges, bottom topography can have a large influence on the flow. The usual way of introucing a variable topography to the shallow water system.) is to a a source term to the equations, obtaining.). Figure 3. illustrates the oneimensional setup. Equation.) is a balance law [6] a conservation law with an aitional source term: 3.) U t + fu) x + gu) y = BU, x) for x = x, y) R. A crucial point of stuy for balance laws is that of steay states. A steay state is a solution of 3.) that is constant in time. Inserting the ansatz U t in 3.), we see that for such a state, the flux terms must exactly balance with the source term. Conversely, if these terms balance, then the solution is a steay state. In the case of the full two-imensional shallow water system with bottom topography, there is no simple criteria that ensures that a solution is a steay state. However, one very simple steay state is available, the so-calle lake at rest 3.) h + b constant, u = v =. This represents a solution that is completely at equilibrium. The lake at rest appears in several contexts. For instance, the worl s oceans are at large at rest, with relatively small perturbations on top of that. When simulating flows with a variable topography, a crucial question is whether the numerical metho will preserve steay states. It is well-known that most stanar schemes such as the Roe an Rusanov schemes o not preserve even the lake.8 u.6 z.4 h. b x Figure 3.. Height soli line) an bottom ashe line) in shallow water with a variable bottom topography. 4

42 4 3. WELL-BALANCED SCHEMES at rest. Given initial ata satisfying 3.), these schemes will prouce incorrect waves an oscillations that are of the orer of iscretization error. On the other han, if a scheme oes compute the lake at rest steay state correctly, then we may expect that slight perturbations of that state will also be correctly compute, an oes not contain unphysical oscillations. Thus, the preservation of the lake at rest is essential for long-time simulations of near-steay flows. The ecisive step in esigning finite volume schemes for the balance law.) is the iscretization of the source term. The general scheme is of the form 3.3) t U i,j = Fi+ /,j F i /,j) y Gi,j+ / G i,j / ) Bi,j, where B i,j is a iscrete version of the source term. In the presence of a steay state, the right-han sie shoul vanish, thus proucing a solution that is constant in time. Definition 3.. A scheme of the form 3.3) is well-balance if it preserves the lake at rest. In other wors, given initial ata 3.4) h i,j + b i,j constant, u i,j = v i,j = for all i, j, the solution compute by the scheme shoul satisfy t U i,j = for all i, j. Remark 3.. The lake at rest is far from being the only steay state of interest for.). For instance, in the one-imensional case, a general steay state is ientifie by hu constant, gh + b) u constant. We refer to [] for well-balance schemes for this steay state Energy preservation an stability. The theory of entropies for balance laws is similar to that of conservation laws. An entropy triple E, Q x, Q y ) for 3.) consists of a convex E : R n R R such that EU, x) t + Q x U, x) x + Q y U, x) y = for all x = x, y) R whenever U is a smooth solution of 3.). Similar to before, this entropy equality is replace by an inequality for weak solutions. The relevant entropy for.) is the total energy 3.5) EU, x) = hu + hv + gh + ghbx) ), which is the sum of potential an kinetic energy. E, Q x, Q y ) by letting We obtain an entropy triple Q x U, x) = hu 3 + huv ) + guh + ghubx) an Q y U, x) = hu v + hv 3) + gvh + ghvbx). Note that these reuce to the energy.6) of the shallow water system with flat topography whenever b. As before, we efine the entropy variables V U, x) = E U U, x), which for the energy 3.5) is u gh + bx)) +v 3.6) V = u, v compare to.7)).

43 3.. A WELL-BALANCED ENERGY PRESERVING SCHEME 43 We stuy finite volume schemes for.) that satisfy iscrete entropy equalities an -inequalities for the energy 3.5). As for the homogeneous equation.), such schemes cannot converge towars entropy violating solutions. What is more, energy preservation an -stability will imply stability for the approximations by bouning the total energy EUx, y, t)) xy uniformly. We will first consier the one-imensional version of.), 3.7) h t + hu) x = ) hu) t + gh + hu = ghb x. In Sections 3. through 3.4, we moify the energy preserving an energy stable schemes evelope in the previous chapter to obtain well-balance schemes. The schemes are generalize to two imensions in Section A well-balance energy preserving scheme The main challenge when esigning schemes for 3.7) lies in the iscretization of the source term. We will employ the iscretization 3.8) B i = g [ ] b h i+ b i b i+ / + h i b i i / we enote b i = bx i )). The EEC scheme with bottom topography is then 3.9) t U i = Fi+ / F i /) B i, where F i+ / is given by.3). We refer to this scheme as the EEC scheme in the remainer, as it reuces to the original EEC scheme in the case b i constant. Theorem 3.3. The EEC scheme has the following properties. i) It is consistent with 3.7) an is secon-orer accurate. ii) It is energy preserving, i.e. it satisfies the iscrete entropy equality t E + Hi+ / H i /) =, where H i+ / = Q b i u i+ + b i+ u i i+ / + gh i+ / an Q is as in.4). H is consistent with Q. iii) It is well-balance. Proof. Consistency in this case simply means that F U, U) = fu), which is satisfie as the EEC scheme without bottom topography is consistent. By a simple Taylor expansion, the iscretization 3.8) is of secon orer. For ii), we left-multiply 3.9) by Vi an get t EU i) = [ ghi u i / ] [ ]) gbi + Fi+/ u i i /) F irectly from the proof of Theorem.3) = Qi+ / Q i /) = Qi+ / Q i /) gb i x Vi B i [ ] gbi Fi+/ i /) F Vi B i hi+ /u i+ / h i /u i /)

