A dynamic p-adaptive Discontinuous Galerkin method for viscous flow with shocks

A dynamic p-adaptive Discontinuous Galerkin method for compressible flows is proposed. The key element is a sensor, which measures the local regularity of the solution. This sensor is designed to preserve the compactness of the method and is easy to implement. In regions where the solution is quasi-uniform and in the vicinity of shocks the degree of the polynomial basis is decreased, while a high-degree basis is used in regions of smooth fluctuations of the flow. Numerical tests carried out in 1D and on 2D unstructured meshes prove that the convergence rate of the hybrid method is equal to the one of the highest degree polynomial basis, but with a significant CPU cost saving.