DISPATCH methods: an approximate, entropy-based Riemann solver for ideal magnetohydrodynamics

With the advance of supercomputers we can now afford simulations with very large ranges of scales. In astrophysical applications, e.g. simulating Solar, stellar and planetary atmospheres, interstellar medium, etc; physical quantities, like gas pressure, density, temperature, plasma $\beta$, Mach, Reynolds numbers can vary by orders of magnitude. This requires a robust solver, which can deal with a very wide range of conditions and be able to maintain hydrostatic equilibrium where it is applicable. We reformulate a Godunov-type HLLD Riemann solver that it would be suitable to maintain hydrostatic equilibrium in atmospheric applications in a range of Mach numbers, regimes where kinetic and magnetic energies dominate over thermal energy without any ad-hoc corrections. We change the solver to use entropy instead of total energy as the primary thermodynamic variable in the system of MHD equations. The entropy is not conserved, it increases when kinetic and magnetic energy is converted to heat, as it should. We propose using an approximate entropy - based Riemann solver as an alternative to already widely used Riemann solver formulations where it might be beneficial. We conduct a series of standard tests with varying conditions and show that the new formulation for the Godunov type Riemann solver works and is promising.