1D MHD Test: Magnetized Sod’s Shock Tube

This is the 1D magnetohydrodynamic shock tube problem [Brio and Wu, 1988]. This problem is often used to verify numerical schemes that solve the magnetohydrodynamic equations.

The test problems provided in MISO run the 1D problem in each of the \(x\), \(y\), and \(z\) directions, but this page describes only the problem in the \(x\) direction.

Location

demo/mhd1d_shock_tube/

Geometry

\(0 \leq x \leq 1\).

Initial Conditions

The initial condition is described by the left and right states separated at \(x=0.5\) and an axial magnetic field \(B_x=0.75\sqrt{4\pi}\). The ratio of specific heats is \(\gamma = 2\).

\[\begin{split} \begin{align*} \begin{pmatrix} \rho_\mathrm{L} \\ p_\mathrm{L} \\ v_\mathrm{L} \\ B_\mathrm{L} \end{pmatrix} &= \begin{pmatrix} 1.0 \\ 1.0 \\ 0.0 \\ -\sqrt{4\pi} \end{pmatrix} \\ \begin{pmatrix} \rho_\mathrm{R} \\ p_\mathrm{R} \\ v_\mathrm{R} \\ B_\mathrm{R} \end{pmatrix} &= \begin{pmatrix} 0.125 \\ 0.1 \\ 0.0 \\ \sqrt{4\pi} \end{pmatrix} \\ \end{align*} \end{split}\]

Here, \(B_\mathrm{L}\) and \(B_\mathrm{R}\) are the transverse magnetic field (in the \(y\) direction).

Boundary Conditions

A symmetric boundary condition is imposed on all variables.

Results

You can visualize the results with Python and compare the results in the \(x\), \(y\), and \(z\) directions. The image files are saved in demo/mhd1d_shock_tube/figs.

cd demo/mhd1d_shock_tube
python plot_data.py