Siddhartha Mishra 1 and Eitan Tadmor 2

Transcription

1 ESAIM: MAN DOI: /man/ ESAIM: Mathematical Moelling an Numerical Analysis CONSTRAINT PRESERVING SCHEMES USING POTENTIAL-BASED FLUXES. III. GENUINELY MULTI-DIMENSIONAL SCHEMES FOR MHD EQUATIONS, Sihartha Mishra 1 an Eitan Tamor Abstract. We esign efficient numerical schemes for approximating the MHD equations in multiimensions. Numerical approximations must be able to eal with the complex wave structure of the MHD equations an the ivergence constraint. We propose schemes base on the genuinely multiimensional GMD framework of [S. Mishra an E. Tamor, Commun. Comput. Phys ; S. Mishra an E. Tamor, SIAM J. Numer. Anal ]. The schemes are formulate in terms of vertex-centere potentials. A suitable choice of the potential results in GMD schemes that preserve a iscrete version of ivergence. First- an secon-orer ivergence preserving GMD schemes are teste on a series of benchmark numerical experiments. They emonstrate the computational efficiency an robustness of the GMD schemes. Mathematics Subject Classification. 65M06, 35L65. Receive September 7, 009. Revise May 3, 010. Publishe online January 11, Introuction Moeling of plasmas lies at the core of many interesting problems in astrophysics, solar physics, electrical an aerospace engineering. Macroscopic plasma ynamics is characterize by the interaction of the moving plasma with the magnetic fiel which is often moele by the equations of ieal Magnetohyroynamics MHD. In Keywors an phrases. Multiimensional evolution equations, magnetohyroynamics, constraint transport, central ifference schemes, potential-base fluxes. To the memory of Davi Gottlieb. The work on this paper was starte 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 grants DMS , DMS 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 Centre of Mathematics for Applications CMA, University of Oslo, P.O. Box 1053, Blinern, 0316 Oslo, Norway. siharm@cma.uio.no. Department of Mathematics, Center of Scientific Computation an Mathematical Moeling CSCAMM, Institute for Physical sciences an Technology IPST, University of Marylan, MD, Marylan, USA. tamor@cscamm.um.eu Article publishe by EDP Sciences c EDP Sciences, SMAI 01

2 66 S. MISHRA AND E. TADMOR two space imensions, the MHD equations are U t + fu x + gu y =0, x, y, t R R R +, 1.1a where ρ ρu 1 ρu ρu U = 3 B, f = 1 B B 3 E ρu 1 ρu 1 + p 1 B 1 ρu 1 u B 1 B ρu 1 u 3 B 1 B 3 0 u B 1 u 1 B u 1 B 3 u 3 B 1 E + pu 1 u BB 1, g = ρu ρu 1 u B 1 B ρu + p 1 B ρu 1 u 3 B 1 B 3 u B 1 u 1 B 0 u B 3 u 3 B E + pu u BB. 1.1b Here, ρ enotes the ensity of the plasma an u =u 1,u,u 3, B =B 1,B,B 3 are, respectively, the velocity an magnetic fiels, E is the total energy an p is the total pressure, p = p + 1 B, 1.1c where p is the thermal pressure ictate by the equation of state of an ieal gas with a gas constant γ, E = p γ ρ u + B. 1.1 The MHD equations combine conservation laws of the mass, momentum an energy with the magnetic inuction equations. The special structure of the equations in their two-imensional setup 1.1 isobservein the 8-vector fluxes which satisfy f 5 = g 6 0, f 6 = g 5 = h, h hu :=u B 1 u 1 B. 1. For an arbitrary 8-vector w, welet w enote the reuce 6-vector w := w 1,w,w 3,w 4,w 7,w 8 ;thenwecan rewrite 1.1, 1. in the equivalent form Ũ t + fu x + gu y =0, 1.3a B 1 t + hu y =0, 1.3b B t hu x =0. 1.3c The last two equations imply that solutions of 1.1 satisfy the two-imensional ivergence constraint iv B 1,B t Similarly, the magnetic fiel in the MHD equations in their three-imensional setup satisfy the special form of the Maxwell s equations, e.g., [36], B t +curlb u =0, x, y, t R R R +, 1.5 which in turn, implies the ivergence constraint: ivb t Since magnetic monopoles have not been observe in nature, the initial magnetic fiel is assume to be ivergence free. The ivergence constraints 1.4, 1.6 imply that the ivergence of the magnetic fiel remains zero. The system of ieal MHD equations is an example for a non-strictly hyperbolic system of conservation laws with an intrinsic constraint [37]. The solutions of a non-linear system like 1.1 evelop generic iscontinuities in the form of shock waves an contact iscontinuities. The lack of strict hyperbolicity an the non-convexity of the MHD equations lea to a complex shock structure, consisting of intermeiate an compoun shocks [38]. Consequently, relatively few theoretical results are currently available for the MHD equations an numerical methos for the approximate solution of these equations are sought.

