The $\nabla \cdot {\bf B}=0$ Constraint

The presence of the $\nabla \cdot {\bf B}=0$ constraint implies that the equations have a semi-elliptic character. It has been shown in [65] that even a small divergence in the magnetic fields can dramatically change the character of results from numerical simulations.

An alternative approach to the magnetic stream function which we used for the incompressible MHD simulations was developed in [62]. This modified the equations by adding a source term proportional to $\nabla \cdot \bf B$:

$${\bf S}_{Powell} = -(\nabla \cdot {\bf B})({0},{B_x},{B_y},{B_z},{u},{v},{w},{{\bf v}\cdot{\bf B}})^{T}$$

to the right hand side of the evolution equation. In [62] this term was incorporated into the Riemann solver for the Euler flux terms. We evaluate this term as a source term, without modifying the Riemann solver. The divergence of the magnetic field is calculated using the DGM derivatives of the magnetic fields.