Physics of some Standard MHD test results using Athena++ code.

The ideal MHD equations in conservative form can be written as

where,

Here is the total energy per unit volume, is the total pressure, and are pressure and density, is the velocity vector, is the magnetic field vector and is the adiabatic index.

In 1D the ideal MHD equations can be written as

where

and

It is clear that owing to the divergence-free condition, .