3 1.1. Finite-volume schemes CONSTRAINT PRESERVING SCHEMES 663 Finite-volume methos are among the most wiely use numerical methos for the approximate solution of systems of conservation laws such as the MHD equations 1.1, see [5, 41] an the references therein. In a finite volume approximation, the computational omain is iscretize into cells an an integral form of the conservation law 1.1 is realize on each cell in terms of cell averages. To this en we cover the x-y plane with 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 =iδx, jδy, i, j Z with fixe mesh sizes Δx, Δy in the x- any-irections. The cell averages of U over C i,j at time t, enote U i,j t, are then evolve by the semi-iscrete scheme [5, 41]: t U i,j = 1 Fi+ 1 Δx,j F 1 i 1,j Gi,j+ 1 Δy G i,j The time epenence of all the quantities in the above expression is suppresse for notational convenience. Classical first-orer schemes employ two-point numerical fluxes of the form F i+ 1,j = FU i,j, U i+1,j, G i,j+ 1 = GU i,j, U i,j A canonical example is provie by the first-orer Rusanov numerical flux: Here, J is the 8 8scalingmatrix: F i+ 1,j = 1 fui,j +fu i+1,j max{ α i,j, α i+1,j }J U i+1,j U i,j, G i,j+ 1 = 1 gui,j +gu i,j+1 max{ β i,j, β i,j+1 } 1.9 U i,j+1 U i,j. J =iag{1, 1, 1, 1, 1, 1, 1, 1}, reflecting the special structure in the opposite signs, f 5 = g 6, reflecte in the fluxes specifie in 1.3b an 1.3c. The α i,j an β i,j in 1.9 are the maximal eigenvalues of the corresponing Jacobians A := U f an B := U g at a given state U i,j, i.e., α i,j = u 1 + c i,j 1, β i,j = u i,j + c ; 1.10a here c k = 1 a a i,j + b i,j + i,j + b i,j 4a i,j b k, k =1,, 1.10b where a i,j := γp i,j, an b i,j = b 1 ρ + b i,j + b i,j 3, b i,j i,j := B i,j 1.10c i,j ρi,j The Rusanov flux 1.9 has been use in [,3] as a simple yet highly effective builing block for solving the ieal MHD equations using high-resolution central schemes. Other popular numerical fluxes use for the solution of MHD equations inclue linearize Roe solvers [9, 36, 37] anhlltypesolvers[7, 17, 1, 6, 3]. Detaile comparisons of ifferent solvers are performe in [18,31]. The Rusanov flux is particularly simple since it is free of the eigenstructure of the Jacobians only a local estimate on the wave spees is neee. But the resolution of these various fluxes is limite by their first-orer accuracy. The first-orer accuracy of the two-point schemes 1.7, 1.8 can be extene to higher orer by employing numericalfluxeswhicharebaseonwier,p-point stencils, I i+ 1 := {i i i 1/ <p} along the x-axis an J j+ 1 := {j j j 1/ <p} along the y-axis, F i+ 1,j = F {U i,j} i Ii+, G i,j+ 1 = G {U 1 i,j } j J j

4 664 S. MISHRA AND E. TADMOR The builing blocks for such extensions are still the two-point numerical fluxes, F, an G,. As a prototype example, we recall the class of secon-orer schemes base on piecewise bilinear MUSCL reconstruction [45], p i,j x, y :=U i,j + U i,j Δx x x i+ U i,j Δy y y j. Here, U an U enote the numerical erivatives 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 { sgnamin{ a, b, c }, minmoa, b, c := 0, otherwise. if sgna =sgnb =sgnc, 1.1a 1.1b 1.1c In this manner, one can reconstruct in each cell C i,j, the point values U E 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, 1.1 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 F i+ 1,j = FUE i,j, UW i+1,j, G i,j+ 1 = GUN i,j, US i,j e The use of minmo limiter ensures the non-oscillatory behavior of the secon-orer schemes 1.7, 1.1. Observe that the resulting secon-orer MUSCL fluxes in 1.1e arebaseon4-pointstencils F i+ 1,j = FU i 1,j, U i,j, U i+1,j, U i+,j, G i,j+ 1 = GU 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 finitevolume semi-iscrete schemes, e.g., [,3,33,40], which coul then be integrate in time using stanar stable high orer Runge-Kutta methos [0]. 1.. Genuinely multi-imensional GMD fluxes Despite their consierable success, finite volume schemes 1.7 are known to be eficient in resolving genuinely multi-imensional GMD waves [5]. Observe that the numerical fluxes F i+ 1,j, G i,j+ 1 in 1.11 arebaseon one-imensional stencils which are supporte in each normal irection but lack explicit transverse information. This coul result in poor approximation of genuinely multi-imensional waves, particularly for complicate systems like the ieal MHD equations 1.1. A characteristic feature of the MHD equations in this context, are the ivergence constraints 1.4, 1.6, which reflect the essential multi-imensional character of the MHD equations. Consierable effort has been evote for evising numerical methos which aress the multi-imensional character of nonlinear system such as 1.1. These methos inclue imensional splitting [5], wave propagation algorithms [4, 5], metho of transport [15, 16, 34], bi-characteristics base evolution Galerkin methos [7, 8] an fluctuation splitting schemes [1]. The absence of an optimal strategy for esigning GMD schemes for constraine systems of conservation laws such as the MHD equations, leaves room for esigning stable GMD schemes that are easy to formulate an coe, have a low computational cost an preserve other esirable properties renere by the multi-imensional structure of the system 1.1 like the ivergence constraint. Their numerical fluxes take a general form } { } F i+ 1 {U,j = F i,j S i+, G 1,j i,j+ 1 = G U i,j S i,j a 1

