A five‐wave Harten–Lax–van Leer Riemann solver for relativistic magnetohydrodynamics

We present a five-wave Riemann solver for the equations of ideal relativistic magneto-hydrodynamics. Our solver can be regarded as a relativistic extension of the five-wave HLLD Riemann solver initially developed by Miyoshi & Kusano for the equations of ideal magnetohydrodynamics. The solution to the Riemann problem is approximated by a five-wave pattern, comprising two outermost fast shocks, two rotational discontinuities and a contact surface in the middle. The proposed scheme is considerably more elaborate than in the classical case since the normal velocity is no longer constant across the rotational modes. Still, proper closure to the Rankine–Hugoniot jump conditions can be attained by solving a non-linear scalar equation in the total pressure variable which, for the chosen configuration, has to be constant over the whole Riemann fan. The accuracy of the new Riemann solver is validated against one-dimensional tests and multidimensional applications. It is shown that our new solver considerably improves over the popular Harten–Lax–van Leer solver or the recently proposed HLLC schemes.

[1]  P. Colella,et al.  Local adaptive mesh refinement for shock hydrodynamics , 1989 .

[2]  William H. Press,et al.  Numerical Recipes in C, 2nd Edition , 1992 .

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

[4]  A. Ferrari,et al.  PLUTO: A Numerical Code for Computational Astrophysics , 2007, astro-ph/0701854.

[5]  Pekka Janhunen,et al.  HLLC solver for ideal relativistic MHD , 2007, J. Comput. Phys..

[6]  W. F. Noh.  Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux , 1985 .

[7]  R. Keppens,et al.  Extragalactic jets with helical magnetic fields: relativistic MHD simulations , 2008, 0802.2034.

[8]  G. Bodo,et al.  An HLLC Riemann solver for relativistic flows – II. Magnetohydrodynamics , 2006 .

[9]  William H. Press,et al.  Numerical recipes in C , 2002 .

[10]  J. Pons,et al.  The exact solution of the Riemann problem in relativistic magnetohydrodynamics with tangential magnetic fields , 2005, Journal of Fluid Mechanics.

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

[12]  Relativistic MHD Simulations of Jets with Toroidal Magnetic Fields , 2005, astro-ph/0511769.

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

[14]  Rony Keppens,et al.  A multidimensional grid-adaptive relativistic magnetofluid code , 2008, Comput. Phys. Commun..

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

[16]  A. Mignone,et al.  Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index , 2007, 0704.1679.

[17]  O. A. Kuznetsov,et al.  An approximate Riemann solver for relativistic magnetohydrodynamics , 2002 .

[18]  M. Aloy,et al.  Relativistic MHD simulations of extragalactic jets , 2005 .

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

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

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

[22]  N. Bucciantini,et al.  Local Kelvin-Helmholtz instability and synchrotron modulation in Pulsar Wind Nebulae , 2006, astro-ph/0603481.

[23]  Katharine Gurski,et al.  An HLLC-Type Approximate Riemann Solver for Ideal Magnetohydrodynamics , 2001, SIAM J. Sci. Comput..

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

[25]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

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

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

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

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

[30]  Shengtai Li.  An HLLC Riemann solver for magneto-hydrodynamics , 2005 .

[31]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[32]  K. Kusano,et al.  A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics , 2005 .

[33]  S. Komissarov.  Numerical simulations of relativistic magnetized jets , 1999 .

[34]  O. Zanotti,et al.  ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics , 2007, 0704.3206.

[35]  S. F. Davis.  Simplified second-order Godunov-type methods , 1988 .

[36]  A. Ferrari,et al.  Formation of dynamical structures in relativistic jets: the FRI case , 2008, 0806.1648.

[37]  Noh's constant-velocity shock problem revisited , 1997 .