论文信息 - A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves

A multiwave approximate Riemann solver for ideal MHD based on relaxation II: numerical implementation with 3 and 5 waves

In the first part of this work Bouchut et al. (J Comput Phys 108:7–41, 2007) we introduced an approximate Riemann solver for one-dimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation, and derive explicit wave speed estimates satisfying the stability conditions of Bouchut et al. (J Comput Phys 108:7–41, 2007). We present a 3-wave solver that well resolves fast waves and material contacts, and a 5-wave solver that accurately resolves the cases when two eigenvalues coincide. A full 7-wave solver, which is highly accurate on all types of waves, will be described in a follow-up paper. We test the solvers on one-dimensional shock tube data and smooth shear waves.

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

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

[3] Timothy J. Barth,et al. Numerical Methods for Gasdynamic Systems on Unstructured Meshes , 1997, Theory and Numerics for Conservation Laws.

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

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

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

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

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

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

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

[11] Christian Klingenberg,et al. A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: theoretical framework , 2007, Numerische Mathematik.