5 CONSTRAINT PRESERVING SCHEMES 665 Here, S i+ 1,j an S i,j+ 1 are genuinely two-imensional stencils which, in contrast to 1.11, allow us to incorporate information from both the normal an transverse irections, S i+ 1,j := { i,j i i 1/ + j j <q }, S i,j+ 1 := { i,j i i + j j 1/ <q }. 1.13b Thus, in contrast to the stanar use of one-imensional stencils in 1.11, we avocate here the use of twoimensional clous, as the basic stencil for conservative GMD schemes. We present such a family of GMD schemes in Section, base on the potential-base framework introuce in our recent papers [9, 30] Divergence preserving schemes A major issue for the numerical approximation of multi-imensional ieal MHD equations 1.1 is the ivergence constraint 1.4, 1.6. Stanar finite volume schemes may not preserve iscrete versions of the constraint, leaing to numerical instabilities [17, 44]. Different approaches have been suggeste to hanle the ivergence constraint in MHD coes an we mention three of the currently available approaches below. i ii The projection metho, [6, 8, 10], is base on the Hoge ecomposition of the solution B of 1.5: the correcte fiel B := B n Δ 1 ivb n is ivergence free. The metho is rather expensive, however, as it requires a global elliptic solver together with a proper set of bounary conitions to be solve at every time step, e.g. [44]. The metho of aing a source term, [35, 36], proportional to the ivergence in 1.5 resultsin B t +curlb u = uivb The form 1.14 is symmetrizable [35]. A variant of this approach is foun in the Generalize Lagrange multiplier metho [13]. Applying the ivergence to both sies, we obtain ivb t +ivuivb =0. Hence, any potential ivergence errors are transporte away from the computational omain by the flow. Recent papers [17,19] have emonstrate that the ae source term in 1.14 nees to be iscretize in a very careful manner for numerical stability. Another ifficulty with this approach lies in the non-conservative form of 1.14 which may result in wrong shock spees [44]. iii The metho of esigning special ivergence operators/staggering is a popular family of methos which consist of staggering the iscretizations of the velocity an magnetic fiels in 1.5. A wie variety of strategies for staggering the meshes has been propose [ 5, 11, 14, 39, 44] an references therein. The presence of ifferent sets of meshes leas to problems when the staggere schemes are parallelize, bounary treatments, etc. Unstaggere variants of this approach have also been propose in [1, 4, 43]. The above iscussion suggests that there is still a room for simple, computationally cheap finite volume schemes for the constraine equations that resolves GMD waves an preserve a iscrete version of the associate constraint. In this paper, we present such a metho with these esire properties for the ivergence-free MHD equations. Our starting point are the GMD finite volume schemes propose in [9, 30]. These schemes moify stanar finite volume fluxes by introucing vertex centere numerical potentials. The potentials incorporate explicit transverse information an lea to a stable an accurate resolution of genuinely multi-imensional waves. A suitable choice of potentials results in GMD schemes that preserve constraints like ivergence in the magnetic inuction equation [9] or vorticity in the system of wave equations [30]. The schemes are very simple to implement an have low computational cost. The extension of the potential-base GMD framework of [30] to the ieal MHD equations 1.1 is carrie out in Section. A suitable choice of numerical potentials outline in Section 3 is shown to yiel schemes that preserves iscrete ivergence. Numerical experiments are reporte in Section 4.

6 666 S. MISHRA AND E. TADMOR. Genuinely multi-imensional GMD schemes Following the presentation of [30], 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 =fu, ψ i+ 1,j+ 1 U,...,U =gu. We nee the following notation for stanar averaging an univie ifference operators, 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,.1 δ x a I,J := a I+ 1,J a I 1,J, δ ya 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.Ineithercase,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,whereasδ 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 t U i,j = 1 Δx δ xμ y φ i,j 1 Δy δ yμ x ψ i,j = 1 Δx 1 Δy 1 1 φ i+ 1,j+ 1 + φ i+ 1,j 1 ψ i+ 1,j+ 1 + ψ 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 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 potential-base schemes.3 isrich: 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 three of them below..1. Symmetric potentials In this approach, the potentials are efine by averaging the finite volume fluxes neighboring a vertex: φ i+ 1,j+ 1 = μ yf i+ 1,j+ 1, ψ i+ 1,j+ 1 = μ xg i+ 1,j+ 1,.4 where F, G are any numerical fluxes consistent with f an g respectively. An explicit computation of.3 with potentials.4 leas to the revealing form, t U i,j = 1 Δx 1 Δy μ y F i+ 1,j+ 1 + μ yf i+ 1,j 1 μ yf i 1,j+ 1 μ yf i 1,j 1. μ x G i+ 1,j+ 1 + μ xg i 1,j+ 1 μ xg i+ 1,j 1 μ xg i 1,j

7 CONSTRAINT PRESERVING SCHEMES 667 Comparing the potential base scheme.5 with the stanar finite volume scheme 1.7, we observe that the potential base scheme moifies 1.7 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 1.7. 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 1.1 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,j+ 1 = θ i+ 1,j+ 1 F i+ 1,j θ i+ 1,j+ 1 F i+ 1,j, ψ i+ 1,j+ 1 = κ i+ 1,j+ 1 G i+1,j κ i+ 1,j+ 1 The weights can be chosen base on the local characteristic spees, G i,j+1/..6 θ i+ 1 κ i+ 1,j+ 1 = max{ β 1 i+ 1,j+ 1 { }, 0} max β 1 i+ 1,j+ 1, 0 +max { } max α 1,j+ 1 = i+ 1,j+ 1, 0 { } max α 1 i+ 1,j+ 1, 0 +max { β 8 i+ 1,j+ 1, 0 }, { α 8 i+ 1,j+ 1, 0 }.7 Here, α l an,β l,l = 1,,...,N are the real eigenvalues of A = U fμ y μ x U i+ 1,j+ 1 an B = U gμ x μ y U i+ 1,j+ 1, sorte in an increasing orer. This choice of weights implies that the potential.6 is upwine..3. Diagonal potentials We efine the iagonal potentials [9], φ i+ 1,j+ 1 = 1 ψ i+ 1,j+ 1 = 1 F + + F, i+ 1,j+ 1 i+ 1,j+ 1 G + + G. i+ 1,j+ 1 i+ 1,j+ 1.8a Here, F ±, G ± are the iagonal fluxes F + i+ 1,j+ 1 G + i+ 1,j+ 1 := F U i,j, U i+1,j+1, F i+ 1,j 1 = F U i,j, U i+1,j 1 := G U i,j, U i+1,j+1, G i 1,j+ 1 := G U i,j, U i 1,j+1..8b which amount to rotating the x- any-axis by angles of π 4 an π 4,whereF, ang, are any two-point numerical fluxes consistent with f an g..4. Isotropic GMD scheme 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

