The exact solution of the Riemann problem in relativistic magnetohydrodynamics

We discuss the procedure for the exact solution of the Rieman n problem in special relativistic magnetohydrodynamics (MHD). We consider both initial stat es leading to a set of only three waves analogous to the ones in relativistic hydrodynamics, as well as generic initial states leading to the full set of seven MHD waves. Because of its generality, the solution presented here could serve as an important test for those numerical codes solving the MHD equations in relativistic regimes†.

[1]  A. H. Taub,et al.  Relativistic Rankine-Hugoniot Equations , 1948 .

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

[3]  Manuel Torrilhon,et al.  Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics , 2003 .

[4]  Relativistic magnetohydrodynamics in dynamical spacetimes: Numerical methods and tests , 2005, astro-ph/0503420.

[5]  Under consideration for publication in J. Fluid Mech. 1 An Improved Exact Riemann Solver for Multidimensional Relativistic Flows , 2008 .

[6]  G. R.S.MYON.  Shock waves and rarefaction waves in magnetohydrodynamics . Part 1 . A model system , 2022 .

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

[8]  S. Komissarov,et al.  On the inadmissibility of non-evolutionary shocks , 1999 .

[9]  S. Komissarov,et al.  On the properties of Alfvn waves in relativistic magnetohydrodynamics , 1997 .

[10]  Ewald Müller,et al.  The analytical solution of the Riemann problem in relativistic hydrodynamics , 1994, Journal of Fluid Mechanics.

[11]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[12]  General relativistic magnetohydrodynamic simulations of monopole magnetospheres of black holes , 2004, astro-ph/0402430.

[13]  Manuel Torrilhon,et al.  Non-uniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamics , 2003 .

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

[15]  P. Roe,et al.  Shock waves and rarefaction waves in magnetohydrodynamics. Part 1. A model system , 1997, Journal of Plasma Physics.

[16]  J. A. Miralles,et al.  Numerical 3+1 General Relativistic Magnetohydrodynamics: A Local Characteristic Approach , 1997 .

[17]  S. Komissarov,et al.  A Godunov-type scheme for relativistic magnetohydrodynamics , 1999 .

[18]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[19]  M. Putten.  A numerical implementation of MHD in divergence form , 1993 .

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

[21]  L. Rezzolla,et al.  New relativistic effects in the dynamics of nonlinear hydrodynamical waves. , 2002, Physical Review Letters.

[22]  Stuart L. Shapiro,et al.  Relativistic hydrodynamic evolutions with black hole excision , 2004 .

[23]  J. Font,et al.  Quasi-periodic accretion and gravitational waves from oscillating 'toroidal neutron stars' around a Schwarzschild black hole , 2002, gr-qc/0210018.

[24]  Charles F. Gammie,et al.  HARM: A NUMERICAL SCHEME FOR GENERAL RELATIVISTIC MAGNETOHYDRODYNAMICS , 2003 .

[25]  L. Baiotti,et al.  Three-dimensional relativistic simulations of rotating neutron-star collapse to a Kerr black hole , 2004, gr-qc/0403029.

[26]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[27]  Masaru Shibata,et al.  Three-dimensional simulations of stellar core collapse in full general relativity: Nonaxisymmetric dynamical instabilities , 2005 .

[28]  B R U N O G I A C O M A Z Z O,et al.  Under consideration for publication in J. Fluid Mech. 1 The Exact Solution of the Riemann Problem in Relativistic MHD , 2008 .

[29]  J. Font,et al.  The runaway instability of thick discs around black holes – I. The constant angular momentum case , 2002, astro-ph/0203403.

[30]  General Relativistic Magnetohydrodynamic Simulations of Collapsars , 2004, astro-ph/0404152.

[31]  L U C I A N O R E Z Z O L L A,et al.  Under consideration for publication in J. Fluid Mech. 1 An Improved Exact Riemann Solver for Relativistic Hydrodynamics , 2008 .

[32]  A. Taub.  Relativistic Fluid Mechanics , 1978 .

[33]  Manuel Torrilhon,et al.  High order WENO schemes: investigations on non-uniform converges for MHD Riemann problems , 2004 .

[34]  The exact solution of the Riemann problem with non-zero tangential velocities in relativistic hydrodynamics , 2000, Journal of Fluid Mechanics.

[35]  L. Smarr,et al.  GENERAL RELATIVISTIC MAGNETOHYDRODYNAMICS , 1983 .

[36]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[37]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[38]  S. A. E. G. Falle,et al.  A multidimensional upwind scheme for magnetohydrodynamics , 1998 .

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

[40]  Paul R. Woodward,et al.  An approximate Riemann solver for ideal magnetohydrodynamics , 1994 .

[41]  A. Jefrey,et al.  Non-Linear Wave Propagation , 1964 .

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

[43]  J. Hawley,et al.  A Numerical Method for General Relativistic Magnetohydrodynamics , 2002, astro-ph/0210518.

[44]  INFN,et al.  The exact solution of the Riemann problem in relativistic magnetohydrodynamics , 2005, Journal of Fluid Mechanics.