Potential based, constraint preserving, genuinely multi-dimensional schemes for systems of conservation laws

Transcription

1 Potential base, constraint preserving, genuinely multi-imensional schemes for systems of conservation laws Sihartha Mishra an Eitan Tamor Abstract. We survey the new framework evelope in [33, 34, 35], for esigning genuinely multi-imensional (GMD) finite volume schemes for systems of conservation laws in two space imensions. This approach is base on reformulating ege centere numerical fluxes in terms of vertex centere potentials. Any consistent numerical flux can be use in efining the potentials. Suitable choices of the numerical potentials yiel finite-volume schemes which preserve iscrete form of constraints such as vorticity an ivergence. The schemes are very simple to coe, flexible an have low computational costs. Numerical examples for the Euler equations of gas ynamics an the ieal MHD equations are presente to illustrate the computational efficiency of the schemes. 1. Introuction Many interesting phenomena in physics, engineering an biology are moele by hyperbolic systems of conservation laws. In two space imensions, these equations take the form (1.1) U t + f(u) x + g(u) y = 0, (x, y, t) R R R +, where U is the vector of unknowns an f, g are the flux vectors in the x- an y- irections respectively. A frequently cite example of the system (1.1) are the Euler equations of gas ynamics, (1.a) ρ t + (ρu 1 ) x + (ρu ) y = 0, (ρu 1 ) t + (ρu 1 + p) x + (ρu 1 u ) y = 0, (ρu ) t + (ρu 1 u ) x + (ρu + p) y = 0, E t + ((E + p)u 1 ) x + ((E + p)u ) y = 0, 1991 Mathematics Subject Classification. 65M06,35L65. Key wors an phrases. multiimensional evolution equations, nonlinear conservation laws, constraint transport, central ifference schemes, potential-base fluxes. Acknowlegment. The work on this paper began when S.M. visite the Center of Scientific Computation an Mathematical Moeling (CSCAMM) an he thanks CSCAMM an all its members for the excellent hospitality an facilities. E. T. Research was supporte in part by NSF grant an ONR grant N He thanks the Centre for Avance Stuy at the Norwegian Acaemy of Science an Letters, for hosting him as part of its international research program on Nonlinear PDEs uring the acaemic year

2 SIDDHARTHA MISHRA AND EITAN TADMOR with ρ being the ensity of the gas, u 1, u are the velocity components in the x- an y-irection respectively an p an E are the pressure an the energy. The equations are augmente by an ieal gas equation of state, which is expresse in terms of the gas constant γ: (1.b) E = p γ (ρu 1 + ρu ). Other interesting examples for hyperbolic systems are the shallow water equations of oceanography an the equations of non-linear elasticity. It is well known that solutions of (1.1) (even in one space imension) evelop iscontinuities in the form of shock waves, even for smooth initial ata. Hence, the solutions sought for (1.1) are efine in a weak sense. Weak solutions are not necessarily unique an (1.1) has to be supplemente by aitional amissibility criteria, the so-calle entropy conitions [1]. The existence an uniqueness theory for multi-imensional scalar conservation laws an for some special cases of onean multi-imensional systems is well evelope. A corresponing theory for multiimensional systems is still work in progress Finite-volume schemes. Explicit formulas for the solution of (1.1) are not available, except in the simplest cases. Consequently, numerical methos are heavily use for approximating (1.1). The most popular numerical methos in this context are the finite volume schemes, see e.g., [8, 45] an references therein for a etaile escription. In a finite volume approximation, the computational omain is iscretize into cells or control volumes an an integral form of the conservation law (1.1) is approximate on each control volume. This metho relies on constructing suitable numerical fluxes in the normal irection, across each cell interface. For simplicity, a uniform Cartesian iscretization of the omain is consiere, with mesh sizes x an y in the x- an y- irections respectively. It consists of the iscrete cells, C i,j := [x i 1, x i+ 1 ) [y j 1, y j+ 1 ), centere at the mesh points (x i, y j ) = (i x, j y), (i, j) Z. The cell average of U over C i,j (at time t), enote U i,j (t), is upate with the semi-iscrete scheme [8, 45]: (1.3) t U i,j = 1 x (F i+ 1,j F i 1,j ) 1 y (G i,j+ 1 G i,j 1 ). The time epenence of all the quantities in the above expression is suppresse for notational convenience. Classical schemes employ two-point numerical fluxes, F i+ 1,j = F(U i,j, U i+1,j ), G i,j+ 1 = G(U i,j, U i,j+1 ). A canonical example is provie by the first-orer Rusanov numerical flux: (1.4) F i+ 1,j = 1 ( f(ui,j ) + f(u i+1,j ) ) max{ (α) i,j, (α) i+1,j } ( ) U i+1,j U i,j, G i,j+ 1 = 1 ( g(ui,j ) + g(u i,j+1 ) ) max{ (β) i,j, (β) i,j+1 } ( ) U i,j+1 U i,j. Here, α i,j an β i,j are the maximal eigenvalues of the Jacobians A = U f an B = U g respectively, for a given state U i,j : α i,j := argmax λ { λ : λ = λ ( A(U i,j ) ) }, β i,j = argmax λ { λ : λ = λ ( B(U i,j ) ) }. Note that the only characteristic information in the Rusanov flux is a local estimate on the wave spees. This flux is almost Jacobian free, very simple to implement