8 668 S. MISHRA AND E. TADMOR consistent numerical fluxes an let F ±, G ± be the corresponing iagonal numerical fluxes in.8b. We efine the isotropic fluxes, F i+ 1,j := 1 F + +F 4 i+ 1,j+ 1 i+ 1,j + F, i+ 1,j 1 G i,j+ 1 := 1.9 G + +G 4 i+ 1,j+ 1 i,j+ 1 + G. i 1,j+ 1 The resulting finite volume scheme reas as t U i,j = 1 Δx δ xf i,j 1 Dy δ yg i,j, = 1 4Δx δ/ F + i,j +δ xf i,j + δ \ F i,j 1 4Δy here, δ / an δ \ enote the iagonal ifference operators, δ/ G + i,j +δ yg i,j δ \ G.10 i,j ; 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 The GMD structure of the scheme is clear from.10: the scheme averages the fluxes along transverse irections. In contrast to the symmetric scheme.5, however, the explicit transverse information in.10 is obtaine by rotating the fluxes. Since the scheme.10 is base on all nine gri points in a cell by involving all four irections, we term it as an isotropic GMD scheme. The isotropic GMD scheme.10 wasshowntobe entropy stable if the unerlying two-point fluxes, F,, G, are, consult[30]. Secon-orer version of the isotropic schemes can be obtaine by the piecewise bilinear reconstruction 1.1. Here, in aition to the point values reconstructe on 1.1e, we also nee the corner point values, an the corresponing iagonal fluxes, U NE i,j := p i,j x i+ 1,y j+ 1 U SE i,j := p i,j x i+ 1,y j 1, U NW i,j := p i,j x i 1,y j+ 1, U SW i,j := p i,j x i 1,y j 1,,.1a F + i+ 1,j+ 1 G + i+ 1,j+ 1 := F U NE i,j, USW i+1,j+1, F := G U NE i,j, USW i+1,j+1 i+ 1,j 1, G i 1,j+ 1 := F U SE i,j, UNW i+1,j+1, := F U NW i,j, U SE i 1,j+1,.1b to efine the secon orer accurate version of the isotropic GMD scheme.10, Divergence preserving schemes The ivergence of the magnetic fiel in 1.1a is preserve 1.4, ue to the special structure of the MHD fluxes in 1.3. Accoringly, to ensure a ivergence-free numerical solution we nee to respect this special structure at the iscrete level by choosing a suitable form of the potential. Let φ i+ 1,j+ 1 an ψ i+ 1,j+ 1 be numerical potentials associate with the MHD fluxes f an g; one may employ here any consistent potential such as the weighte symmetric, iagonal or isotropic potentials constructe in Section. Letη an ζ be the reuce 6-vector potentials consistent with the reuce fluxes f an g: η i+ 1,j+ 1 := φ 1 i+ 1,j+ 1,...,φ 4 i+ 1,j+ 1, φ 7 i+ 1,j+ 1, φ 8 i+ 1,j+ 1, 3.1 ζ i+ 1,j+ 1 := ψ 1 i+ 1,j+ 1,...,ψ 4 i+ 1,j+ 1, ψ 7 i+ 1,j+ 1, ψ 8 i+ 1,j+ 1.

9 CONSTRAINT PRESERVING SCHEMES 669 Our scheme will evolve the cell averages of the reuce vector Ũ := ρ, ρu 1,ρu,ρu 3,B 3,E together with the x- any-components of the magnetic fiel, B 1 an B. The potential-base approximation of 1.3 reas = 1 tũi,j Δx δ xμ y η i,j 1 Δy δ yμ x ζ i,j, t B 1 i,j = 1 Δy δ yμ x χ i,j, t B i,j = 1 Δx δ xμ y χ i,j. Here, χ i+ 1,j+ 1 is any numerical potential consistent with hu =u 1B u B 1. The ivergence preserving property of 3. is summarize below. 3. Lemma 3.1. Let η i+ 1,j+ 1,ζ i+ 1,j+ 1 an χ i+ 1,j+ 1 be arbitrary numerical potentials consistent with f, g an, respectively, h. LetB i,j be the approximate magnetic fiels compute with the corresponing potential-base GMD scheme 3.. Then, their iscrete ivergence iv, iv B 1,B 1 i,j := Δx μ yδ x B 1 i,j + 1 Δy μ xδ y B i,j, 3.3a is preserve in time, iv B 1,B t i,j 0, i, j. 3.3b Verification of 3.3b is straightforwar: the ifference operators δ x,δ y an the averaging operators μ x,μ y commute with each other. We apply the iscrete ivergence operator iv to the numerical scheme 3. to fin ΔxΔy t iv B 1,B i,j =μx δ y δ x μ y μ y δ x δ x μ x χ i,j 0. Remark 3.. A similar treatment of ivergence-preserving potential-base schemes for the 3D magnetic transport 1.5 was carrie out in [9], Section.5. This coul be serve as the builing block for ivergence-preserving potential-base scheme for the 3D MHD equations along the lines of our treatment of D MHD equations in Lemma 3.1. Remark 3.3. One approach in esigning constraint preserving schemes is to satisfy that constraint approximately: for example, the iscrete statement of the ivergence constraint coul be interprete as a secon-orer approximation of the ifferential ivergence, iv B 1,B i,j =iv B1,B x i,y j +OΔx + Δy. This, however, requires the smoothness of the unerlying solution. Instea, a key feature of constraint preserving schemes base on numerical potentials is that they satisfy exactly a iscrete constraint, so that their numerical solution remains on a iscrete sub-manifol, inepenent of the unerlying smoothness. Similarly, a relate potential-base GMD scheme which preserves a iscrete vorticity was escribe in [30]. The scalar potential χ in 3. can be chosen in ifferent ways. We mention two possible choices below Divergence preserving symmetric GMD scheme The potentials η, ζ are efine as in.4. A natural choice [9] of the potential χ is the symmetric potential: χ i+ 1,j+ 1 = 1 4 F 6 i+ 1,j +F 6 i+ 1,j+1 +G 5 i,j+ 1 +G 5 i+1,j

