Authors: Gabor Toth (University of Michigan), Bart van der Holst (University of Michigan)

Extended magnetohydrodynamic (XMHD) involves extra variables and/or terms compared to ideal magnethohydrodynamics (MHD). For example, the solar wind can be modeled with XMHD that includes separate electron pressure and/or anisotropic ion pressure. While the XMHD equations are well understood, the non-adiabatic heating at a shock in an XMHD simulation is not easy to describe. Conservation laws do not determine how much the non-adiabatic heating should be distributed among various thermal energies, such as parallel and perpendicular ion and/or electron energy densities. Using non-conservative pressure equations in fact results in non-adiabatic shock heating that depends on the numerical scheme and does not converge even with increasing grid resolution.

We describe a new approach that provides a well-behaved numerical solution that converges with grid refinement. In addition, the new method allows adjusting the distribution of non-adiabatic heating among the various energy densities with simple model parameters. For example, we can prescribe that one quarter of the non-adiabatic heating goes to the electrons and three quarter to the ions, which is consistent with observations and kinetic simulations.