Shu and Osher's Shocktube Test

Reference:

Shu, C and Osher, S., "Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes, II", J. Computational Physics, 83, 32-78 (1989). The test is Example 8.

Description:

This test is a hydrodynamic shocktube where the left and the right states are given as follows. Left: (ρ=3.857143; Vx= 2.629369, P = 10.33333) Right: (ρ=1 + 0.2 sin(5 π x); Vx=0; P=1). Essentially it is a Mach=3 shock interacting with a sine wave in density.

What's important about this test?

This test shows the difficulty of capturing both small-scale smooth flow, and shocks.

Results:

Results for the density at time=0.47 computed with Athena using the third-order Roe solver on an array of 800 cells (solid line) and 200 cells (squares) with a Courant number of 0.8 and γ = 1.4. Note that the plot is normalized to a spatial extent of [-1,1] as opposed to the [-4,4] shown in the reference. By comparing to Fig. 14c in the text, it is possible to see the significant improvement in performance per zone at low resolution. At a resolution of 200 cells, all of the short-wavelength extrema are captured, albeit with a lower amplitude than the 800 cell calculation. The velocity and pressure profiles do not contain as much interesting structure and are not shown.