10 670 S. MISHRA AND E. TADMOR with F 5,6, G 5,6 being components of any consistent numerical fluxes F, G. Let F =F 1,...,F 4, F 7, F 8, G =G1,...,G 4, G 7, G 8 be the corresponing reuce fluxes associate with any consistent F, G. The potential-base symmetric GMD scheme 3. takes the explicit form: = 1 tũi,j Δx 1 Δy μ y Fi+ 1,j+ 1 + μ y F i+ 1,j 1 μ y F i 1,j+ 1 μ y F i 1,j 1 μ x Gi+ 1,j+ 1 + μ x G i 1,j+ 1 μ x G i+ 1,j 1 μ x G i 1,j 1 t B 1 i,j = 1 4Δy μ xf 6 i,j+1 μ x F 6 i,j 1 1 4Δy t B i,j = 1 4Δx μ yg 5 i+1,j μ y G 5 i 1,j + 1 4Δx, δ y μ x G 5 i+ 1,j+ 1 + μ xg 5 i 1,j+ 1 δ x μ y F 6 i+ 1,j+ 1 + μ yf 6 i+ 1,j 1, Divergence preserving isotropic GMD scheme We efine a iagonal form of the potential χ: χ i+ 1,j+ 1 = 1 F G + i+ 1,j+ 1 5 i+ 1,j+ 1 + F 6 + G i+ 1,j+ 1 5 i+ 1,j for iagonal fluxes F ±, G ± efine in.8b. Denote F ± = F ± 1,...,F± 4, F± 7, F± 8, G± = G ± 1,...,G± 4, G± 7, G± 8. The potential-base moification of the isotropic GMD scheme.10 base on the potential 3.6 reas = 1 δ / F+ tũi,j i,j 4Δx +δ x F i,j + δ \ F i,j 1 4Δy t B 1 i,j = 1 F + μ x δ y 4Δy 6 i,j + F 6 i,j + G + 5 i,j + G 5 t B i,j = 1 F + μ y δ x 4Δx 6 i,j + F 6 i,j + G + 5 i,j + G 5 i,j δ / G+ i,j +δ y G i,j δ \ G i,j, i,j,. 3.7 Remark 3.4. Observe that the isotropic scheme 3.7 is not a potential-base scheme: the reuce vector Ũi,j is compute using the GMD isotropic fluxes. Instea, only the x- any-components of the magnetic fiel are evaluate using the iagonal-base potential χ i+ 1,j+ 1, consistent with 1.3b, 1.3c which in turn, imply the esire ivergence-preserving property. 4. Numerical results All the potential base GMD schemes escribe in the previous section are semi-iscrete. We efine a fully iscrete version of the first-orer GMD schemes by using stanar forwar Euler time integration. Secon-orer strong stability preserving Runge-Kutta metho [0] efines fully iscrete versions of the secon-orer accurate GMD schemes. The time step is etermine by a stanar CFL conition. A CFL number of 0.45 is use in all the subsequent simulations.

11 CONSTRAINT PRESERVING SCHEMES 671 a SYM b SCP c ISO ICP Figure 1. The pressure p for the Orszag-Tang vortex compute at t = π on a mesh with first-orer GMD schemes. 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.10. SCP SCP First secon-orer version of the ivergence preserving symmetric GMD scheme 3.5. ICP ICP First secon-orer version of the ivergence preserving isotropic GMD scheme Orszag-Tang vortex The Orszag-Tang vortex is a wiely reporte benchmark for multi-imensional MHD equations [44]. The initial ata is ρ, u 1,u,u 3,B 1,B,B 3,p= γ, siny, sinx, 0, siny, sinx, 0,γ, in the computational omain: x, y, t [0, π] [0,π] with perioic bounary conitions. Although the exact solution is not known, qualitative features have been reporte [44]. 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 1 an.

12 67 S. MISHRA AND E. TADMOR a SYM b SCP c ISO ICP Figure. The pressure p for the Orszag-Tang vortex compute at t = π on a mesh with secon-orer GMD schemes. Figure 1 shows the approximate pressure compute with the first-orer GMD schemes. The solution is smeare at this resolution, but the qualitative features are capture without any spurious oscillations an other numerical artifacts. The ivergence preserving SCP an ICP schemes are clearly more accurate than the SYM an ISO schemes, inicating that preserving a iscrete version of the constraint leas to a gain in accuracy. The results for the secon-orer schemes are plotte in Figure an show a consierable improvement in the resolution. The gain in accuracy is pronounce, 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. In the absence of an exact formula for the solution, the maximum pressure [17, 44] has been suggeste as a measure of accuracy. The maximum pressure at time t = π, compute on a sequence of meshes, is presente in Table 1. The table provies a quantitative comparison between the schemes an vinicates the conclusions from the plots. The gain in resolution with the secon-orer schemes is consierable. As the initial ata is ivergence free, the ivergence constraint 1.6 implies that the ivergence shoul remain zero uring the evolution. We show the errors in the iscrete ivergence operator iv 3.3a, measure in the