44 44 3. WELL-BALANCED SCHEMES gu i h i+ / [b] + ) i+ / h i / [b] i / after some cancellations) = ) b i u i+ + b i+ u i Qi+/ + gh i+/ ) ) b i u i + b i u i Q i / + gh i / = Hi+ / H i /). Lastly, assume that the lake at rest conitions u i an h + b) i constant are satisfie. Then the first term on the right-han sie of 3.9) is g [ ] h i+ / h i / Using the secon ientity in.8), we get that t U i = g [ ] h i+ / [h] + h i+ / i / [h] i / g [ ] h i+ / [b] + h i+ / i / [b] i / = g [ ] h i+ / [h + b] + h i+ / i / [h + b] i / =, since [[h + b] i± / = Numerical experiments. We test the EEC scheme on a problem that has been feature in [4, 8, 5], among others. The bottom topography is given by { 4 x ) 3.) bx) = if x < else, an we impose the lake at rest initial conition h i + b i, u i. We compute on a mesh of gri points up to time t =, an we use g = 9.8 as the gravitational constant. The result is isplaye in Figure 3.. There is no visible x a) Water level h+b soli line) an bottom topography otte line) b) Relative change in energy over time Figure 3.. Lake at rest at t = using gri points

45 3.3. A WELL-BALANCED ENERGY STABLE SCHEME 45 change in the solution over time. Inee, the relative change in energy is of the orer of only, as seen in Figure 3.b). This tiny error is solely ue to floating point an temporal iscretization errors. Numerical experiment: Perturbe lake at rest. With the success of the EEC scheme in computing the lake at rest problem, we can expect that small perturbations on top of this steay state will also be compute correctly. We perturb the initial height by +. in the region x 6 < /4. The perturbation shoul break up into two smaller waves, one left- an one right-going. As seen in Figure 3.3, this is exactly what happens. However, in between the waves there are unphysical oscillations that are ue to a lack of numerical viscosity. 6 x a) Water level h+b soli line) an bottom topography otte line) b) Deviation from the steay state. Figure 3.3. Lake at rest with perturbation at t = A well-balance energy stable scheme Recall that the ERoe scheme has numerical flux 3.) F ERoe i+ / = F i+ / R Λ R [V ] i+ /, where R an Λ are evaluate at the average state h = h i+ /, u = u i+ /, an V is the vector of energy variables.). To generalize this scheme to the shallow water system with bottom topography, we replace the energy variables V with the energy variables for 3.7), [ ] gh + bx)) u 3.) V =. u We refer to the resulting scheme as the ERoe scheme, as it reuces to.) whenever b constant. We have the following result for ERoe. 3.3) Theorem 3.4. The ERoe scheme satisfies the following properties. i) It is consistent with 3.7) an is first-orer accurate. ii) It satisfies the iscrete energy issipation estimate t E i + Ĥi+ / Ĥi /) = 4 [V ] i+ / R i+ / Λ i+ / R i+ / [V ] i+ / 4 [V ] i / R i / Λ i / R i / [V ] i /,

46 46 3. WELL-BALANCED SCHEMES where the numerical energy flux Ĥ is 3.4) Ĥ i+ / = V i+ /F i+ / Ψ i+ / g 4 h i+ / [u] i+ / [b] i+ / + V i+ /R i+ / Λ i+ / R i+ / [V i+ / ]. iii) It is well-balance. Proof. The proof of i) is straightforwar. The proof of 3.3) follows the proof of the corresponing result in Theorem.9 an we omit the etails here. To prove iii), we assume that the ata satisfies u i an h i + b i constant. Then we have [u] an [h + b] i+ / i+ for all i. / Consequently, by the efinition of the energy variables, [V ] i+ for all i. / Hence the iffusion operator in 3.) rops out, an the scheme reuces to the EEC scheme. Thus, by Theorem 3.3 iii), we have t h i an t h iu i ) for all i A well-balance, secon-orer accurate scheme We continue with the secon-orer accurate ERoe scheme. Recall from Chapter that this scheme uses a linear reconstruction of the energy variables to obtain left an right states Vi W an Vi E in each gri cell. The resulting flux is then 3.5) Fi+ ERoe = F U / i, U i+ ) R Λ R Vi+ W Vi E ). The motivation for performing reconstruction in the energy variables rather than the conserve variables has been postpone until now: By reconstructing in the energy variables 3.), the ERoe scheme is well-balance. Theorem 3.5. The ERoe scheme 3.5) is consistent with 3.7), secon-orer accurate an well-balance. Proof. Consistency is trivial. Using a Taylor expansion proves the seconorer accurate local truncation error. To prove that the ERoe scheme preserves the lake at rest, we observe that when the ata satisfies u i an h i + b i constant, we have u i an [h + b] i+ /, whence [V ] i+. Therefore, by the non-oscillatory property of the slope limiter, / we obtain σ i, so Vi E = Vi W = V i = constant for all i. As a consequence, the term Vi+ W V ) i E in the iffusion operator in 3.5) vanishes, an we follow the argument in the proof of iii) in Theorem 3.3 to conclue that t h i an Hence, the iscrete lake at rest is preserve. t h iu i ) for all i.