3 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 3 an has a very low computational cost. But its resolution is limite by the firstorer accuracy. But the first-orer schemes (1.3),(1.4) can be extene to higher orer accuracy by employing numerical fluxes base on wier, p-point stencils, I i+ 1 := {i i i 1/ < p} an J j+ 1 := {j j j 1/ < p} along the x- an y-axis, respectively, (1.5) F i+ 1,j = F ({U i,j} i Ii+ 1 ) (, G i,j+ 1 = G {U i,j } j J j+ 1 ). The builing blocks for such extensions are still the -point numerical fluxes, F(, ) an G(, ). As a prototype example, we recall the class of secon-orer schemes base on piecewise bilinear MUSCL reconstruction [7] (1.6a) p i,j (x, y) := U i,j + U i,j x (x x i) + U i,j y (y y j); Here, U an U enote the numerical erivatives (1.6b) U i,j = minmo(u i+1,j U i,j, 1 (U i+1,j U i 1,j ), U i,j U i 1,j ), U i,j = minmo(u i,j+1 U i,j, 1 (U i,j+1 U i,j 1 ), U i,j U i,j 1 ), which utilize the minmo limiter (1.6c) { sgn(a) min{ a, b, c }, if sgn(a) = sgn(b) = sgn(c), minmo(a, b, c) = 0, otherwise. In this manner, one can reconstruct in each cell C i,j, the point values { U E (1.7a) i,j := p i,j (x i+ 1, y j), U W i,j := p i,j(x i 1, y j), U N i,j := p i,j(x i, y j+ 1 ), US i,j := p i,j(x i, y j 1 ), from the given neighboring cell averages U i,j, U i±1,j an U i,j, U i,j±1. The resulting secon-orer fluxes are then given by (1.7b) F i+ 1,j = F(U E i,j, U W i+1,j), G i,j+ 1 = G(UN i,j, U S i,j+1). The use of minmo limiter ensures the non-oscillatory behavior of the secon-orer schemes (1.3),(1.6). Observe that the secon-orer MUSCL fluxes (1.7b) are base on 4-point stencils F i+ 1,j = F(U i 1,j, U i,j, U i+1,j, U i+,j ), G i,j+ 1 = F(U i,j 1, U i,j, U i,j+1, U i,j+ ) Similar reconstructions together with upwin or central averaging yiel a large class of high-resolution finite-volume semi-iscrete schemes, e.g., [3],[44],[5], which coul then be integrate in time using stanar stable high orer Runge- Kutta methos []. 1.. Genuinely multi-imensional (GMD) schemes. Despite their consierable success in many applications, finite volume schemes (1.3) are known to be eficient [8] in resolving genuinely multi-imensional waves in the solution of (1.1). A possible explanation lies in the structure of the scheme (1.3). The numerical fluxes F i+ 1,j, G i,j+ 1 are efine in each normal irection an lack explicit transverse information. Consierable effort has been evote to evising genuinely multi-imensional (GMD) finite volume schemes for approximating (1.1). We provie a very brief summary of some of the available methos:

4 4 SIDDHARTHA MISHRA AND EITAN TADMOR (i) Dimensional splitting. This proceure is base on sequentially upating the cell average with flux F i+ 1,j (in the x-irection) an then upating with the numerical flux G i,j+ 1 (in the y- irection). Secon orer accuracy results from Strang splitting [8]. Despite the splitting, the resulting metho may still fail to resolve genuinely multi-imensional waves (examples are provie in [9]). (ii) Multi-imensional wave propagation. This metho is base on the Corner Transport Upwin (CTU) metho [11] for linear equations. Contributions from waves in the transverse irection are explicitly calculate. It was extene to non-linear systems in [9] by solving transverse Riemann problems. The metho is implemente in the CLAWPACK software package [8]. A relate scheme was propose in [8]. (iii) Metho of Transport. In [17, 18, 38], the non-linear conservation law (1.1) is reformulate locally as a system of transport equations. Explicit solutions of the transport equations efine a genuinely multi-imensional scheme. Complicate formulas for specific wave moels may be a major isavantage of this metho. (iv) Finite volume Evolution Galerkin (FVEG) methos. In [30, 31] (an other references therein), the conservation law (1.1) is linearize locally an the linearize system is solve in terms of bi-characteristics. The resulting evolution operator efines genuinely multi-imensional finite volume fluxes by a Galerkin type approximation. The task of eriving explicit solutions in terms of bi-characteristics for specific moels may be quite complicate. (v) Resiual istribution/fluctuation-splitting schemes. Genuinely multi-imensional methos for unstructure meshes were propose in [14,, 37]. They involve computing a cell resiual at each time step an istributing it to the cell noes by using some suitable upwining proceure, base on local flow irections. The absence of an optimal strategy for genuinely multi-imensional schemes leaves room for esigning stable GMD schemes that are easy to formulate an coe, have a low computational cost an preserve other esirable properties share by the multiimensional structure of the system (1.1). Their numerical fluxes take a general form (1.8a) F i+ 1,j = F({U (i,j ) S i+ 1,j }), G i,j+ 1 = G({U (i,j ) S i,j+ 1 }). Here, S i+ 1,j an S i,j+ 1 are two-imensional stencils which, in contrast to (1.5), allow us to incorporate information from both the normal an transverse irections, (1.8b) S i+ 1,j := { (i, j ) i i 1/ + j j < q }, S i,j+ 1 := { (i, j ) i i + j j 1/ < q } We present such a family of GMD schemes in section Conservation laws with constraints. Many interesting multi-imensional systems of conservation laws also involve intrinsic constraints. A representative example for such a system are the magnetohyroynamic (MHD) equations of plasma

