Rotated Shock Tube Tests

These tests involve the same initial conditions as described in Tables 2.1 and 2.2 but run on a 2D domain with the initial discontinuity at an angle to the grid axes. The only test which has different initial conditions is the Alfvén wave test which is described separately below. If the angle is not zero or 45$^\circ$ then the fluxes in the $x$ and $y$ directions are different and it is a genuinely multi-dimensional problem. To avoid boundary effects (and avoiding setting up complicated special boundaries) we run the 2D simulations in a large domain and take results from a unit area square subdomain with $200\times 200$ grid cells which is unaffected by boundary waves. For the 2D tests the discontinuity normal direction is set to an angle of $\theta=\arctan(2)$ or $\theta=\arctan(0.5)$ from the $x$-axis. All points in the subdomain are plotted in the figures below, with positions and vector quantities rotated to the shock normal axes. For the angles used here most of the points project on top of each other, but the projection means the number of points appearing within a shock is obviously larger than the actual number required to resolve the shock (by a factor of 2). Hydro shock tube results are shown in figure 2.3, and MHD results in figure 2.4. The results are comparable to the 1D simulations at equivalent resolution (see figures 2.1 and 2.2) .

Figure 2.3: 2D shock tube results for adiabatic hydrodynamics. The solid line shows the results using 10,000 grid cells on a 1D grid, and the points show results in 2D on a domain with $200\times 200$ cells. The figures show results for Toro's tests 1-5 from top to bottom (see Table 2.1).

Figure 2.4: 2D shock tube results for adiabatic MHD. The solid line shows the results using 10,000 grid cells on a 1D grid, and the points show results in 2D on a domain with $200\times 200$ cells. The figures show results for the tests in Table 2.2 and are from top to bottom: BW, FS, SS, FR, SR. Note the Alfvén Wave test is not included in this plot, and gas pressure is plotted in the second panel from the left instead of total pressure used in figure 2.2.

Alfvén Wave Test in 2D

This test is set up in the same way as in Stone (2008) using the initial conditions from Tóth (2000). We have run the test with a travelling wave on a periodic domain of size $1\times2$ with $N\times2N$ cells until $t=5$ when the wave has crossed the domain five times. The transverse magnetic field (in the $x$-$y$ plane) at the final state for models with $N=[16,32,64,128]$ is shown in the left plot of figure 2.5. The initial wave amplitude is 0.1, so the 64 and 128 cell results have very little degradation. Comparing with fig. 19 in Stone (2008) the low resolution results presented here are clearly worse than for Athena. This is partly because they use a third order reconstruction; piecewise parabolic is a much better approximation to the extrema than piecewise linear at low resolution for this test. The right plot shows the L1 error:

$$L_1 = \frac{1}{N}\sum_{i=0}^{N-1}\left\vert\phi_i-\phi_i^0\right\vert$$ (2.1)

for $N$ cells, where $\phi_i^0$ is the reference state for each cell and $\phi_i$ is the approximate numerical solution. This clearly converges quadratically with resolution, as it should for a second order algorithm with no discontinuities.

Figure 2.5: Circularly Polarised Alfvén Wave test. The left plot shows the transverse magnetic field after advecting five times across the domain (initial amplitude 0.1) for models with resolutions $N_x=[16,32,64,128]$. The right plot shows the L1 error (see equation 2.1) as a function of resolution for this test, obtained by comparing the initial to the final solution at each resolution. The second order convergence is clearly seen.