Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations
ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 661-680.

We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688-710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023-1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.

DOI : 10.1051/m2an/2011059
Classification : 65M06, 35L65
Mots-clés : multidimensional evolution equations, magnetohydrodynamics, constraint transport, central difference schemes, potential-based fluxes 
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Mishra, Siddhartha; Tadmor, Eitan. Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations. ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 661-680. doi : 10.1051/m2an/2011059. http://www.numdam.org/articles/10.1051/m2an/2011059/

[1] R. Artebrant and M. Torrilhon, Increasing the accuracy of local divergence preserving schemes for MHD. J. Comput. Phys. 227 (2008) 3405-3427. | MR

[2] J. Bálbas and E. Tadmor, Non-oscillatory central schemes for one and two-dimensional magnetohydrodynamics II : High-order semi-discrete schemes. SIAM. J. Sci. Comput. 28 (2006) 533-560. | Zbl

[3] J. Bálbas, E. Tadmor and C.C. Wu, Non-oscillatory central schemes for one and two-dimensional magnetohydrodynamics I. J. Comput. Phys. 201 (2004) 261-285. | Zbl

[4] D.S. Balsara, Divergence free adaptive mesh refinement for magnetohydrodynamics. J. Comput. Phys. 174 (2001) 614-648. | Zbl

[5] D.S. Balsara and D. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149 (1999) 270-292. | MR | Zbl

[6] J.B. Bell, P. Colella and H.M. Glaz, A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85 (1989) 257-283. | MR | Zbl

[7] F. Bouchut, C. Klingenberg and K. Waagan, A multi-wave HLL approximate Riemann solver for ideal MHD based on relaxation I- theoretical framework. Numer. Math. 108 (2007) 7-42. | MR | Zbl

[8] J.U. Brackbill and D.C. Barnes, The effect of nonzero DivB on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35 (1980) 426-430. | MR | Zbl

[9] M. Brio and C.C. Wu, An upwind differencing scheme for the equations of ideal MHD. J. Comput. Phys. 75 (1988) 400-422. | MR | Zbl

[10] A.J. Chorin, Numerical solutions of the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. | Zbl

[11] W. Dai and P.R. Woodward, A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comput. Phys. 142 (1998) 331-369. | MR | Zbl

[12] H. Deconnik, P.L. Roe and R. Struijs, A multi-dimensional generalization of Roe's flux difference splitter for Euler equations. Comput. Fluids 22 (1993) 215. | Zbl

[13] A. Dedner, F. Kemm, D. Kröner, C.D. Munz, T. Schnitzer and M. Wesenberg, Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175 (2002) 645-673. | MR | Zbl

[14] C. Evans and J.F. Hawley, Simulation of magnetohydrodynamic flow : a constrained transport method. Astrophys. J. 332 (1998) 659.

[15] M. Fey, Multi-dimensional upwingding. (I) The method of transport for solving the Euler equations. J. Comput. Phys. 143 (1998) 159-180. | Zbl

[16] M. Fey, Multi-dimensional upwingding.(II) Decomposition of Euler equations into advection equations. J. Comput. Phys. 143 (1998) 181-199. | Zbl

[17] F. Fuchs, S. Mishra and N.H. Risebro, Splitting based finite volume schemes for ideal MHD equations. J. Comput. Phys. 228 (2009) 641-660. | MR | Zbl

[18] F. Fuchs, A. Mcmurry, S. Mishra, N.H. Risebro and K. Waagan, Finite volume methods for wave propagation in stratified magneto-atmospheres. Commun. Comput. Phys. 7 (2010) 473-509. | MR | Zbl

[19] F. Fuchs, A.D. Mcmurry, S. Mishra, N.H. Risebro and K. Waagan, Approximate Riemann solver and robust high-order finite volume schemes for the MHD equations in multi-dimensions. Commun. Comput. Phys. 9 (2011) 324-362. | MR

[20] S. Gottlieb, C.W. Shu and E. Tadmor, High order time discretizations with strong stability property. SIAM. Rev. 43 (2001) 89-112. | MR | Zbl

