ShuVortex convergence tests interpretation
At 28. Sept Michael Dumbser wrote about the convergence rates of the Euler ShuVortex simulation:
I think the convergence results are not bad at all. The "systematic artefacts" are not artefacts, but we clearly understand where they come from and the code would be wrong if we could not see them :-) To get a clean and proper convergence table for Exahype without periodic boundary conditions, you must simulate the ShuVortex
- only until a rather small final time, say t=0.5 or t=1.0. The vortex must not touch the boundary at all, and since the vortex is an exponential function, the high order DG scheme will even see very small errors of the order 1e-7 or 1e-8, hence the Gaussian must really be perfectly flat on the boundary, well below this accuracy. Not only at the initial, but also at the final time. In numerical analysis, it is not common practice to report convergence order over simulation time. Usually, one reports the error in a certain norm (L1, L2, Linf) on a given mesh for a fixed output time, which satisfies all the necessary theoretical criteria for getting high order (sufficiently smooth solution). If convergence order drops due to boundary effects, at least one of the necessary criteria (regularity) is violated, so it is clear that no convergence at the expexted rate is observed.
- If you want larger simulation times, you must increase the domain size (even bigger than the canonical [0,10]^2 domain, say [-10,20]^2), in order to make sure that the Gaussian is perfectly flat on the boundary, up to the desired error threshold.
- A nice test case which works for arbitrarily large times and without periodic boundaries is the large Amplitude Alfven wave for SRMHD. Olindo can send you the setup. The exact solution is a travelling sine wave, and if you impose the exact solution on the boundaries, you should be able to simulate the problem even without periodic boundaries and without any artefact.