Shu-Osher shock test

This test problem demonstrates the ability of the code to resolve fine features and shocks simultaneously. The resolution of our code is comparable to the ENO-RF-3 scheme shown in Figure 14b of (Shu & Osher, 1989)1.

Parameters

The left- (\(x < 1.0\)) and right-side (\(x \ge 1.0\)) initial conditions are:

\[\begin{aligned} \rho_0 = 3.857143 \\ v_{x,0} = 2.629369 \\ P_0 = 10.33333 \\ \rho_1 = 1 + 0.2 \sin(5 x) \\ v_{x,1} = 0 \\ P_1 = 1 \end{aligned}\]

Solution

We use the RK2 integrator with a fixed timestep of \(10^{-4}\) and a mesh of 400 equally-spaced zones, evolving until time \(t=1.8\). There are some subtle stair-step artifacts similar to those seen in the sawtooth linear advection test, but these converge away as the spatial resolution is increased.

These artefacts can be eliminated by projecting into characteristic waves and reconstructing the interface states in the characteristic variables, as done in §4 of (Shu & Osher, 1989)1. The reference solution is computed using Athena++ with PPM reconstruction in the characteristic variables on a grid of 1600 zones.

The density is shown as the solid blue line. There is no exact solution for this problem.

  1. Shu, C.-W., & Osher, S. (1989). Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes, II. Journal of Computational Physics, 83(1), 32–78. https://doi.org/10.1016/0021-9991(89)90222-2 ↩↩