[21] K.F. Gurski, An HLLC-type approximate Riemann solver for ideal Magneto-hydro dynamics. SIAM. J. Sci. Comput. 25 (2004) 2165-2187. | MR | Zbl

[22] A. Harten, B. Engquist, S. Osher and S.R. Chakravarty, Uniformly high order accurate essentially non-oscillatory schemes. J. Comput. Phys. 71 (1987) 231-303. | MR | Zbl

[23] A. Kurganov and E. Tadmor, New high resolution central schemes for non-linear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. | MR | Zbl

[24] R.J. Leveque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys. 131 (1997) 327-353. | Zbl

[25] R.J. Leveque, Finite volume methods for hyperbolic problems. Cambridge university press, Cambridge (2002). | MR | Zbl

[26] T.J. Linde, A three adaptive multi fluid MHD model for the heliosphere. Ph.D. thesis, University of Michigan, Ann-Arbor (1998).

[27] M. Lukacova-Medvidova, K.W. Morton and G. Warnecke, Evolution Galerkin methods for Hyperbolic systems in two space dimensions. Math. Comput. 69 (2000) 1355-1384. | MR | Zbl

[28] M. Lukacova-Medvidova, J. Saibertova and G. Warnecke, Finite volume evolution Galerkin methods for Non-linear hyperbolic systems. J. Comput. Phys. 183 (2003) 533-562. | MR | Zbl

[29] S. Mishra and E. Tadmor, Constraint preserving schemes using potential-based fluxes. I. Multi-dimensional transport equations. Commun. Comput. Phys. 9 (2010) 688-710. | MR

[30] S. Mishra and E. Tadmor, Constraint preserving schemes using potential-based fluxes. II. Genuinely multi-dimensional systems of conservation laws. SIAM J. Numer. Anal. 49 (2011) 1023-1045. | MR | Zbl

[31] A. Mignone et al., Pluto : A numerical code for computational astrophysics. Astrophys. J. Suppl. 170 (2007) 228-242.

[32] T. Miyoshi and K. Kusano, A multi-state HLL approximate Riemann solver for ideal magneto hydro dynamics. J. Comput. Phys. 208 (2005) 315-344. | MR | Zbl

[33] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | MR | Zbl

[34] S. Noelle, The MOT-ICE : A new high-resolution wave propagation algorithm for multi-dimensional systems of conservation laws based on Fey's method of transport. J. Comput. Phys. 164 (2000) 283-334. | MR | Zbl

[35] K.G. Powell, An approximate Riemann solver for magneto-hydro dynamics (that works in more than one space dimension). Technical report, ICASE, Langley, VA (1994) 94-24.

[36] K.G. Powell, P.L. Roe, T.J. Linde, T.I. Gombosi and D.L. De zeeuw, A solution adaptive upwind scheme for ideal MHD. J. Comput. Phys. 154 (1999) 284-309. | Zbl

[37] P.L. Roe and D.S. Balsara, Notes on the eigensystem of magnetohydrodynamics. SIAM. J. Appl. Math. 56 (1996) 57-67. | MR | Zbl

[38] J. Rossmanith, A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics. Ph.D. thesis, University of Washington, Seattle (2002). | MR

[39] D.S. Ryu, F. Miniati, T.W. Jones and A. Frank, A divergence free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J. 509 (1998) 244-255.

[40] C.W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory schemes - II. J. Comput. Phys. 83 (1989) 32-78. | MR | Zbl

[41] E. Tadmor, Approximate solutions of nonlinear conservation laws, in Advanced Numerical approximations of Nonlinear Hyperbolic equations, edited by A. Quarteroni. Lecture notes in Mathematics, Springer Verlag (1998) 1-149. | MR | Zbl

[42] M. Torrilhon, Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comput. 26 (2005) 1166-1191. | MR | Zbl

[43] M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations. SIAM. J. Numer. Anal. 42 (2004) 1694-1728. | MR | Zbl

[44] G. Toth, The DivB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comput. Phys. 161 (2000) 605-652. | MR | Zbl

[45] B. Van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov's method. J. Comput. Phys. 32 (1979) 101-136. | MR | Zbl

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