MHD Blast Wave

A blast wave problem is a good test of the robustness of a code in modelling 3D shocks and rarefactions. We ran the test problem in Stone (2009, section 6.5) in 3D at the same numerical resolution ( $200\times300\times200$). The domain has size $1\times1.5\times1$ units; the initial background has a density $\rho_0=1$, pressure $p_0=0.1$, magnetic field $\ensuremath{\mathbf{B}}_0=[1/\sqrt{2},1/\sqrt{2},0]$, and is at rest. Within a radius $R=0.1$ from the centre of the domain, the initial pressure is set to $p=100p_0$ (note I think their paper has a typo - they say $p=100$). The results obtained are almost identical to theirs, and are shown in figure 2.11. The major difference is that discontinuities are not quite as sharp, indicating that the Stone (2009) method is less diffusive than ours. Because of the robustness of our algorithm, we are able to run the same model with a field $10\times$ stronger, i.e. $\beta _0=0.002$, at which point the Stone (2009) algorithm fails for $t\gtrsim0.05$. Results are shown at $t=0.2$ in figure 2.12. Note that by this stage the leading outward moving MHD waves have left the domain due to the much faster wave propagation speeds.

Figure 2.11: Plots of gas density and magnetic pressure at time $t=0.2$ for the MHD blast wave test problem with $\beta _0=0.2$. We show slices in the $x-y$ plane through the centre of the domain, on a linear scale. This figure can be compared to figure 8 in Stone (2009).

Figure 2.12: As figure 2.11: plots of gas density and magnetic pressure at time $t=0.2$, but this time with $\beta _0=0.002$.