MHD Shock Tubes

The shock-tube tests presented in Falle (1998) were used to test the MHD Riemann solver. These tests consist of the Brio (1988) test problem and a number of other tests which focus on a specific MHD wave, as listed in Table 2.2. All tests use $\gamma=5/3$ except the Brio and Wu test with $\gamma=2$, and all are dimensionless and run on a domain of unit length. Note the Alfvén wave test is not a discontinuity so the left and right states are the same. The wave is set up to have width 0.3 and the velocity and magnetic fields rotate by $2\pi$ in the $y-z$ plane over this length. This test also uses periodic boundary conditions to advect the wave across the domain five times. Other tests use zero-gradient boundary conditions and only the small startup waves leave the domain before the end of the simulation. The results are shown in Fig. 2.2. All of the results are very similar to those shown in Falle (1998) showing that our MHD approximate Riemann solver correctly captures all of the waves generated. Note that Falle (1998) evolve the Alfvén wave three times its width for their test whereas we have evolved it 5 times across the domain; this is why our results have degraded significantly more than theirs.

Table 2.2: MHD Shock Tube Test Problems from Falle (1998). State variables are $[\rho ,p_g,\{v_x,v_y,v_z\},\{B_x,B_y,B_z\}]$. BW is the Brio & Wu test problem, AW is an Alfvén wave, FS and SS are fast and slow switch-on shocks, FR and SR are fast and slow rarefaction waves.

Test	Left State	Right State	$t_f$	$x_0$
BW	$[1,1,\{0,0,0\},\{0.75,1,0\}]$	$[0.125,0.1,\{0,0,0\},\{0.75,-1,0\}]$	0.12	0.5
AW	$[1,1,\{0,1,1\},\{1,1,0\}]$	$[1,1,\{0,1,1\},\{1,1,0\}]$	5.0	0.65
FS	$[3,16.33,\{-0.732,-1.333,0\},\{3,2.309,0\}]$	$[1,1,\{-4.196,0,0\},\{3,0,0\}]$	0.4	0.3
SS	$[1.368,1.769,\{0.269,1,0\},\{1,0,0\}]$	$[1,1,\{0,0,0\},\{1,1,0\}]$	0.5	0.3
FR	$[1,2,\{0,0,0\},\{1,3,0\}]$	$[0.2641,0.2175,\{3.6,-2.551,0\},\{1,0,0\}]$	0.1	0.5
SR	$[1,2,\{0,0,0\},\{1,0,0\}]$	$[0.2,0.1368,\{1.186,2.967,0\},\{1,1.6405,0\}]$	0.3	0.5

Figure 2.2: 1D shock tube results for adiabatic MHD. The solid line shows the results using 10,000 grid cells, and the points show results using 200 cells. The figures show results for the tests in Table 2.2 and are from top to bottom: BW, AW, FS, SS, FR, SR. Note for the Alfvén wave the solid black line is the initial state and the cyan symbols show the high resolution results.