Slow-moving shock test

This test problem demonstrates the extent to which post-shock oscillations are controlled in slowly-moving shocks. This effect can be exhibited in all Godunov codes, even with first-order methods, for sufficiently slow-moving shocks across the computational grid (Jin & Liu, 1996).

The shock flattening method of (Colella & Woodward, 1984) (implemented in our code in modified form) reduces the oscillations, but does not completely suppress them. Adding artificial viscosity according to the method of (Colella & Woodward, 1984), even to the level of smoothing the contact discontinuity by 5-10 cells, does not cure the problem.

Parameters

The left- and right-side initial conditions are (Quirk, 1994):

The shock moves to the right with speed \(s = 0.1096\).

Solution

We use the RK2 integrator with a fixed timestep of \(10^{-3}\) and a mesh of 100 equally-spaced cells. The contact discontinuity is initially placed at \(x=0.5\).

The density is shown as the solid blue line. The exact solution is the solid orange line.