5 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 5 physics: ρ t + (ρu 1 ) x + (ρu ) y = 0, (1.9) (ρu 1 ) t + (ρ(u 1 ) + p 1 (B 1) ) x + (ρu 1 u B 1 B ) y = 0, (ρu ) t + (ρu 1 u B 1 B ) x + (ρ(u ) + p 1 (B ) ) y = 0, (ρu 3 ) t + (ρu 1 u 3 B 1 B 3 ) x + (ρu u 3 B B 3 ) y = 0, (B 1 ) t + (u B 1 u 1 B ) y = 0, (B ) t + (u 1 B u B 1 ) x = 0, (B 3 ) t + (u 1 B 3 u 3 B 1 ) x + (u B 3 u 3 B ) y = 0, E t + ((E + p)u 1 (u B)B 1 ) x + ((E + p)u (u B)B ) y = 0, where the ensity of the plasma is enote by ρ, u = (u 1, u, u 3 ) an B = (B 1, B, B 3 ) are the velocity an magnetic fiels, respectively. E is the total energy an p := p + 1 B is the total pressure, with p being the thermal pressure. The unknowns are relate by an ieal gas equation of state similar to (1.b). The ieal MHD equations (1.9) form a (non-strictly) hyperbolic system of conservation laws, which combine the conservation laws for mass, momentum an energy with the magnetic inuction equations (a special form of the Maxwell s equations): (1.10) B t + curl(b u) = 0, (x, y, t) R R R +, which implies the ivergence constraint, (1.11a) iv(b) t 0. In the particular two-imensional setup of (1.9), the ivergent constraint is reuce to the two-component statement (1.11b) iv ( (B 1, B ) ) t 0. Since magnetic monopoles have not been observe in nature, the initial magnetic fiel is assume to be ivergence free. The ivergence constraint (1.11) implies that the ivergence of the magnetic fiel remains zero. Hence, the ieal MHD equations are an example for multi-imensional systems of conservation laws with an intrinsic constraint. Other interesting examples for systems with constraints are the system wave equation [36] (with vorticity as the constraint) an the Einstein equations of general relativity. A major issue for the numerical approximation of multi-imensional ieal MHD equations (1.9) is the ivergence constraint (1.11). The failure of stanar finite volume schemes to preserve iscrete versions of that constraint may lea to numerical instabilities [48, 19]. Different approaches have been suggeste to hanle the ivergence constraint in MHD coes. We escribe some of them briefly. (i) Projection metho. This metho [10, 7, 6] is base on the Hoge ecomposition of the magnetic fiel B. The upate B n, at each time step, may not be ivergence free an is correcte by the ecomposition: B n = Ψ+curlΦ. Applying the ivergence operator to the ecomposition leas to the elliptic equation: Ψ = iv(b n ).

6 6 SIDDHARTHA MISHRA AND EITAN TADMOR The correcte fiel B = B n Ψ is ivergence free. This metho can be very expensive computationally as an elliptic equation has to be solve at every time step. (ii) Source terms. Aing a source term, proportional to the ivergence, in (1.10) results in B t + curl(b u) = u(iv(b)). Applying the ivergence operator to both sies: iv(b) t + iv(u(ivb)) = 0. Hence any potential ivergence errors are transporte away from the computational omain by the flow. This proceure for cleaning the ivergence was introuce in [39, 40] an it nees to be iscretize in a very careful manner in orer to avoi two main ifficulties: to keep numerical stability [19, 0] an to avoi a wrong shock spee ue to the non-conservative form of the source term [48]. A variant of this approach is the Generalize Lagrange multiplier metho [15]. (iii) Design of special ivergence operators/staggering. This popular metho consists of staggering the iscretizations of the velocity an magnetic fiels in (1.10). A wie variety of strategies for staggering the meshes has been propose, [16, 5, 13, 43, 48, 4] an references therein. The presence of ifferent sets of meshes leas to problems when parallelizing this metho an using aaptive mesh refinement. Unstaggere variants of this approach have also been propose in [47, 46, 1]. The above iscussion suggests there is ample scope for a simple, computationally cheap finite volume scheme for the MHD equations that resolves genuinely multi-imensional waves an preserves a iscrete version of the ivergence constraint. Our aim in this paper is to summarize the results of recent papers [33, 34, 35] an present a new framework for approximating the two-imensional conservation law (1.1) in a genuinely multi-imensional manner. The GMD scheme is esigne by rewriting the stanar finite volume scheme (1.3) in terms of vertex centere numerical potentials. Stanar ege centere numerical fluxes serve as builing blocks of the GMD scheme as the numerical potential is efine in terms of them. The choice of potentials is very general an a specific choice of potential results in an entropy stable GMD scheme [34, 3]. In particular, in section 4 we iscuss potential-base GMD schemes that preserve a iscrete version of the ivergence in the MHD equations. Numerical experiments illustrating the robustness of the schemes in approximating the Euler equations (1.a) an the ieal MHD equations (1.9) are presente.. Genuinely multi-imensional (GMD) schemes Following the presentation of [34], we introuce the numerical potentials φ i+ 1,j+ 1 an ψ i+ 1,j+ 1 at each vertex (x i+ 1, y j+ 1 ), with the sole requirement that these potentials are consistent with the ifferential fluxes, i.e, φ i+ 1,j+ 1 (U,, U) = f(u), ψ i+ 1,j+ 1 (U,, U) = g(u).