47 3.5. GENERALIZATION TO TWO SPATIAL DIMENSIONS 47 N Roe EEC ERoe ERoe 5.76e- 6.7e-4.9e-8 3.7e-6 7.6e-3.6e-3.4e e-7.e e e-8.34e e-4.76e-.e-7.4e-5 Table 3.. The L error in height for the lake at rest with ifferent schemes on a sequence of meshes at time t = Numerical experiments. We repeat the lake at rest problem of Section 3.. an compare the Roe, EEC, ERoe an ERoe schemes on a sequence of meshes. As we have prove that the latter three schemes are well-balance, they shoul preserve the steay state exactly. We isplay the ifference at t = from the initial ata in the L norm in Table 3.. The error in the well-balance schemes are of the orer of machine precision, as expecte. The Roe scheme has errors of the orer of truncation error, ue to its lack of well-balancing. Numerical experiment: Perturbe lake at rest. In the perturbe lake at rest problem, the EEC scheme prouce an unacceptable amount of oscillations. These were cause by a lack of numerical iffusion. We repeat the experiment with the ERoe an ERoe schemes to see the effect that the well-balance iffusion operators have on the EEC scheme. The result is isplaye in Figure 3.4a). Comparing ERoe ERoe a) ERoe an ERoe b) Roe Figure 3.4. Deviation from the lake at rest steay state at t =.5 on a mesh of mesh points. to Figure 3.3a), we see that all oscillations are gone. The secon-orer accurate ERoe scheme resolves the two waves much more sharply than the ERoe scheme. Due to the well-balancing of both schemes, the small-scale perturbation is resolve correctly without any isturbance from spurious waves. Contrast this with the unbalance Roe scheme Figure 3.4b)), for which unphysical waves ruin the solution Generalization to two spatial imensions Next we generalize the ieas of the previous sections to obtain well-balance, energy preserving an energy stable schemes for the full two-imensional shallow water equations.). The two-imensional counterparts of the EEC, ERoe an ERoe schemes are very similar to the one-imensional schemes, an so the iscussion will be brief.

48 48 3. WELL-BALANCED SCHEMES The natural generalization of the source term iscretization 3.8) is 3.6) B i,j = g h i+ /,j bi+,j bi,j + h i /,j bi,j bi,j. b h i,j+ b i,j b i,j+ / y + h i,j b i,j i,j / y Nothing is altere in the flux of the two-imensional EEC scheme. In the ERoe an ERoe schemes we replace the vector of energy variables V by 3.6). The schemes are then of the form 3.3), with either the EEC, ERoe or ERoe numerical fluxes in place of F an G. Two-imensional counterparts of theorems 3.3, 3.4 an 3.5 are easily obtaine. Theorem 3.6. The two-imensional EEC scheme has the following properties. i) It is consistent with.) an is secon-orer accurate. ii) It is energy preserving, i.e. it satisfies the iscrete entropy equality t E + Hx i+ /,j H i /,j) x + Hy y i,j+ H ) y / i,j =, / where an H x i+ /,j = Q x i+ /,j + gh i+ /,j b i,j u i+,j + b i+,j u i,j H y i,j+ = Q y / i,j+ + gh b i,j u i,j+ + b i,j+ u i,j / i,j+ /, an Q x an Q y are as in.9). Hx an H y are consistent with Q x an Q y, respectively. iii) It is well-balance. Theorem 3.7. The two-imensional ERoe scheme satisfies the following properties. i) It is consistent with.) an is first-orer accurate. ii) It satisfies the iscrete energy issipation estimate t E + Ĥx i+ /,j Ĥx i /,j) + ) Ĥy y i,j+ Ĥy / i,j / where Ĥx an Ĥy are consistent with Q x an Q y, respectively iii) It is well-balance., Theorem 3.8. The two-imensional ERoe scheme is consistent with.), secon-orer accurate an well-balance Numerical experiments. We test the well-balancing of the three schemes in a two-imensional lake at rest problem that has been feature in [5, 3]. The bottom topography is given by bx, y) =.8 exp 5x.9) 5y.5) ) an we use the lake at rest initial ata h i,j + b i,j, u i,j v i,j for all i, j. This setup is illustrate in Figure 3.5. We set g = 9.8 an compute in the omain x, y) [, ] [, ], using stanar Neumann bounary conitions. We compute with the Roe, EEC, ERoe an ERoe schemes on a sequence of meshes up to time t =. The eviation from the steay state is isplaye in Table 3.. Clearly the three well-balance schemes preserve the initial ata to machine precision, whereas the Roe scheme prouces spurious waves in the solution.

49 3.6. CONCLUSION 49 Figure 3.5. Water level an bottom topography for the twoimensional lake at rest. N Roe EEC ERoe ERoe Table 3.. The L error in height for the two-imensional lake at rest with ifferent schemes on a sequence of N N meshes at time t =. Numerical experiment: Perturbe lake at rest. Lastly, we compute a perturbe version of the lake at rest. We a +. to the initial height in the region x [.,.]. This perturbation shoul break up into two smaller waves moving in either irection. The left-going wave hits the bounary at approximately t =.3; this will test how well the schemes hanle the bounary conition. The right-going wave will go over the bump in the bottom topography, creating a complex wave pattern. Figure 3.6 shows the compute height with the ERoe an ERoe schemes on a mesh of 6 3 mesh points. At t =., the left-going wave has cleanly left the omain without any trace of bounce-back waves. In the remaining snapshots, the ERoe scheme clearly gives the sharpest resolution of the flow. The results coincie well with the results of [5] an [3] Conclusion We have esigne novel well-balance, energy preserving an energy stable finite volume schemes for the shallow water equations with bottom topography in one an two space imensions. The only ifference from their homogeneous counterparts is a simple iscretization of the source term, an so they inherit many of the properties of the schemes of Chapter, incluing simplicity an accuracy.

