This originally two-dimensional test case aims especially at challenging the shock capturing properties of the numerical scheme. The presented three dimensional extension of this problem is achieved easily, since the simple initial conditions only need to be changed slightly. The initial setting is the following:
At the origin of the [-0.5 , 0.5] x [-0.5 , 0.5] x [-0.5 , 0.5] domain, a spherical “bubble” of radius r = 0.1 with initial pressure p = 10 is located. The ambient fluid has a pressure of p= 0.1. The magnetic field is set to
Using this setting, the magnetic field lines are diagonally aligned, resulting in a real three-dimensional characteristic of this test case. The initial density is set to unity and the velocity remains zero. In our case, the problem is set up with periodic boundaries, although other boundary conditions are possible as well. Shock capturing is performed using our artificial viscosity approach.
In the following images one can easily see that the propagation of the density shock-wave is stretched along the magnetic field lines, as the calculation continues.