7 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 7 We nee the following notation for stanar averaging an (univie) ifference operators, (.1) µ x a I,J := a I+ 1,J + a I 1,J, µ y a I,J := a I,J+ 1 + a I,J 1 δ x a I,J := a I+ 1,J a I 1,J, δ y a I,J := a I,J+ 1 a I,J 1. A wor about our notations: we note that the above iscrete operators coul be use with inexes I, J which are place at the center or at the ege of the computational cells, e.g., I = i or I = i + 1. In either case, we tag the resulting iscrete operators accoring to the center of their stencil; thus, for example, µ x w i+ 1 employs gri values place on the integer-inexe eges, w i an w i+1, whereas δ y w j employs the half-integer inexe centers, w j± 1. We now set the numerical fluxes: (.) F i+ 1,j = µ y φ i+ 1,j, G i,j+ 1 = µ xφ i,j+ 1. The resulting finite volume scheme written in terms of the numerical potentials reas (.3) t U i,j = 1 x δ xµ y φ i,j 1 y δ yµ x ψ i,j, = 1 ( 1 (φ i+ 1 x 1 y,j+ 1 + φ i+ 1,j 1 ) 1 ) (φ i 1,j+ 1 + φ i 1,j 1 ) ( 1 (ψ i+ 1,j+ 1 + ψ i 1,j+ 1 ) 1 ) (ψ i+ 1,j 1 + ψ i 1,j 1 ). The potential base scheme (.3) is clearly conservative as well as consistent as the potentials φ, ψ are consistent. The genuinely multi-imensional nature of the scheme is evient from (.3): the potentials are ifference in the normal irection but average in the transverse irection. We claim that the family of potentialbase schemes (.3) is rich: any stanar finite volume flux can be use as a builing block for constructing the numerical potentials in (.), an the resulting potential-base scheme inherits the accuracy of the unerlying numerical flux. There are several ways to pursue the construction of numerical potentials an we outline four of them below..1. Symmetric potentials. In this approach, the potentials are efine by averaging the finite volume fluxes neighboring a vertex: (.4) φ i+ 1,j+ 1 = µ yf i+ 1,j+ 1, ψ i+ 1,j+ 1 = µ xg i+ 1,j+ 1, where F i+ 1,j an G i,j+ 1 are any numerical fluxes consistent with f an g respectively. An explicit computation of (.3) with potentials (.4) leas to the revealing form, (.5) t U i,j = 1 x (µ yf i+ 1,j+ 1 + µ yf i+ 1,j 1 µ yf i 1,j+ 1 µ yf i 1,j 1 ) 1 y (µ xg i+ 1,j+ 1 + µ xg i 1,j+ 1 µ xg i+ 1,j 1 µ xg i 1,j 1 ).,

8 8 SIDDHARTHA MISHRA AND EITAN TADMOR Comparing the potential base scheme (.5) with the stanar finite volume scheme (1.3), we observe that the potential base scheme moifies (1.3) by averaging the fluxes in the transverse irection. Hence, it incorporates explicit transverse information in each irection. When employing two-point fluxes, the local stencil for the GMD scheme (.5) consists of nine points instea of the stanar five point stencil for the finite volume scheme (1.3). One can use wier stencils to achieve higher-orer of accuracy; for example, the symmetric potential-base scheme base on secon-orer four-point MUSCL flux (1.6) yiels a secon-orer GMD scheme base on a stencil of twenty-three points... Weighte symmetric potentials. Weighte averages of the neighboring fluxes can be consiere in place of the simple averaging use in (.4). For prescribe θ i+ 1,j+ 1, κ i+ 1,j+ 1 (0, 1), the weighte potential is efine as φ i+ 1 (.6),j+ 1 = θ i+ 1,j+ 1 F i+ 1,j+1 + (1 θ i+ 1,j+ 1 )F i+ 1,j, ψ i+ 1,j+ 1 = κ i+ 1,j+ 1 G i+1,j+ 1 + (1 κ i+ 1,j+ 1 )G i,j+1/. The weights can be chosen base on the local characteristic spees, (.7) θ i+ 1,j+ 1 = max{ (β 1 ) i+ 1,j+ 1, 0} max{ (β 1 ) i+ 1,j+ 1, 0} + max{(β N) i+ 1,j+ 1, 0}, κ i+ 1,j+ 1 = max{ (α 1 ) i+ 1,j+ 1, 0} max{ (α 1 ) i+ 1,j+ 1, 0} + max{(α N) i+ 1,j+ 1, 0}. Here, α l an, β l, l = 1,,, N are the real eigenvalues of A = U f(µ y µ x U i+ 1,j+ 1 ) an B = U g(µ x µ y U i+ 1,j+ 1 ), sorte in an increasing orer. This choice of weights means that the potential (.6) is upwine i.e, takes local flow irections into account..3. Staggere potentials. We efine the numerical potential as (.8) φ i+ 1,j+ 1 = F(µ yu i,j+ 1, µ yu i+1,j+ 1 ), ψ i+ 1,j+ 1 = G(µ xu i+ 1,j, µ x U i+ 1,j+1 ) for any consistent numerical fluxes F, G..4. Diagonal potentials. We efine the iagonal potentials [33], φ i+ 1 (.9a),j+ 1 = 1 ( F + + F ) i+ 1,j+ 1 i+, 1,j+ 1 ψ i+ 1,j+ 1 = 1 (G+ + G ). i+ 1,j+ 1 i+ 1,j+ 1 Here, F ±, G ± are the iagonal fluxes F + i+ 1,j+ 1 := F(U i,j, U i+1,j+1 ), F = F(U i+ (.9b) 1,j 1 i,j, U i+1,j 1 ) G + := G(U i+ 1,j+ 1 i,j, U i+1,j+1 ), G := G(U i 1,j+ 1 i,j, U i 1,j+1 ). which amount to rotating the x- an y-axis by angles of π 4 an π 4 an F(, ) an G(, ) are any two-point numerical fluxes consistent with f an g. We conclue our list for recipes of GMD schemes with an example which is not renere by a numerical potential, but nevertheless, highlights the use of a GMD stencil. Let F(, ) an G(, ) are any two-point consistent numerical fluxes an let F ±, G ±

