Spherical Blast Waves Test
Various papers have presented results from similar spherical blast wave tests, e.g. Zachary, Malagoli, A., & Colella,P., SIAM J. Sci. Comp., 15, 263 (1994); Balsara, D., & Spicer, D., JCP 149, 270 (1999); Londrillo, P. & Del Zanna, L., ApJ 530, 508 (2000).
Different authors set this problem up in different ways. To test Athena, we used a rectangular domain, -0.5 ≤ x ≤ 0.5; -0.75 ≤ y ≤ 0.75. The boundary conditions are periodic everywhere. This non-square domain and periodic boundary conditions produces complex shock-shock and shock-CD interactions at late times.
The initial density is 1.0, the pressure is 0.1, and the gas constant is γ = 5/3. Initial velocities are zero everywhere. Within the region r < 0.1, the pressure is set to 10.0 (that is, 100 times the ambient pressure). For the MHD problem, the initial magnetic field is uniform everywhere with Bx / (4π)1/2 = By / (4π)1/2 = 1/√2.
To be honest, this test is not very quantitative, but makes great movies!
At early times, it is important that the out-going blast wave is spherical and shows no grid alignment effects.
At late times, the interaction of the blast wave with the CD at the edge of the evacuated bubble in the center produces filaments of dense gas by the Richtmyer-Meshkov instability. It is important these fingers are sharp and not diffused away. Moreover, for the hydrodynamical problem, the pattern of the fingers should be EXACTLY symmetric top-to-bottom and left-to-right. For the MHD problem, the Richtmyer-Meshkov instability is suppressed, and no fingers are evident.
Results computed with Athena using the HLLC solver and the third order algorithm on a 400x600 grid are shown below. The images show the density on a linear color map between 0.08 and 6.5. The image on the left is at t = 0.2, the image on the right is at t = 1.5
Note the pattern of dense fingers in the interior produced by the Richtmyer-Meshkov instability is exactly symmetric.
Click on the right image to download a .gif movie (39.5MB).
Results computed with Athena using the Roe solver and the third order algorithm on a 400x600 grid are shown below. The images show the density on a linear color map between 0.08 and 6.5. The image on the left is at t = 0.2, the image on the right is at t = 1.0
Note the Richtmyer-Meshkov instability is supressed by the magnetic field, and no fingers are evident in the interior of the bubble. Also note the bubble is strongly aligned along the magnetic field.
Click on the right image to download a .gif movie (32MB).