Three-Dimensional Magnetohydrostatic Example

We modified the two-dimensional test case, used in the previous section, to be a three-dimensional test case for the compressible MHD component of the code. Again we used a steady irrotational magnetic field and zero velocity solution to the MHD equations. 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. By construction, the magnetic field is irrotational and 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.

The domain and discretization we used is depicted in figure 8.2. We also show that the approximation decreases exponentially with increasing expansion order.

  

Figure 8.2: Three-dimensional magnetohydrostatic test case for the compressible code. Iso-contours of the magnetic and energy fields are shown at time t=1. Top Left: Tetrahedral mesh used, Top Right: x-component of the magnetic field, Middle Left: y-component of the magnetic field, Middle Right: z-component of the magnetic field, Bottom: Convergence plot showing exponential decrease in L2 error with increasing expansion order.