A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework

We present a relaxation system for ideal magnetohydrodynamics (MHD) that is an extension of the Suliciu relaxation system for the Euler equations of gas dynamics. From it one can derive approximate Riemann solvers with three or seven waves, that generalize the HLLC solver for gas dynamics. Under some subcharacteristic conditions, the solvers satisfy discrete entropy inequalities, and preserve positivity of density and internal energy. The subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate Riemann solver itself. The 7-wave version of the solver is able to resolve exactly all material and Alfven isolated contact discontinuities. Practical considerations and numerical results will be provided in another paper.

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

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

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

[4]  François Bouchut,et al.  Entropy satisfying flux vector splittings and kinetic BGK models , 2003, Numerische Mathematik.

[5]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

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

[7]  Gérard Gallice,et al.  Positive and Entropy Stable Godunov-type Schemes for Gas Dynamics and MHD Equations in Lagrangian or Eulerian Coordinates , 2003, Numerische Mathematik.

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

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

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

[11]  Derek M. Causon,et al.  On the Choice of Wavespeeds for the HLLC Riemann Solver , 1997, SIAM J. Sci. Comput..

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

[13]  C. Munz,et al.  Hyperbolic divergence cleaning for the MHD equations , 2002 .

[14]  Paris Vi,et al.  Lagrangian systems of conservation laws Invariance properties of Lagrangian systems of conservation laws, approximate Riemann solvers and the entropy condition , 2001 .

[15]  Kenneth G. Powell,et al.  AN APPROXIMATE RIEMANN SOLVER FOR MAGNETOHYDRODYNAMICS (That Works in More than One Dimension) , 1994 .

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

[17]  C. D. Levermore,et al.  Hyperbolic conservation laws with stiff relaxation terms and entropy , 1994 .

[18]  Gérard Gallice,et al.  Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws , 1997 .

[19]  Kun,et al.  Gas-kinetic Theory Based Flux Splitting Method for Ideal Magnetohydrodynamics , 2022 .

[20]  Athanasios E. Tzavaras,et al.  Materials with Internal Variables and Relaxation to Conservation Laws , 1999 .

[21]  B. Perthame,et al.  Some New Godunov and Relaxation Methods for Two-Phase Flow Problems , 2001 .

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

[23]  Bruno Després,et al.  An Entropic Solver for Ideal Lagrangian Magnetohydrodynamics , 1999 .

[24]  Bruno Després,et al.  Lagrangian systems of conservation laws , 2001, Numerische Mathematik.

[25]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

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

[27]  Yann Brenier,et al.  Averaged Multivalued Solutions for Scalar Conservation Laws , 1984 .