A 5-Wave Relaxation Solver for the Shallow Water MHD System

The shallow water magnetohydrodynamic system describes the thin layer evolution of the solar tachocline. It is obtained from the three dimensional incompressible magnetohydrodynamic system similarly as the classical shallow water system is obtained from the incompressible NavierStokes equations. The system is hyperbolic and has two additional waves with respect to the shallow water system, the Alfven waves. These are linearly degenerate, and thus do not generate dissipation. In the present work we introduce a 5-wave approximate Riemann solver for the shallow water magnetohydrodynamic system, that has the property to be non dissipative on Alfven waves. It is obtained by solving a relaxation system of Suliciu type, and is similar to HLLC type solvers. The solver is positive and entropy satisfying, ensuring its robustness. It has sharp wave speeds, and does not involve any iterative procedure.