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 Navier-Stokes 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.
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