50 5 3. WELL-BALANCED SCHEMES a) t =. b) t =.4 c) t =.36 ) t =.48 e) t =.6 Figure 3.6. A simulation of the two-imensional lake at rest with perturbation using the ERoe an ERoe scheme with 6 3 mesh points. Left column: ERoe, right column: ERoe. Note that the scales in the figures at each point in time are ifferent.)

51 CHAPTER 4 Vorticity preservation 4.. Introuction When numerically approximating physical moels, orer of convergence is important. However, just as important is the path of convergence. Approximate solutions may lie near the correct solution in some L p norm, but still may lie in an entirely physically incorrect part of the phase space. Thus, it is esirable that the scheme converges along a path over which the properties of the approximations resemble the true solution. This is especially important in simulations of noneterministic systems such as weather systems. In short-range simulations, gri size an orer of convergence are the key factors in the accurate resolution of the flow fiel. However, in simulations over longer perios of time it is the overall statistical properties of the flow that are important []. We have alreay consiere consistency of numerical methos with respect to the correct preservation/issipation of energy. In this chapter we consier the correct resolution of vorticity in the flow. Recall that the essence of the finite volume metho is a partition of the computational omain into polygonal control volumes. By approximating the solution of the ifferential equation by an average over each control volume, the focus is shifte towars computing the correct flux through the eges of the control volume. This ecoupling into normal irections may lea to a poor resolution of tangential flows or forces. As a result, or perhaps as a cause, the vorticity, which satisfies its own evolution equation, is not resolve correctly. Of the more recent approaches to resolving vortical flows are the projection methos []. These were originally evelope by Chorin [9] an Bell, Colella an Glaz [6] to impose the ivergence constraint in the incompressible Navier-Stokes equations, but have also been stuie by Brackbill an Barnes [7] an Toth [39] in the context of the MHD equations. The projection metho introuces a correction term to a finite volume scheme to control the unphysical generation of vorticity. At each time step we solve simultaneously for the flow fiel an the vorticity. By applying a projection of the flow fiel in the form of an elliptic operator, we obtain to within a iscretization error a solution with a correct vorticity fiel. Denoting the vorticity of.) by ω = v x u y, one may reaily calculate that smooth solutions of the shallow water equations satisfy 4.) ω t + uω) x + vω) y =. Stanar finite volume schemes might not respect this relation, which might lea to an accumulation of errors over time []. However, the relation 4.) is har to impose numerically, so instea we aim to control the pseuo-vorticity of the shallow water equations. The pseuo-vorticity of a flow fiel is efine as the curl of the momentum, in contrast to vorticity, which is the curl of velocity. As momentum, an not velocity, is a conserve variable, it will be easier to control errors in pseuo-vorticity, rather than vorticity. For the rest of this chapter we refer to pseuo-vorticity simply as vorticity. We enote vorticity by Ω = m ) x m ) y, where m = hu an m = hv are the momentum in the x an y irections, 5

52 5 4. VORTICITY PRESERVATION respectively. Then, the change of vorticity over time is Ω t = m ) xt m ) yt = m ) t ) x m ) t ) y, an inserting m ) t an m ) t from.) an rearranging, we fin that the vorticity of the shallow water equations satisfies the evolution equation 4.) Ω t + uω) x + vω) y + m u x + v y )) x m u x + v y )) y u + v ) u + v ) + h y h x =. x Before presenting the projection metho for the shallow water equations, we will consier a simpler, linearize version of the equations, the system wave equation. This system exhibits a particularly simple evolution equation for vorticity. Thus, the exposition will be as transparent as possible. If the solution of a conservation law or any other ifferential equation) consists of relatively small perturbations on top of a steay state, then the flow is mainly governe by linear effects; see LeVeque [6]. By linearizing the equations aroun the unerlying steay state, we obtain a much simpler, linear conservation law that is still a goo approximation of the original problem. Thus, assume that the solution of.9) is of the form U = U + Ũ, where U is a constant, steay state an Ũx, t) = [ h, m, m ] is small. Inserting U into.9) an removing terms of orer Ũ gives the conservation law Ũ t + f U )Ũx + g U )Ũy =. Assuming that the velocity of the backgroun flow is zero, we have f U ) = gh, g U ) =. gh Denote the wave spee in the flui by c = gh, an let ζ = c h, m = m an m = m. Then [ζ, m, m ] solves the equivalent) conservation law 4.3) ζ m + c c ζ m + c ζ m =. m m c m This is the system wave equation. equation t x y The connection between this an the wave u tt = c u xx + u yy ) is seen by letting ζ = u t, m = cu x an m = cu y ; this gives precisely 4.3). The Jacobians of the flux functions of 4.3) both have eigenvalues λ = c, λ =, λ 3 = c, so the equation is hyperbolic. In the remainer, we rop mention of the backgroun flow U an consier 4.3) as a conservation law in itself with solution U = [ζ, m, m ]. The system wave equation has a very simple expression for vorticity. Writing Ω = m ) x m ) y as before, we have an so Ω is constant in time. Ω t = cζ y ) x cζ x ) y ) =, y

