MHD Test Problems

Summary

This is a repository of some MHD test problems. Most of them are taken from Athena++ and Gardiner & Stone 2008

Linear Waves

I’ll be using the wave tests from Athena++ which are detailed in Gardiner & Stone 2008. The initial conditions are given by

\[\vec{U} = \vec{\bar{U}} + A \vec{R_k} \cos\left(\frac{2\pi x}{\lambda}\right).\]

Where \( \vec{U} \) is the initial condition in conserved variables and follows the usual density, momentum, magnetic field, energy ordering, \(\vec{\bar{U}} \) is the constant background state that is being pertubed by the wave, \( A \) is the amplitude of the wave/perturbation, \( R_k \) is the right eigenvector and depends on the wave, \( x \) is the position in space and \( \lambda \) is the wavelength; \(\lambda = 1 \) for the initial conditions and waves listed here. Magnetic fields will have to be implemented slightly differently than the other components because they’re at slightly different positions (cell faces rather than centers). \( R_k \) is in conserved variables with the order \( \left( \rho, \rho v_1, \rho v_2, \rho v_3, B_1, B_2, B_3, E \right) \)

Here are the values of the parameters

\[A = 10^{-6}\]

In primitive variable \( \vec{\bar{U}} \) is

\[\vec{\bar{U}}_{primitive} = \begin{bmatrix} \rho \\ P \\ v_x \\ v_y \\ v_z \\ B_x \\ B_y \\ B_z \end{bmatrix} = \begin{bmatrix} 1 \\ 1/\gamma \\ 0 \text{(or 1 for the contact/entropy wave)} \\ 0 \\ 0 \\ 1 \\ 3/2 \\ 0 \end{bmatrix}.\]

\( \vec{R}_k \) for fast magnetosonic waves with \( c_f = 2 \)

\[\vec{R}_{\pm c_f} = \frac{1}{2\sqrt{5}} \begin{bmatrix} 2 \\ \pm 4 \\ \mp 2 \\ 0 \\ 0 \\ 4 \\ 0 \\ 9 \end{bmatrix}.\]

\( \vec{R}_k \) for slow magnetosonic waves with \( c_s = 0.5 \)

\[\vec{R}_{\pm c_s} = \frac{1}{2\sqrt{5}} \begin{bmatrix} 4 \\ \pm 2 \\ \pm 4 \\ 0 \\ 0 \\ -2 \\ 0 \\ 3 \end{bmatrix}.\]

\( \vec{R}_k \) for Alfvén waves with \( c_a = 1 \)

\[\vec{R}_{\pm c_a} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \mp 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}.\]

\( \vec{R}_k \) for contact/entropy waves, make sure to set \( v_x=1 \) in \( \vec{\bar{U}} \) for this wave, with \( c_c = v_x \)

\[\vec{R}_{\pm c_c} = \frac{1}{2} \begin{bmatrix} 2 \\ 2 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}.\]

Rotated Linear Waves

Gardiner & Stone 2008 use the following rotation for rotated waves. \( \alpha \) is the pitch angle (rotation about \( y \)) and \( \beta \) is the yaw angle (rotation about \( z \)).

\[\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x_1 \cos{\alpha} \cos{\beta} - x_2 \sin{\beta} - x_3 \sin{\alpha} \cos{\beta} \\ x_1 \cos{\alpha} \sin{\beta} - x_2 \cos{\beta} - x_3 \sin{\alpha} \sin{\beta}\\ x_1 \sin{\alpha} - x_3 \cos{\alpha} \end{bmatrix}.\]

At an angle where \( \sin{\alpha} = 2/3 \) and \( \sin{\beta} = 2/\sqrt{5} \) Stone & Gardiner use a domain of \( 0 \leqslant x \leqslant 3.0 \), \( 0 \leqslant y \leqslant 1.5 \), and \( 0 \leqslant z \leqslant 1.5 \)

MHD Riemann Problems

See the MHD Riemann Problems post for info on MHD Riemann Problems.