9 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 9 be the corresponing iagonal numerical fluxes in (.9a). We efine the isotropic fluxes, ( F i+ 1 F + + F i+ 1,j+ 1 i+ 1,j + F ) i+, 1,j 1 (.10a),j = 1 4 G i,j+ 1 = 1 4 The resulting finite volume scheme reas as (.10b) t U i,j = 1 x δ F x i,j 1 y δ G y i,j, = 1 4 x ( G + + G i+ 1,j+ 1 i,j+ 1 + G i 1,j+ 1 ( δ/ F + i,j + δ xf i,j + δ \ F i,j) 1 4 y here, δ / an δ \ enote the iagonal ifference operators, ). ( δ/ G + i,j + δ yg i,j δ \ G ) i,j ; (.11) δ / a I,J := a I+ 1,J+ 1 a I 1,J 1, δ \a I,J := a I+ 1,J 1 a I 1,J+ 1. The GMD structure of the scheme is clear from (.10b): the scheme averages the fluxes along transverse irections. In contrast to the symmetric scheme (.5), however, the explicit transverse information in (.10b) is obtaine by rotating the fluxes. We term (.10b) as the isotropic GMD scheme. The stencil of the isotropic scheme consists of nine points. Secon-orer accuracy can be obtaine by the piecewise bilinear reconstruction (1.6). In aition to (1.7), we also nee the corner point values, U NE i,j := p i,j (x i+ 1 (.1a), y j+ 1 ), UNW i,j := p i,j (x i 1, y j+ 1 ), U SE i,j := p i,j (x i+ 1, y j 1 ), USW i,j := p i,j (x i 1, y j 1 ), an the corresponing iagonal fluxes, (.1b) F + i+ 1,j+ 1 G + i+ 1,j+ 1 := F(U NE i,j, U SW i+1,j+1), F i+ 1,j 1 := F(U SE i,j, U NW i+1,j+1), := G(U NE i,j, U SW i+1,j+1), G i 1,j+ 1 := F(U NW i,j, U SE i 1,j+1), to efine the secon orer accurate version of the isotropic GMD scheme. Remark.1. The isotropic GMD scheme (.10b) is a esirable form of the GMD scheme as we can prove that it is entropy stable provie that the builing block numerical fluxes F, G in (.10b) are entropy stable. A precise statement of the stability theorem an etails of the proof are presente in [34]. 3. Numerical Experiments The semi-iscrete first (secon) orer GMD schemes (.5), (.10b) are integrate in time with the stanar forwar Euler (strong stability preserving Runge- Kutta []) metho. The time step is etermine by a stanar CFL conition. All simulations reporte here, are performe with a CFL number of We test the following schemes: SYM (SYM) First (secon)-orer version of the symmetric GMD scheme (.5). ISO (ISO) First (secon)-orer version of the isotropic GMD scheme (.10b).