53 4.. INTRODUCTION Vorticity of the Rusanov scheme. As the wave equation is hyperbolic, all the theory of finite volume schemes from Section.3 apply to this system. We will use the Roe an the Rusanov schemes. Recall that the Rusanov scheme uses a rather coarse) approximation of the local wave spee, the maximum of the neighboring eigenvalues. For the wave equation the maximum eigenvalue is always c, an so the Rusanov flux in each irection is an F Rus i+ /,j = fui,j ) + fu i+,j ) ) c U i+,j U i,j ) G Rus i,j+ / = gui,j ) + gu i,j+ ) ) c U i,j+ U i,j ). We investigate the vorticity preserving properties of the Rusanov scheme. For iscretizing vorticity we use the secon-orer approximation 4.4) Ω i,j = D x m ) i,j D y m ) i,j, where D x an D y are the central ifferences 4.5a) D x U i,j = µ xδ x U i,j = U i+,j U i,j an 4.5b) D y U i,j = y µ yδ y U i,j = U i,j+ U i,j, y an δ x, δ y, µ x an µ y are the ifference an average operators δ x u i+ /,j = u i+,j u i,j, δ y u i,j+ / = u i,j+ u i,j, µ x u i+ /,j = u i,j + u i+,j, µ y u i,j+ / = u i,j + u i,j+. Central ifferences are use to avoi the nee to stagger variables. With this iscretization of vorticity, we have the following issipation estimate for the Rusanov scheme. Lemma 4.. Solutions of 4.3) compute by the Rusanov scheme satisfy 4.6) t Ω i,j = c δ xω i,j + ) y δ yω i,j in the interior of the omain. In particular, if the initial vorticity is zero, then it stays zero at all times. Moreover, if Ω i,j = at the bounary, then the energy of Ω is non-increasing, y Ω i,j = c y Ω i+,j Ω i,j ) + Ω i,j+ Ω i,j ) ) t i,j i,j, with equality only when Ω. Proof. Inserting the efinition of F Rus an G Rus into the finite volume scheme.) an rearranging, we fin that t m ) i,j = c ) δ x δ x m ) i,j µ x ζ i,j + ) y δ ym ) i,j, t m ) i,j = c ) y δ y δ y m ) i,j µ y ζ i,j + ) δ xm ) i,j. Inserting this into t Ω i,j = t D xm ) i,j D y m ) i,j ) an using the fact that δ an µ commute, we get 4.6). If Ω i,j ), then 4.6) reas tω =, so we get Ω.

54 54 4. VORTICITY PRESERVATION For the energy issipation estimate, we remark that for a one-imensional gri function u i we have N N 4.7) u i δ u i = u i+ u i ) + u N u N+ u N ) u u u ). i= This implies that y t i,j Ω i,j i= = y i,j = c y i,j Ω i,j t Ω i,j Ω i,j δ xω i,j + ) y δ yω i,j by 4.7); the bounary terms rop out) = c y Ω i+,j Ω i,j ) + Ω i,j+ Ω i,j ) ). i,j The vorticity of the wave equation shoul be constant in time, an by the above lemma, the Rusanov scheme respects this whenever the initial vorticity is zero. However, if it is nonzero at the initial time, then the scheme will incorrectly issipate vorticity. What is more, the energy issipation estimate relies on having zero vorticity at the bounary. If this is not the case, then in many cases there will be an excessive amount of vorticity prouction at the borer. A prime example is when using Neumann bounary conitions, simply copying the conserve variables onto the bounary. As we will see later, the Rusanov scheme will prouce a large amount of vorticity along the bounary. In the case of a perioic bounary, the bounary terms in 4.7) vanish an vorticity will be issipate. In either case, vorticity is not kept constant, as it shoul be. The projection metho, which we escribe next, will correct this. 4.. Vorticity projection for the wave equation We motivate the iscrete projection metho by first consiering the continuous problem. Let Ω = Ωx, t) be the vorticity at the initial time step. Given a solution U n = [h n, m n, m n ] at time t n, we compute a caniate solution Ũ n+ = [ h, m, m ] at the next time step t n+ with any metho available not necessarily numerical. We project the momentum fiel m = m, m ) onto the space of functions with curl equal to Ω as follows. Let ψ be the solution of the Poisson equation ψ = Ω Ω, where is the Laplace operator an Ω = m ) x m ) y is the vorticity of Ũ. Define U n+ = [ζ, m, m ] as ζ = ζ, m = m ψ y, m = m + ψ x. Observe that this makes ψ the stream function of the fiel m m. In other wors, streamlines of m m coincie with level curves of ψ. The vorticity of U n+ is then m ) x m ) y = Ω + ψ = Ω, an so U n+ is an approximate solution at the next time step with correct vorticity. This process is repeate at each time step. At the bounary of the computational omain we set U n+ = Ũ. As ψ is the stream function of m m, this makes the bounary of the omain a level curve of

