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Arbitrary high-order discontinuous Galerkin schemes for the magnetohydrodynamic equations. (English) Zbl 1176.76075
Summary: We propose a discontinuous Galerkin scheme with arbitrary order of accuracy in space and time for magnetohydrodynamic equations. It is based on the Arbitrary order using DERivatives (ADER) methodology: the high order time approximation is obtained by a Taylor expansion in time. In this expansion, all the time derivatives are replaced by space derivatives via the Cauchy-Kovalevskaya procedure. We propose an efficient algorithm of the Cauchy-Kovalevskaya procedure in the case of the three-dimensional magneto-hydrodynamic (MHD) equations. Parallel to the time derivatives of the conservative variables, the time derivatives of the fluxes are calculated. This enables analytic time integration of the volume integral as well as the surface integral of the fluxes through the grid cell interfaces which occur in discrete equations. At the cell interfaces, the fluxes and all their derivatives may jump. Following the finite volume ADER approach the break up of all these jumps into the different waves are taken into account to get proper values of the fluxes at the grid cell interfaces. The approach under considerations is directly based on the expansion of the flux in time in which the leading order term may be any numerical flux calculation for the MHD-equation. Numerical convergence results for these equations up to 7th order of accuracy in space and time are presented.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76W05 Magnetohydrodynamics and electrohydrodynamics
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##### References:
 [1] Dumbser M., Munz C.-D. (2005). Building blocks for arbitrary high order discontinuous Galerkin schemes. J. Sci. Comput. ISSN: 0885-7474 (Paper) 1573-7691 (Online), DOI: 10.1007/s10915-005-9025-0, Issue: Online First. · Zbl 1210.65165 [2] Toro E., Titarev V. (2003). Solution of the generalized Riemann problem for advection-reaction equations. Proc. Roy. Soc. A: Mathematical, Physical and Engineering Sciences. 458 (2018): 271–281 · Zbl 1019.35061 [3] Titarev V.A., Toro E.F. (2005). ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys. 204: 715–736 · Zbl 1060.65641 [4] Becker J. (2000). Entwicklung eines effizineten Verfahrens zur LĂ¶sung hyperbolischer Differentialgleichungen. http://www.freidok.uni-freiburg.de/volltexte/123 (October 2000) [5] Dumbser M. (2005). Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains. Shaker Verlag, Aachen [6] Cockburn B., Shu C.W. (1989). TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52: 411–435 · Zbl 0662.65083 [7] Cockburn B., Shu C.W. (1998). The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Computat. Phys. 141: 199–224 · Zbl 0920.65059 [8] Stroud A.H. (1971). Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, NJ · Zbl 0379.65013 [9] Toro E.F., Millington R.C., and Nejad L. A. M. (2001). Towards very high order Godunov schemes, in: Godunov Methods. Theory and Applications, Kluwer/Plenum Academic Publishers, pp. 905–938. · Zbl 0989.65094 [10] Titarev V.A., Toro E.F. (2002). ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17(1–4): 609–618 · Zbl 1024.76028 [11] Dyson R. W. (2001). Technique for very high order nonlinear simulation and validation. Tech. Rep. TM-2001-210985, NASA. [12] Balsara D.S. (1998). Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics. Astrophys. J. Suppl. Ser. 116: 119–131 [13] Cargo P., Gallice G. (1997). Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws. J. Comput. Phys. 136(2): 446–466 · Zbl 0919.76053 [14] Roe P.L., Balsara D.S. (1996). Notes on the eigensystem of magnetohydrodynamics. IMA J. Appl. Math. 56(1): 57–67 · Zbl 0845.35092 [15] Balsara D.S. (2001). Total variation diminishing scheme for relativistic magnetohydrodynamics. Ap. J. Supp. 132: 1 · Zbl 1157.76369 [16] Ryu D., Jones T.W. (1995). Numerical magnetohydrodynamics in astrophysics: algorithm and tests for one-dimensional flow. Astrophys J. 442: 228–258 [17] Dai W., Woodward P. R. (1994). Extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics. J. Comput. Phys. 115(2): 485–514 · Zbl 0813.76058 [18] Balsara D. S., A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes. ArXiv Physics e-prints arXiv:physics/0411160, Provided by the Smithsonian/NASA Astrophysics Data · Zbl 1124.65072 [19] Qiu J., Shu C.W. (2003). Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys. 193: 115–135 · Zbl 1039.65068
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