10 10 SIDDHARTHA MISHRA AND EITAN TADMOR 3.1. Numerical experiment #1. We begin with a test case for the Euler equations, reporte in [34]. The two-imensional raially symmetric version of the stanar So shock tube [8] consiers (1.a) with initial ata: (3.1) ρ(x, y, 0) = p(x, y, 0) = u(x, y, 0) = v(x, y, 0) 0. { 1.0 if x + y < 0.4, 0.15 otherwise, in the computational omain [, ] [, ]. The initial raial iscontinuity breaks into an outwar propagating shock wave, a contact iscontinuity an a rarefaction wave. The waves are raially symmetric an the stanar finite volume scheme is known to be eficient, [8]. We plot the approximate ensity at time t = 0., on a mesh in figure 1. The first-orer SYM an ISO schemes are iffusive, particularly at the contact iscontinuity. The raially symmetric structure is retaine an no gri aligne effects or spurious waves are observe. The secon-orer SYM an ISO schemes are much more accurate with goo resolution at the shock an the contact. The SYM scheme leas to small oscillations at the outer shock, inicating that the scheme oesn t contain enough iffusion (similar examples were presente in [33]). The secon-orer ISO scheme results in non-oscillatory an resolves the circular waves quite well. The results are comparable to those presente in [31] an references therein. 3.. Numerical experiment #. As a secon example for the Euler equations, we consier a benchmark test escribe in [8, 17, 9, 31] an references therein. The two imensional initial Riemann ata is (3.) ρ = , u = 0, v = 0, p = 0.4, if x > 0, y > 0, ρ = 1.0, u = 0, v = 0.776, p = 1.0, if x > 0, y < 0, ρ = 1.0, u = 0.776, v = 0, p = 1.0, if x < 0, y > 0, ρ = 0.8, u = 0, v = 0, p = 1.0, if x < 0, y < 0, in the computational omain, [ 1, 1] [ 1, 1]. The exact solution consists of two forwar moving shocks, two slip lines an a Mach reflection. Some stanar finite volume schemes approximate a regular reflection instea of a Mach reflection, [17]. The approximate ensity at time t = 0.5, on a mesh, is plotte in figure. The results are very similar to the previous numerical experiment. The first-orer SYM an ISO schemes resolve the multi-imensional features with some iffusion. The secon-orer ISO an SYM schemes attain consierably better resolution, particularly at the slip lines an at the reflection. The SYM scheme has a slight overshoot at the top right corner, inicating the absence of sufficient iffusion. The ISO scheme is very stable an accurate. The results are comparable to the ones obtaine in [17, 9, 31]. The above numerical experiments emonstrate that the GMD schemes presente in this paper are robust. The first orer schemes can be iffusive. A possible reason is the use of the Rusanov flux (1.4). Experiments with more accurate fluxes like the Roe flux le to a reuction in the amount of numerical iffusion. We prefer the Rusanov flux as it is very simple to coe an is computationally cheap. Furthermore, accuracy is recovere at secon-orer.

11 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 11 (a) SYM (b) ISO (c) SYM () ISO Figure 1. Approximate solutions of ensity for numerical experiment #3 at t = 0. on a mesh compute with the GMD schemes. 4. Divergence preserving schemes The ivergence of the magnetic fiel in the MHD equations (1.9) is preserve (1.11a). Despite incorporating explicit transverse information, the GMD schemes (.5) an (.10b) may not necessarily preserve a iscrete version of the ivergence constraint. A possible explanation lies in the special structure of the 8-vectors fluxes f an g in (1.9). Note that f 5 = g 6 0, f 6 = g 5 = u B 1 u 1 B. This interaction between the fluxes f, g is responsible for the ivergence constraint (1.11a). We must incorporate this information in the structure of the numerical potentials. Let φ, ψ be the potentials. Following [35], we require that potential components φ 5, φ 6, ψ 5 an ψ 6 satisfy: (4.1) ( ) φ 5 = ( ) ( ) ψ 6 0, φ6 = ( ) ψ 5 = χ i+ 1,j+ 1 i+ 1,j+ 1 i+ 1,j+ 1 for some consistent scalar potential χ, i.e, i+ 1,j+ 1 χ(u,, U) = u 1 B u B 1. i+ 1,j+ 1

12 1 SIDDHARTHA MISHRA AND EITAN TADMOR (a) SYM (b) ISO (c) SYM () ISO Introucing Figure. Approximate ensity for numerical experiment #4 at t = 0.5 on a mesh compute with the GMD schemes. V = {ρ, u 1, u, u 3, B 3, E}, Φ = {φ 1,, φ 4, φ 7, φ 8 }, Ψ = {ψ 1,, ψ 4, ψ 7, ψ 8 } for any consistent potentials φ, ψ. The potential base scheme (.3) with the choice of potential (4.1) reas as t V i,j = 1 x δ xµ y Φ i,j 1 y δ yµ x Ψ i,j, ( (4.) B1 t )i,j = 1 y δ yµ x χ i,j, ( B t )i,j = 1 x δ xµ y χ i,j. The constraint preserving property of the scheme is escribe in following lemma. Lemma 4.1. Define the iscrete ivergence operator: (4.3) iv ( (B 1, B ) ) := 1 i,j x µ ( ) yδ x B1 + 1 i,j y µ ( ) xδ y B. i,j Then, the potential base GMD scheme (4.) satisfies the iscrete ivergence constraint, analogous to (1.11b) t iv ( (B 1, B ) ) 0, i, j. i,j