55 4.. VORTICITY PROJECTION FOR THE WAVE EQUATION 55 ψ. Hence, by aing a constant if necessary, we may assume that ψ vanishes at the bounary Discrete projection metho. We iscretize the projection metho so that the iscretize vorticity 4.4) will be preserve exactly. As Ω i,j shoul be constant in time, we nee only make sure that the vorticity is equal to Ω i,j = Ω i,j ) at all times. Given a solution at time t n, the solution at time t n+ is compute as follows. 4.8) Compute a caniate solution Ũ n+ i,j volume scheme. Let Ω n+ i,j at time t n+ with any consistent finite be its vorticity, compute with the formula 4.4). Fin the solution ψ = ψ i,j ) of the iscrete Poisson problem { ) Dx + Dy n+ ψi,j = Ω i,j Ω i,j in the interior, ψ i,j = at the bounary, where D x an D y are the iscrete erivatives 4.5). Define the solution at the next time step U n+ i,j = [ζ i,j, m ) i,j, m ) i,j ] as ζ i,j = ζ i,j, m ) i,j = m ) i,j D y ψ i,j, m ) i,j = m ) i,j + D x ψ i,j. Proposition 4.. The solution U = ζ, m, m ) compute by the projection metho satisfies D x m ) i,j D y m ) i,j = Ω i,j. Hence, the solution has the correct amount of vorticity. Proof. Simply inserting the efinition of m an m gives D x m ) i,j D y m ) i,j = D x m ) i,j D y m ) i,j + D x + D y) ψi,j = Ω i,j. We enote the projection metho using the Rusanov scheme in the preiction step as the VPRus scheme, an similarly for VPRoe. Recall from Section. that the energy E = U U is an entropy for conservation laws with linear an symmetric fluxes. As the wave equation 4.3) is precisely this, we have an energy for the wave equation. Next, we show the stability result that in the case of zero initial vorticity, the projection step of the VP methos oes not introuce aitional energy to the solution. ) Proposition 4.3. Let E = ζ + m + m be the energy of a solution compute by the projection metho, an let Ẽ = ζ + m + m ) be the energy of the solution obtaine in the preiction step. Then E i,j Ẽ i,j Ω i,j ψ i,j. i,j i,j i,j In particular, if the initial vorticity is zero, then E i,j Ẽ i,j. i,j i,j Proof. We use summation by parts extensively. As ψ = at the bounary, all bounary terms will rop out. For notational convenience, we leave out variable inexes. Then ζ + m D y ψ) + m + D x ψ) ) i,j E = i,j

56 56 4. VORTICITY PRESERVATION = i,j Ẽ i,j m D y ψ m D x ψ) + i,j Dx ψ) + D y ψ) ). The secon term is i,j m D y ψ m D x ψ) = i,j ψd y m D x m ) = i,j ψ Ω D x + D y)ψ ) = i,j Ωψ i,j Dx ψ) + D y ψ) ). Hence, Dx ψ) + D y ψ) ) i,j E = Ẽ i,j i,j Ẽ i,j Ωψ i,j Ωψ. If Ω i,j, then the last term rops out Solving for the stream function. The iscrete Poisson equation 4.8) must be solve at every time step, so a fast solver is essential to the computational efficiency of the scheme. Assume we are computing on an N M uniform Cartesian gri. If we write C i,j = as 4.9) Ω n+ i,j Ω i,j, then the problem can be rewritten 4 T NΨ + 4 y ΨT M = C, where Ψ = ψ i,j ) i,j an T N is the N N symmetric, positive efinite matrix 4.) T N = We solve this equation with a metho from [8] that requires ON 3 ) floating point operations. Let R N be the matrix of eigenvectors an D N = iagλ N,..., λn N ) the matrix of eigenvalues of T N. R N may be chosen such that RN = I N, the ientity matrix in R N N. Multiplying 4.9) by R N on the left an R M on the right, we fin that 4 D NX + 4 y XD M = R N CR M, where X = R N ΨR M. The i, j) entry of the left-han sie of this equation is Hence, X = R N CR M )./S, where λ i N 4 X i,j + λj M 4 y X i,j. S i,j = λi N 4 + λj M 4 y an./ enotes component-wise ivision. The solution of 4.9) is therefore Ψ = R N XR M.

57 4.. VORTICITY PROJECTION FOR THE WAVE EQUATION Perioic bounary conitions. The above iscussion i not rely on the specifics of the bounary conition on U. In the special case of a perioic bounary, U,j = U N,j, U,j = U N,j etc., we can improve the metho by applying the same conition on ψ. This will result in the matrix equation 4.) 4 P NΨ + 4 y ΨP M = C, with ) P N = We were unable to fin the general expression of the eigenvectors an eigenvalues of P N, so instea of using the metho escribe in the previous section, we will employ the conjugate graient metho [8]. The conjugate graient metho searches for the solution of a matrix equation Az = b along orthogonal search paths s k R NM, an it fins the exact solution after at most NM iterations. As this number grows large quaratically, we will use the metho as an iterative metho, halting the process when s k l is less than some ε >. Given a right-han sie b, we set ε = α b l for an α >, so that the allowe error in the solution is proportional to b. We chose α = 5 in this thesis. The matrix-matrix equation 4.) can be rewritten as a matrix-vector equation Az = b, with z = vecψ), b = vecc) an A = 4 P N I M + 4 y I N P M, enoting the Kroenecker prouct an vecψ) the column-first vectorize version of Ψ [8]. Contrary to T N, the matrix P N is only positive semiefinite; its kernel is spanne by the vectors rn =, rn = R N.. if N is even, an by rn = RN. if N is o. Hence, the matrix A will be positive semiefinite with a kernel spanne by the vectors) rn k rm l. As a consequence, the equation Az = b oes not have a solution whenever b has a nonzero component in ker A. However, the equation Az = b, where b = projker A) b) = proj ImA b)),

