Nonlinear Stability Meets High-Order Flux Reconstruction

We utilized an extended version of the methods described in the 2012 work by Wang and Shu, incorporating modifications from Liter and Roe's upwind dissipation. Our initial test involved the Leblanc shock tube case, often referred to as the "shock tube from hell" due to the extreme pressure difference of 10^9 across the shockwave.

In our experiments with the discontinuous Galerkin (DG) method, we observed that lowering the CFL condition did not resolve the failure at P4. However, for the C+ and adaptive methods, we were able to successfully run the cases, even while the DG method failed at P4. It is important to note that the Leblanc shock tube case does not employ a TVVD limiter, resulting in oscillations, yet it remains operational.