13 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 13 Proof. The ifference operators δ x, δ y an the averaging operators µ x, µ y commute with each other. Applying the iscrete ivergence operator iv to the numerical scheme (4.), x y t iv ( (B 1, B ) ) i,j = (µ xδ y δ x µ y µ y δ x δ x µ x )χ i,j 0. A similar scheme preserves a iscrete version of vorticity for the system wave equation [34]. The scalar potential χ can be chosen in the following ways Divergence preserving symmetric GMD scheme. The potentials Φ, Ψ are efine by (.4). A natural choice [35] of the potential χ is the symmetric potential: (4.4) χ i+ 1,j+ 1 = 1 ( (F6 4 )i+ + ( F 1,j 6 )i+ + ( ) G 1,j ( ) ) G i,j+ 1 5 i+1,j+ 1 with F 5,6, G 5,6 being components of any consistent numerical fluxes F, G. Let H = {F 1,, F 4, F 7, F 8 }, K = {G 1,, G 4, G 7, G 8 } for any consistent fluxes F, G. The ivergence preserving symmetric GMD scheme has the explicit form: (4.5) t V i,j = 1 x (µ yh i+ 1,j+ 1 + µ yh i+ 1,j 1 µ yh i 1,j+ 1 µ yh i 1,j 1 ) 1 y (µ xk i+ 1,j+ 1 + µ xk i 1,j+ 1 µ xk i+ 1,j 1 µ xk i 1,j 1 ), t (B 1) i,j = 1 ( ) (µ x F6 4 y µ ( ) ) i,j+1 x F6 1 ( ) ( ) (δ i,j 1 y (µ x G5 + µ 4 y i+ 1,j+ 1 x G5 t (B ) i,j = 1 4 x (µ y ( G5 ) i+1,j µ y i 1,j+ 1 ( G5 )i 1,j ) x (δ ( ( ) x(µ y F6 + µ )i+ 1,j+ 1 y F6 i+ 1,j Divergence preserving isotropic GMD scheme. Following [33], we efine a iagonal form of the potential χ: (4.6) χ i+ 1,j+ 1 = 1 ( (F ) ( G + ) i+ 1,j ( F ) i+ 1,j ( G ) ) i+ 1,j+ 1 5 i+ 1,j+ 1 for iagonal fluxes F ±, G ± efine in (.9b). Denote H ± = {F ± 1,, F± 4, F± 7, F± 8 }, K± = {G ± 1,, G± 4, G± 7, G± 8 } The ivergence preserving moification of the isotropic GMD scheme (.10b) base on the potential (4.6) is (4.7) t V i,j = 1 ( δ/ H + i,j 4 x + δ xh i,j + δ \ H 1 i,j) 4 y t (B 1) i,j = 1 ( (F ) + (µ x δ y 4 y + ( 6 F ) i,j 6 + ( G + ) i,j 5 + ( G ) i,j 5 t (B ) i,j = 1 ( (F ) + (µ y δ x 4 x + ( 6 F ) i,j 6 + ( G + ) i,j 5 + ( G ) i,j 5 i,j ( δ/ K + i,j + δ yk i,j δ \ K ) i,j, i,j )), )) Numerical Experiments. In aition to the SYM (SYM) an ISO (ISO) schemes of the last section, we also test the following two ivergence preserving GMD schemes: ) ). ) ),

14 14 SIDDHARTHA MISHRA AND EITAN TADMOR SCP (SCP) First (secon)-orer version of the ivergence preserving symmetric GMD scheme (4.5); ICP (ICP) First (secon)-orer version of the ivergence preserving isotropic GMD scheme (4.7). (a) SYM (b) SCP (c) ISO () ICP Figure 3. The pressure p for the Orszag-Tang vortex compute at t = π on a mesh with first-orer GMD schemes Orszag-Tang vortex. The Orszag-Tang vortex is a wiely reporte benchmark for multi-imensional MHD equations [48]. The initial ata is (ρ, u 1, u, u 3, B 1, B, B 3, p) = ( γ, sin(y), sin(x), 0, sin(y), sin(x), 0, γ ), in the computational omain, (x, y, t) [0, π] [0, π] with perioic bounary conitions. Although the exact solution is not known, some qualitative features have been reporte [48]. The solution consists of shocks along the iagonals an interesting smooth features incluing a vortex near the center of the omain. The approximate pressures, compute on a mesh, are shown in figures 3 an 4. Figure 3 shows the approximate pressure compute with the first-orer GMD schemes. The solution is smeare at this resolution, but the qualitative features are capture quite well. The shocks an the central vortex are approximate, without any spurious waves or oscillations. The ivergence preserving SCP an ICP schemes are clearly more accurate than the SYM an ISO schemes. The

15 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 15 (a) SYM (b) SCP (c) ISO () ICP Figure 4. The pressure p for the Orszag-Tang vortex compute at t = π on a mesh with secon-orer GMD schemes. results for the secon-orer schemes are plotte in figure 4. There is a consierable improvement in the resolution with secon-orer schemes. The gain in accuracy is evient, both at the shocks an at the central vortex. The ivergence preserving SCP an ICP are slightly more accurate than the SYM an ISO schemes. The initial ata is ivergence free an the ivergence constraint (1.11a) implies that it remains zero uring the evolution. We show the errors in the iscrete ivergence operator iv (4.3) in Table 1. The stanar GMD schemes lea to M SYM ISO SCP ICP SYM ISO SCP ICP e-1 4.4e e e e-1 1.7e e e e-1 6.9e e e e-1 6.0e e e-13 Table 1. Discrete ivergence iv (4.3) in L 1 for the Orszag-Tang vortex with all the GMD schemes on a M M mesh at time t = π. O(1) ivergence errors, with large amounts of iscrete ivergence being generate near the shocks. The ivergence error is even larger for the secon-orer SYM an ISO schemes. This behavior is to be expecte as the secon-orer schemes resolve the shocks more sharply. The SCP, SCP, ICP an ICP schemes preserve