58 58 4. VORTICITY PRESERVATION oes have a unique solution in ker A). What is more, the conjugate graient metho is well-efine an converges for this moifie equation. Therefore, we propose using the solution Ψ of this moifie equation. Note that this is a sort of preconitioning of the problem: We have moifie the ill-conitione problem Az = b by multiplying on both sies by the projection matrix of ker A), thus obtaining Az = b Numerical experiments. In this section we test the VP schemes for the wave equation 4.3) in two numerical experiments. The main emphasis will be on how well the schemes preserve vorticity. Also consiere is the runtime of the VP schemes, compare to the runtime of the unerlying preictor scheme. Furthermore, we compare the accuracy in the conserve variables ζ, m, m ). In both experiments we use c = an a CFL number of.45. Numerical experiment: Perioic waves. The first experiment features a perioic bounary an a nonzero initial vorticity. The perioic bounary conition will ri us of any unwante prouction of vorticity along the bounary. The initial conitions are given by ζ, It is reaily checke that m = m = cosπx + y)) cosπx y)). ζx, y, t) = sinπx + y)) sin πt), m x, y, t) = m x, y, t) = cosπx + y)) cos πt) cosπx y)) is the solution of this initial value problem. The corresponing vorticity is Ωx, y, t) = m ) x m ) y = π sinπx y)). As expecte, the expression for Ω is constant in time. The initial ata is plotte in Figure 4.a). We compute for x, y) [, ] [, ] up to time t =. The solution at the final time step compute with the Rusanov, VPRus, Roe an VPRoe schemes are plotte in Figure 4., along with the exact solution. While ζ is left untouche by the VP schemes, the vorticity fiel is resolve much more sharply than in the preictor schemes. This is verifie in Table 4. an 4., where we show relative errors Ω Ωexact L Ωexact L for Ω an ζ on a sequence of meshes. The VP schemes preserve vorticity up to machine precision, while the Rusanov an Roe schemes have errors in vorticity of the orer of iscretization error. The L issipation estimate of the Rusanov scheme is verifie in Figure 4.. The VPRus an VPRoe, not shown) scheme preserves vorticity exactly, while the Roe an Rusanov issipate it excessively. Momentum is shown in the thir column of Figure 4.. Clearly, the projection methos have a positive effect on the accuracy, preventing too much iffusion in m. Inee, as shown in Table 4.3, the error in momentum is about 4 per cent lower than in the preiction solvers. The conjugate graient metho use in the elliptic solver converges rapily, with only 5 to iterations neee to get below the error threshol. Thus, the overhea is low, an the VP schemes takes only about.5 times more time to run than the preictor schemes. Numerical experiment: Expaning wave. This experiment features a smooth solution with an open Neumann) bounary conition The initial ata is given by 4.3) ζ = c exp 5x + y ) ), m = m =. As the initial vorticity is zero, it shoul stay zero at all later times. The exact solution is plotte in Figure 4.3.

59 4.. VORTICITY PROJECTION FOR THE WAVE EQUATION 59 L norm of Ω over time 8 6 VPRus Rusanov Roe Figure 4.. L norm of vorticity over time on a mesh of 6 6 gri points. Rusanov VPRus Roe VPRoe Table 4.. Relative error in Ω. Rusanov VPRus Roe VPRoe Table 4.. Relative error in ζ. Rusanov VPRus Roe VPRoe Table 4.3. Relative error in m, the momentum. We solve for x, y) [, ] [, ] up to time t =. The spatial omain was iscretize with N = M = 5,, 5 an gri points in each irection. Vorticity at the final time step is shown in Figure 4.4. The Rusanov scheme preserves the initial vorticity exactly in the mile of the omain, but there is a large amount of vorticity prouction along the borer. This vorticity propagates into the omain at a spee of one gri cell per time step. As the ratio t is kept constant, this spee is invariant with respect to gri size. This is clear in Table 4.4, where we show vorticity errors for the four schemes. The error for the Rusanov scheme is about 3, irrespective of gri size. The VPRus scheme clears out these errors, an only noise of the orer of machine precision is left note the scaling of the figures).

60 6 4. VORTICITY PRESERVATION The vorticity of the Roe scheme is shown in Figure 4.4c). The figure shows that the initial constant vorticity is not at all preserve. Again, the projection metho clears out these errors completely. The projection metho gives no gain in accuracy for the conserve variables in this experiment. As the runtime of the metho lies between an 4 times that of the preictor solver, there is little use in vorticity projection when the main interest is in an accurate solution of conserve variables. However, the success of the metho in clearing out vorticity errors gives a motivation for applying it to the shallow water equations, where nonlinearity creates a close interconnection between flow in the x an y irections. Rusanov VPRus Roe VPRoe Table 4.4. Ω L in the expaning wave problem Vorticity projection for the shallow water system Next we exten the projection metho to the shallow water system.). Instea of having constant vorticity, solutions of this system satisfy the more complex evolution equation 4.). This complicates the metho somewhat. Given a solution U n at time t n, we compute a caniate solution Ũ n+ at time t n+ with any consistent finite volume scheme. Let Ω n an Ω n+ be their respective vorticity. Now, solve the equation 4.) with Ω n as initial ata by using any stanar finite volume scheme. This gives the correct vorticity Ω n+ at time t n+. Let ψ be the solution of ψ = Ω n+ Ω n+. Then, letting h = h, m = m ψ y, m = m + ψ x, we get a solution U n+ = [h, m, m ] that satisfies the correct equation for vorticity. To obtain Ω n+, we employ the Nessyahu-Tamor NT) scheme [, 9]. This is a secon-orer version of the Lax-Frierichs scheme that solves for Ω n+ i+ /,j+, / an approximation of the staggere cell average Ωx, y, t) xy for J i,j = [x i, x i+ ) [y j, y j+ ). y J i,j Write the vorticity flux as fω, U) = uω+m +sh y an gω, U) = vω m sh x, where = u x + v y an s = u +v compare with 4.)). The expression for Ω n+ i+ /,j+ is / Ω n+ i+ /,j+ = Ω n x, y) xy / y J i,j tn+ fω, U) x + gω, U) y xyt y t n J i,j Ω n 4 i,j + Ω n i+,j + Ω n i,j+ + Ω n i+,j+) t ) ) f Ω n+ /, U n+ / + g Ω n+ /, U n+ / xy, y J i,j x y

