On the stability of a split form CPR scheme in under-resolved flows

The correction procedure via reconstruction (CPR) method is a compact and efficient high-order method suitable for unstructured grids. However, when discretizing the nonlinear convection term, it is easy to cause numerical instability due to the accumulation of aliasing errors. In the present work, we study the stability of the split form CPR method based on LG (Legendre-Gauss) points in under-resolved flows, and combine the method with the subcell limiting technique to solve under-resolved flows with shock waves. First, numerical tests are carried out to verify that the split form CPR method based on LG points with the boundary flux correction can satisfy the discrete conservation law, and such conservation is still preserved under subcell limiting. In the simulation of under-resolved flows without shock waves, compared to the divergence form CPR method, the split form significantly improves the stability of the calculation, and has smaller numerical errors than the split form CPR method using LGL (Legendre-Gauss-Lobatto) points. When solving under-resolved flows with shock waves, compared to the subcell limiting strategy of discontinuous Galerkin spectral element method based on LGL points, the subcell limiting strategy of split form CPR scheme based on LG points developed in this paper has higher resolution and less oscillation.