13 CONSTRAINT PRESERVING SCHEMES 673 Table 1. Maximum pressure for the Orszag-Tang vortex with all the GMD schemes on a M M mesh at time t = π. M SYM ISO SCP ICP SYM ISO SCP ICP Table. Discrete ivergence iv 3.3a inl 1 for the Orszag-Tang vortex with all the GMD schemes on a M M mesh at time t = π. M SYM ISO SCP ICP SYM ISO SCP ICP e e e e e e e e e e e e e-1 1.8e e e-1 Table 3. Discrete ivergence iv c 4.1 inl 1 for the Orszag-Tang vortex with all the GMD schemes on a M M mesh at time t = π. M SYM ISO SCP ICP SYM ISO SCP ICP L 1 norm, in Table. The table shows that the stanar GMD schemes lea to O1 ivergence errors. The ivergence error is larger for the secon-orer SYM an ISO schemes than the first-orer schemes. This is to be expecte as the secon-orer schemes resolve the shocks sharply. On the other han the SCP, SCP, ICP an ICP schemes preserve this iscrete ivergence to machine precision. It is natural to question whether controlling one iscrete version of ivergence, iv, will imply control of a ifferent iscrete version of the ivergence operator. The stanar central iscrete ivergence operator is efine by, iv c B 1,B 1 i,j := Δx δ xb 1 i,j + 1 Δy δ yb i,j. 4.1 Note that the central iscrete ivergence operator iv c iffers from the iscrete ivergence operator iv by OΔx + Δy provie that the unerlying magnetic fiel is smooth. However, since the magnetic fiel in this example has shocks an other iscontinuities, it is not clear that preserving iv will lea to some control in iv c. We explore this question by presenting errors in iv c at time t = π on a sequence of meshes in Table 3. A closer investigation of Table 3 an comparison between Tables an 3 reveal an interesting picture. As with iv errors, the errors in the iscrete ivergence iv c increase when the secon-orer schemes are use. Furthermore, the errors for both iscrete ivergence operators with the first- an secon-orer SYM an ISO schemes are of comparable magnitue. On the other han, controlling iv errors to machine precision oes not imply that the errors in iv c with the SCP an ICP schemes are very low. In fact, the errors are O1 ue to the formation of iscontinuities in this problem. However, the errors in iv c with the SCP an ICP schemes are consistently lower than the corresponing errors with the SYM an ISO schemes. The ifference ranges from a factor of about four for the secon-orer schemes to almost an orer of magnitue for the first-orer schemes. Furthermore, the errors with the SCP an

14 674 S. MISHRA AND E. TADMOR a SYM b SCP Figure 3. The iscrete ivergence iv c 4.1 for the Orszag-Tang vortex compute at t = π on a mesh with secon-orer SYM an SCP schemes. ICP schemes are reuce by ecreasing the mesh size inicating some convergence but at a very slow rate as the mesh is refine. In-orer to explain the above observations, we plot the iscrete ivergence iv c for the SYM an SCP schemes on a mesh in Figure 3. The figure clearly shows that errors in the iscrete ivergence operator iv c with the SCP scheme are create at shocks an sharp graients in the solution, see Figure for a comparison. However, the errors in the SYM scheme are not localize. The largest magnitue of error is still at the iscontinuities but there are large errors away from them. Figure 3 an Tables an 3 suggest that preserving one version of iscrete ivergence implies leas to some control on another version of iscrete ivergence in the smooth parts of the solution an localization of ivergence errors at shocks. Therefore, the overall ivergence error with the SCP an ICP schemes is lower than that of the SYM an ISO schemes. 4.. Rotor problem Another benchmark test for the MHD equations is the rotor problem [44]. The computational omain is x, y, t [0, 1] [0, 0.95] with artificial Neumann type bounary conitions. The initial ensity is 10.0 if r<0.1, ρ = 1+9fr if 0.1 r<0.115, 1.0 otherwise, with rx, y = x, y 0.5, 0.5 an fr = 3 00r 3 The other initial variables are, 10y 5ρ, 10x 5ρ if r<0.1, ρu 1,ρu = 10y 5frρ, 10x 5frρ if 0.1 r<0.115, 0.0, 0.0 otherwise, ρu 3,B 1,B,B 3,p= 0.0,.5/ π,0.0, 0.0, 0.5.

15 CONSTRAINT PRESERVING SCHEMES 675 a SYM b ISO c SCP ICP Figure 4. The pressure p for the rotor problem compute at t =0.95 on a mesh with first-orer schemes. The initial velocity an magnetic fiels are such that the variables are rotate in the omain. The pressure rops to very low values in the center, an this test case is set up in orer to etermine how a scheme hanles low pressures. The approximate pressure compute with the first-orer GMD schemes, on a mesh, is shown in Figure 4. The figure shows that all the schemes are stable at this resolution an the low pressure at the center is resolve. The first-orer schemes are iffusive. The ivergence preserving SCP an ICP schemes are more accurate in this case. The results for the secon-orer schemes areplotteinfigure5. They reveal a significant gain in resolution with the secon-orer schemes, particularly at shocks. The errors in the iscrete ivergence iv are isplaye in Table 4. The ivergence errors generate by the SYM an ISO schemes an their secon-orer versions are again O1. These errors increase with increasing resolution, i.e., either by reucing mesh size or by increasing the orer of accuracy of the scheme, inicating that the bulk of the ivergence errors are generate near the shocks. The SCP, ICP, SCP an ICP schemes preserve iscrete ivergence to machine precision.

