Linear Stability

[tommbendall]

I have been investigating the stability of various configurations with large Courant numbers, and thought it would be useful to record some interesting results here.

Several years ago in Gungho we saw stability improvements from using an "advective-then-flux" approach to transporting the density field -- this ensures that increments to the density field are linear in the divergence (as-opposed-to being non-linear). This is achieved when using Runge-Kutta timestepping by only applying the flux form on the final RK stage. It wasn't obvious whether Gusto was also benefiting from this formulation...

I was also interested in the results of Wood, Kent & Thuburn: https://rmets.onlinelibrary.wiley.com/doi/abs/10.1002/qj.70060, which show that shallow-water dispersion relations are improved by including a "predictor" in the transporting of the depth field. This means transporting a density that includes a contribution from the divergence.

Here are some results from Gusto investigating these options with large-ish Courant numbers (which accentuate any issues).

Thermal Shallow-Water

Here are some results from the thermal shallow-water gravity wave test. The case with advective-then-flux but no predictor crashes.

Compressible Euler

Here are plots from the Skamarock-Klemp gravity wave in a vertical slice. Now just using the predictor appears to be unstable.