A high-order kinetic flux-splitting method for the relativistic magnetohydrodynamics

In this paper we extend the special relativistic hydrodynamic (SRHD) equations [L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, New York, 1987] and as a limiting case the ultra-relativistic hydrodynamic equations [M. Kunik, S. Qamar, G. Warnecke, J. Comput. Phys. 187 (2003) 572-596] to the special relativistic magnetohydrodynamics (SRMHD). We derive a flux splitting method based on gas-kinetic theory in order to solve these equations in one space dimension. The scheme is based on the direct splitting of macroscopic flux functions with consideration of particle transport. At the same time, particle ''collisions'' are implemented in the free transport process to reduce numerical dissipation. To achieve high-order accuracy we use a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. For the direct comparison of the numerical results, we also solve the SRMHD equations with the well-developed second-order central schemes. The 1D computations reported in this paper have comparable accuracy to the already published results. The results verify the desired accuracy, high resolution, and robustness of the kinetic flux splitting method and central schemes.

[1]  Jean-Pierre Croisille,et al.  Numerical simulation of the MHD equations by a kinetic-type method , 1995 .

[2]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[3]  I. Mirabel,et al.  A superluminal source in the Galaxy , 1994, Nature.

[4]  P. Londrillo,et al.  High-Order Upwind Schemes for Multidimensional Magnetohydrodynamics , 1999, astro-ph/9910086.

[5]  F. Jüttner.  Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie , 1911 .

[6]  E. Muller,et al.  GENESIS: A High-Resolution Code for Three-dimensional Relativistic Hydrodynamics , 1999, astro-ph/9903352.

[7]  M. Brio,et al.  An upwind differencing scheme for the equations of ideal magnetohydrodynamics , 1988 .

[8]  M. Norman,et al.  ZEUS-2D : a radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II : The magnetohydrodynamic algorithms and tests , 1992 .

[9]  José A. Font,et al.  Morphology and Dynamics of Highly Supersonic Relativistic Jets , 1995 .

[10]  C. Richard DeVore,et al.  Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics , 1989 .

[11]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[12]  Gerald Warnecke,et al.  Kinetic schemes for the ultra-relativistic Euler equations , 2003 .

[13]  Angelo Marcello Anile,et al.  Relativistic fluids and magneto-fluids , 2005 .

[14]  Dinshaw Balsara,et al.  Divergence-free adaptive mesh refinement for Magnetohydrodynamics , 2001 .

[15]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[16]  André Lichnerowicz,et al.  Relativistic Hydrodynamics And Magnetohydrodynamics , 1967 .

[17]  M. Rees,et al.  Tidal Heating and Mass Loss in Neutron Star Binaries: Implications for Gamma-Ray Burst Models , 1992 .

[18]  P. Londrillo,et al.  An efficient shock-capturing central-type scheme for multidimensional relativistic flows. II. Magnetohydrodynamics , 2002 .

[19]  Antonio Marquina,et al.  Capturing Shock Reflections , 1996 .

[20]  Paul R. Woodward,et al.  Extension of the Piecewise Parabolic Method to Multidimensional Ideal Magnetohydrodynamics , 1994 .

[21]  Claus-Dieter Munz,et al.  New Algorithms for Ultra-relativistic Numerical Hydrodynamics , 1993 .

[22]  Kun Xu,et al.  Gas-Kinetic Theory-Based Flux Splitting Method for Ideal Magnetohydrodynamics , 1999 .

[23]  Kun Xu,et al.  A high-order gas-kinetic method for multidimensional ideal magnetohydrodynamics , 2000 .

[24]  A. M. Anile,et al.  Fluid models for relativistic electron beams , 1989 .

[25]  M. Norman,et al.  ZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. I - The hydrodynamic algorithms and tests. II - The magnetohydrodynamic algorithms and tests , 1992 .

[26]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[27]  N. Bucciantini,et al.  An efficient shock-capturing central-type scheme for multidimensional relativistic flows , 2002 .

[28]  Shamsul Qamar,et al.  A BGK-Type Flux-Vector Splitting Scheme for the Ultrarelativistic Euler Equations , 2004, SIAM J. Sci. Comput..

[29]  Martin J. Rees,et al.  Theory of extragalactic radio sources , 1984 .

[30]  Shamsul Qamar,et al.  Kinetic schemes for the relativistic gas dynamics , 2004, Numerische Mathematik.

[31]  Dinshaw S. Balsara,et al.  Notes on the Eigensystem of Magnetohydrodynamics , 1996, SIAM J. Appl. Math..

[32]  Francesco Miniati,et al.  A Divergence-free Upwind Code for Multidimensional Magnetohydrodynamic Flows , 1998 .

[33]  J. Hawley,et al.  Simulation of magnetohydrodynamic flows: A Constrained transport method , 1988 .

[34]  G. Mellema,et al.  General Relativistic Hydrodynamics with a Roe solver , 1994, astro-ph/9411056.

[35]  Dinshaw Balsara,et al.  Second-Order-accurate Schemes for Magnetohydrodynamics with Divergence-free Reconstruction , 2003, astro-ph/0308249.

[36]  Dinshaw S. Balsara,et al.  Total Variation Diminishing Scheme for Relativistic Magnetohydrodynamics , 2001 .