16 676 S. MISHRA AND E. TADMOR Table 4. Discrete ivergence iv 3.3a inl 1 for the rotor problem with all the eight schemes on a M M mesh at time t =0.95. M SYM ISO SCP ICP SYM ISO SCP ICP e e e e e e e e e e e e e-13.49e e e-13 a SYM b ISO c SCP ICP Figure 5. The pressure p for the rotor problem compute at t =0.95 on a mesh with secon-orer schemes Clou-Shock interaction The next 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 [38] consists of a shock locate at x = 0.05 with ρ, u 1,u,u 3,B 1,B,B 3,p = { , , 0, 0, 0,.18618,.18618, , if x< , 0, 0, 0, 0, , , 1.0, if x<

17 CONSTRAINT PRESERVING SCHEMES 677 a SYM b ISO c SCP ICP Figure 6. The ensity ρ for the clou-shock interaction compute at t = 0.06 on a mesh with first-orer schemes. an a circular clou of ensity ρ = 10 with raius 0.15, centere at x, y =0.5, 0.5. The computational omain is [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 features in the center. We plot the approximate ensity, on a mesh, at time t =0.06 in Figures 6 an 7. The first-orer results in Figure 6 show that the first-orer GMD schemes are stable but quite iffusive. The ivergence preserving SCP an ICP schemes are again more accurate than the SYM an ISO schemes. The secon-orer results are plotte in Figure 7 an show a ramatic increase in resolution. Both the bow shock an the trailing shock are capture accurately. The smooth region with turbulent like features is also resolve quite well. The ivergence errors for iscrete ivergence iv are shown in Table 5. The table shows large ivergence errors for the SYM SYM an ISO ISO schemes. On the other han, the constraint preserving SCP SCP an ICP ICP schemes preserve iscrete ivergence to machine precision. Remark 4.1. The first-orer GMD schemes were quite iffusive. A possible reason is the use of the Rusanov flux 1.9. This flux is known to prouce excessive smearing at the shocks. However, we avocate the use of the

18 678 S. MISHRA AND E. TADMOR Table 5. Discrete ivergence iv 3.3a inl 1 for clou shock interaction with all the eight schemes on a M M mesh at time t =0.06. 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 Figure 7. The ensity ρ for the clou-shock interaction compute at t = 0.06 on a mesh with secon-orer schemes. Rusanov flux as the accuracy is recovere at secon-orer. The Rusanov flux is very easy to implement, uses minimal characteristic information an has a low computational cost. It fits into the black box framework of our GMD schemes. The three numerical experiments show that the GMD schemes are quite robust. There oes not appear to be a strong connection between the ivergence errors an stability of a GMD scheme. The GMD structure of the

19 CONSTRAINT PRESERVING SCHEMES 679 schemes incorporates stability. However, there is a gain in accuracy at least at first-orer when the ivergence preserving versions of the scheme are use. Physicists are generally reluctant to use numerical schemes that prouce ivergence errors. Hence, we avocate the use of the ivergence preserving GMD schemes. Furthermore, the computational cost of a ivergence preserving GMD scheme is virtually ientical to the cost of other GMD schemes. 5. Conclusion The ieal MHD equations 1.1a are consiere. The equations are non-strictly hyperbolic an posses a complex shock structure. Design of stable an accurate numerical methos for the MHD equations in multiimensions is complicate on account of its genuinely multi-imensional structure an the ivergence constraint. We exten the potential base GMD framework of recent papers [9, 30] to the MHD equations. The finite volume schemes are formulate in terms of vertex centere numerical potentials. Symmetric.5 an isotropic.10 versions of the potential base GMD schemes are escribe. The GMD schemes are moifie with a suitable choice of potentials to yiel ivergence preserving GMD schemes. Secon-orer versions are obtaine by employing non-oscillatory piecewise bilinear reconstructions. The schemes are constraint preserving GMD extensions of the central schemes of Kurganov an Tamor [3]. Benchmark numerical experiments for the MHD equations are presente. They show that the first-orer GMD schemes resolve the waves with some iffusion. There is a gain in accuracy when the ivergence preserving versions are use. The gain in resolution with the secon-orer schemes is consierable. The multi-imensional shocks are vortices are capture, with goo accuracy. The non ivergence preserving versions of the GMD schemes can generate large ivergence errors, particularly at shocks. These errors o not seem to affect the stability of the schemes, at least in our tests. But large ivergence errors might create instabilities at finer resolutions. Hence, we avocate using the ivergence preserving versions of the GMD schemes. The GMD approach is simple to implement, robust an has a very low computational cost. It will be extene to higher than secon-orer of accuracy an to unstructure meshes in future papers. Other future projects inclue using the ivergence preserving GMD schemes to compute realistic flows in solar physics an astrophysics. References [1] R. Artebrant an M. Torrilhon, Increasing the accuracy of local ivergence preserving schemes for MHD. J. Comput. Phys [] J. Bálbas an E. Tamor, Non-oscillatory central schemes for one an two-imensional magnetohyroynamics II: High-orer semi-iscrete schemes. SIAM.J.Sci.Comput [3] J. Bálbas, E. Tamor an C.C. Wu, Non-oscillatory central schemes for one an two-imensional magnetohyroynamics I. J. Comput. Phys [4] D.S. Balsara, Divergence free aaptive mesh refinement for magnetohyroynamics. J. Comput. Phys [5] D.S. Balsara an D. Spicer, A staggere mesh algorithm using high orer Gounov fluxes to ensure solenoial magnetic fiels in magnetohyroynamic simulations. J. Comput. Phys [6] J.B. Bell, P. Colella an H.M. Glaz, A secon-orer projection metho for the incompressible Navier-Stokes equations. J. Comput. Phys [7] F. Bouchut, C. Klingenberg an K. Waagan, A multi-wave HLL approximate Riemann solver for ieal MHD base on relaxation I- theoretical framework. Numer. Math [8] J.U. Brackbill an D.C. Barnes, The effect of nonzero DivB on the numerical solution of the magnetohyroynamic equations. J. Comput. Phys [9] M. Brio an C.C. Wu, An upwin ifferencing scheme for the equations of ieal MHD. J. Comput. Phys [10] A.J. Chorin, Numerical solutions of the Navier-Stokes equations. Math. Comput [11] W. Dai an P.R. Woowar, A simple finite ifference scheme for multi-imensional magnetohyroynamic equations. J. Comput. Phys [1] H. Deconnik, P.L. Roe an R. Struijs, A multi-imensional generalization of Roe s flux ifference splitter for Euler equations. Comput. Fluis [13] A. Dener, F. Kemm, D. Kröner, C.D. Munz, T. Schnitzer an M. Wesenberg, Hyperbolic ivergence cleaning for the MHD equations. J. Comput. Phys