16 16 SIDDHARTHA MISHRA AND EITAN TADMOR the iscrete ivergence to machine precision. We woul like to point out that preserving iv to machine precision woul imply that other iscrete versions of the ivergence operator like the stanar central ivergence will be of the orer of machine precision for smooth solutions. Since, the above example contains strong shocks, large values for the stanar central ivergence may be observe even if iv is preserve to machine precision. Numerical stability (particularly on fine meshes) for the MHD equations is elicate [19]. Stanar schemes (even those with ivergence cleaning) may crash ue to instabilities an negative pressures on fine resolutions [19]. In spite of the large ivergence errors, the SYM (SYM) an ISO (ISO) schemes are stable. The genuinely multi-imensional structure of the schemes imparts aitional numerical stability Clou-Shock Interaction. Another benchmark test case for the MHD equations involves the interaction of a high ensity clou with a shock. The initial ata for this clou-shock interaction problem [4] consists of a shock locate at x = 0.05 with (4.8) (ρ, u 1, u, u 3, B 1, B, B 3, p) { ( , , 0, 0, 0,.18618,.18618, ), if x < 0.05 = (1.0, 0, 0, 0, 0, , , 1.0), if x < an a circular clou of ensity ρ = 10 with raius 0.15, centere at (x, y) = (0.5, 0.5) in the computational omain [0, 1] [0, 1]. The test is configure in such a way that a right moving shock violently interacts with a high ensity clou. The solution has a extremely complex structure, consisting of bow shock at the left, trailing shocks at the right an a complicate smooth region with turbulent like features in the center. We plot the approximate ensity, on a mesh, at time t = 0.06 in figures 5 an 6. The first-orer results in figure 5 show that although iffusive, the first-orer GMD schemes are stable an resolve the shock structure in the correct qualitative manner. The ivergence preserving SCP an ICP schemes are more accurate than the SYM an ISO schemes. The secon-orer results are plotte in figure 6 an show a ramatic increase in resolution. Both the bow shock an the trailing shock are capture accurately. The smooth region with turbulent features is also resolve quite well. In fact, a clear ifference between the first- an secon-orer schemes lies in the fact that the secon-orer schemes resolve some of the turbulent features on very coarse meshes. The ivergence errors for iscrete ivergence iv are shown in Table an emonstrate quite large ivergence errors for the SYM (SYM) an ISO (ISO) schemes. The ivergence errors increase with reuction in mesh size, inicating prouction of ivergence at the shocks. The SCP (SCP) an ICP (ICP) schemes preserve iscrete ivergence to machine precision. 5. Conclusion The structure of solutions of conservation laws in several space imensions is very rich an consists of complex multi-imensional waves. Stanar finite volume methos are base on ege centere fluxes an o not incorporate any explicit transverse information. Consequently, they are eficient in resolving genuinely

17 CONSTRAINT PRESERVING GENUINELY MULTI-DIMENSIONAL SCHEMES 17 (a) SYM (b) ISO (c) SCP () ICP Figure 5. The ensity ρ for the clou-shock interaction compute at t = 0.06 on a mesh with first-orer schemes. M SYM ISO SCP ICP SYM ISO SCP ICP e-1.1e e-13.7e e-1 8.7e e e e e e e e e e-13.e-13 Table. Discrete ivergence iv (4.3) in L 1 for clou shock interaction with all the eight schemes on a M M mesh at time t = multi-imensional waves. These eficiencies are particularly evient for conservation laws with intrinsic constraints like vorticity an ivergence. Finite volume schemes may not preserve iscrete versions of the constraint an may lea to spurious numerical waves an oscillations. We summarize the results of a recent series of papers [33, 34, 35] where a new framework for genuinely multi-imensional (GMD) schemes was presente. These schemes are base on vertex centere numerical potentials. Stanar ege centere fluxes are use to efine the potentials. A particular version of the GMD schemes, he isotropic GMD scheme (.10b) is entropy stable if its builing block numerical fluxes are entropy stable.

18 18 SIDDHARTHA MISHRA AND EITAN TADMOR (a) SYM (b) SCP (c) ISO () ICP Figure 6. The ensity ρ for the clou-shock interaction compute at t = 0.06 on a mesh with secon-orer schemes A suitable choice of potentials leas to a GMD scheme that preserves iscrete version of the ivergence constraint for the ieal MHD equations (1.9). Higher orer of spatial accuracy is obtaine by employing the non-oscillatory reconstruction proceure of [5]. A choice of the Rusanov flux as the builing block for the GMD schemes leas to genuinely multi-imensional an constraint preserving versions of the popular central schemes of Kurganov an Tamor [5]. Numerical experiments for the Euler an the MHD equations are presente. They show that the GMD schemes are robust an resolve the multi-imensional waves with high accuracy. Preserving the ivergence constraint leas to higher resolution, particularly at first-orer. The computational cost of the schemes are very low an they are very simple to implement in a coe. Hence, the GMD framework constitutes an unifie an highly effective strategy for approximating multi-imensional conservation laws. Future papers consier higher than seconorer versions of the GMD schemes on unstructure gris. References [1] R. Artebrant an M.Torrilhon. Increasing the accuracy of local ivergence preserving schemes for MHD. J. Comp. Phys., 7 (6), , 008. [] R. Abgrall an P. L. Roe. High-orer fluctuation schemes on triangular meshes. SIAM J. sci. comput., 198, 3-36, 003.

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