61 4.3. VORTICITY PROJECTION FOR THE SHALLOW WATER SYSTEM 6 where we have use the quarature rule for the time integral. Ω n+ / an U n+ / are approximations of Ω an U at time t n+ /. We select U n+ / = U n + Ũ n+ ) an Ω n+ / = Ω n t f x n + gy n ), where fx n an gy n are the graients of the flux at time t n. To go from staggere values Ω n+ i+ /,j+ / piecewise linear reconstruction of Ω n+ i+ /,j+ / to mesh values Ωn+ i,j, we apply a Ω n+ x, y) = Ω n+ i+ /,j+ / + σ ix x i+ /) + γ j y y j+ /) for x, y) J i,j an average over I i,j to get Ω n+ i,j = y I i,j Ω n+ x, y) xy. All erivatives use in this metho are obtaine using the maxmo slope limiter Ω i,j = mm Ω i,j Ω i,j, Ω ) i+,j Ω i,j Ω i+,j Ω i,j,, with mm as in.5). This slope limiter gives a better resolution of iscontinuities than the minmo limiter.4); see [9] Numerical experiment: Vorticity avection. We test the projection metho on a problem where the exact solution is known. It is reaily checke that where hx, y, t) = c 4c g ef ux, y, t) = M cosα) + c y y Mt sinα))e f vx, y, t) = M sinα) c x x Mt cosα))e f f = fx, y, t) = c x x Mt cosα)) + y y Mt sinα)) ), gives a smooth solution U = [h, hu, hv] of.) for any choice of constants M, c, c, α, x an y. The solution consists of a vortex traveling at a constant velocity M in a irection specifie by the angle α. We let M = /, g =, c, c ) =.4,.) an x, y ) =, ). To test the schemes ability of resolving flows that are not aligne with the computational gri, we let α = π 6. We compute for x, y) [ 5, 5] [ 5, 5] up to time t =. The exact solution is shown in Figure 4.5. The compute solutions are plotte in Figure 4.7. The Rusanov scheme issipates the solution by a large amount, an the vortex is barely visible in the plot. The VPRus scheme, on the other han, preserves both the magnitue an the symmetry of the vorticity well, an the solution resembles the exact solution closely. The Roe scheme solves the vortex avection problem poorly; the solution looks malforme an unsymmetric. The projection metho corrects this to a goo extent. The VP schemes take about twice as long to run as the preiction solver on the same gri, an as such, a comparison of error versus gri size between the schemes woul be unfair. Instea, we compute error versus runtime over a sequence of meshes, from 5 5 to 5 5 gri points. These are plotte in Figure 4.6. Clearly, the VPRus scheme gives the best error per runtime ratio of the schemes consiere. The VPRoe scheme also performs well, but it seems like the nonsymmetry of the Roe scheme pollutes the solution of the height variable an makes the solution look unsymmetric. As a final remark, we note that using secon-orer versions of the Rusanov an Roe schemes gives similar results: Vorticity projection greatly enhances the accuracy of both schemes.

62 6 4. VORTICITY PRESERVATION 4.4. Conclusion The metho of vorticity projection was esigne to reuce or eliminate errors in variables other than the conserve variables. We have emonstrate that with an efficient implementation of the metho, the resolution of the conserve variables may actually be better, at the same computational cost, than when using more traitional schemes. At the same time, a significant improvement in the resolution of vorticity is achieve. The efficiency an accuracy of the metho for solving the equation for vorticity is vital to the projection metho. In the case of the wave equation, this was an easy task since vorticity is constant in time. For the shallow water equations, we restricte ourselves to a secon-orer scheme for vorticity. Using other schemes than the NT scheme may give a more computationally efficient scheme.

63 4.4. CONCLUSION 63 a) Initial ata b) Exact solution c) Rusanov ) VPRus e) Roe f) VPRoe Figure 4.. Solutions at t = compute by the four schemes on a mesh of 6 6 gri points.

64 64 4. VORTICITY PRESERVATION a) t = b) t = Figure 4.3. Exact solution of the expaning wave problem at the initial an final time steps. a) Rusanov b) VPRus c) Roe ) VPRoe Figure 4.4. Vorticity at t = for the four schemes, compute on a mesh of 5 5 gri points. Note the scaling of each figure.

65 4.4. CONCLUSION 65 a) t = b) t = Figure 4.5. Exact solution of the vorticity avection problem at t = an t =. Height Momentum Error 4 Error Rusanov Roe VPRoe VPRus Runtime Vorticity 3 Rusanov Roe VPRoe VPRus Runtime.. Error.5.8 Rusanov Roe VPRoe VPRus Runtime Figure 4.6. Runtime x-axis) versus relative L errors in height, momentum an vorticity y-axis).

66 66 4. VORTICITY PRESERVATION a) Rusanov b) VPRus c) Roe ) VPRoe Figure 4.7. Height an vorticity at t = compute by the four schemes on a mesh of gri points.