20 680 S. MISHRA AND E. TADMOR [14] C. Evans an J.F. Hawley, Simulation of magnetohyroynamic flow: a constraine transport metho. Astrophys. J [15] M. Fey, Multi-imensional upwinging.i The metho of transport for solving the Euler equations. J. Comput. Phys [16] M. Fey, Multi-imensional upwinging.ii Decomposition of Euler equations into avection equations. J. Comput. Phys [17] F. Fuchs, S. Mishra an N.H. Risebro, Splitting base finite volume schemes for ieal MHD equations. J. Comput. Phys [18] F. Fuchs, A. McMurry, S. Mishra, N.H. Risebro an K. Waagan, Finite volume methos for wave propagation in stratifie magneto-atmospheres. Commun. Comput. Phys [19] F. Fuchs, A.D. McMurry, S. Mishra, N.H. Risebro an K. Waagan, Approximate Riemann solver an robust high-orer finite volume schemes for the MHD equations in multi-imensions. Commun. Comput. Phys [0] S. Gottlieb, C.W. Shu an E. Tamor, High orer time iscretizations with strong stability property. SIAM. Rev [1] K.F. Gurski, An HLLC-type approximate Riemann solver for ieal Magneto-hyro ynamics. SIAM.J.Sci.Comput [] A. Harten, B. Engquist, S. Osher an S.R. Chakravarty, Uniformly high orer accurate essentially non-oscillatory schemes. J. Comput. Phys [3] A. Kurganov an E. Tamor, New high resolution central schemes for non-linear conservation laws an convection-iffusion equations. J. Comput. Phys [4] R.J. LeVeque, Wave propagation algorithms for multi-imensional hyperbolic systems, J. Comput. Phys [5] R.J. LeVeque, Finite volume methos for hyperbolic problems. Cambrige university press, Cambrige 00. [6] T.J. Line, A three aaptive multi flui MHD moel for the heliosphere. Ph.D. thesis, University of Michigan, Ann-Arbor [7] M. Lukacova-Meviova, K.W. Morton an G. Warnecke, Evolution Galerkin methos for Hyperbolic systems in two space imensions. Math. Comput [8] M. Lukacova-Meviova, J. Saibertova an G. Warnecke, Finite volume evolution Galerkin methos for Non-linear hyperbolic systems. J. Comput. Phys [9] S. Mishra an E. Tamor, Constraint preserving schemes using potential-base fluxes. I. Multi-imensional transport equations. Commun. Comput. Phys [30] S. Mishra an E. Tamor, Constraint preserving schemes using potential-base fluxes. II. Genuinely multi-imensional systems of conservation laws. SIAM J. Numer. Anal [31] A. Mignone et al., Pluto: A numerical coe for computational astrophysics. Astrophys. J. Suppl [3] T. Miyoshi an K. Kusano, A multi-state HLL approximate Riemann solver for ieal magneto hyro ynamics. J. Comput. Phys [33] H. Nessyahu an E. Tamor, Non-oscillatory central ifferencing for hyperbolic conservation laws. J. Comput. Phys [34] S. Noelle, The MOT-ICE: A new high-resolution wave propagation algorithm for multi-imensional systems of conservation laws base on Fey s metho of transport. J. Comput. Phys [35] K.G. Powell, An approximate Riemann solver for magneto-hyro ynamics that works in more than one space imension. Technical report, ICASE, Langley, VA [36] K.G. Powell, P.L. Roe, T.J. Line, T.I. Gombosi an D.L. De zeeuw, A solution aaptive upwin scheme for ieal MHD. J. Comput. Phys [37] P.L. Roe an D.S. Balsara, Notes on the eigensystem of magnetohyroynamics. SIAM. J. Appl. Math [38] J. Rossmanith, A wave propagation metho with constraine transport for shallow water an ieal magnetohyroynamics. Ph.D. thesis, University of Washington, Seattle 00. [39] D.S. Ryu, F. Miniati, T.W. Jones an A. Frank, A ivergence free upwin coe for multiimensional magnetohyroynamic flows. Astrophys. J [40] C.W. Shu an S. Osher, Efficient implementation of essentially non-oscillatory schemes II. J. Comput. Phys [41] E. Tamor, Approximate solutions of nonlinear conservation laws, in Avance Numerical approximations of Nonlinear Hyperbolic equations, eite by A. Quarteroni. Lecture notes in Mathematics, Springer Verlag [4] M. Torrilhon, Locally ivergence preserving upwin finite volume schemes for magnetohyroynamic equations. SIAM. J. Sci. Comput [43] M. Torrilhon an M. Fey, Constraint-preserving upwin methos for multiimensional avection equations. SIAM.J.Numer. Anal [44] G. Toth, The DivB = 0 constraint in shock capturing magnetohyroynamics coes. J. Comput. Phys [45] B. van Leer, Towars the ultimate conservative ifference scheme, V. A secon orer sequel to Gounov s metho. J. Comput. Phys