Two-Dimensional Magnetohydrostatic Example

A simple test for the compressible MHD component of the code is to consider a steady irrotational magnetic field and zero velocity. The test was performed as an initial value problem and the following exact solution:

was used as the boundary conditions and as the initial condition. This solution was derived by Priest [84]. The irrotational magnetic field implies that the Lorentz force is zero so the momentum equations are trivially satisfied. The magnetoviscous term is zero and the

$${\bf v}\times{\bf B}$$

term is also zero. Thus, the compressible MHD equations are satisfied.

The domain and discretization we used is depicted in figure 8.1. We also show that the approximation decreases exponentially with increasing expansion order.

  

Figure 8.1: Magnetohydrostatic test case for the compressible code. Top Left: Mesh KTri = 38, KQuad = 22 Top Right: Magnetic streamlines of steady solution at N=12, Bottom: dependence of steady state error on expansion order.