Implicit method and slope limiter in AHMR procedure for high order discontinuous Galerkin methods for compressible flows

Abstract : In this paper the robustness and the performance of adaptive hierarchical mesh refinement (AHMR) for high order Discontinuous Galerkin (DG) finite element method with slope limiting procedure combined with an implicit time scheme for the 2D non-linear Euler equations are shown. A slope limiting procedure based on triangular meshes is implemented and has been extended and amended accordingly to suit quadrilateral elements. The combination of DG methods and slope limiters is generally used with explicit time schemes. Here, the slope limiter implemented is incorporated into a quasi implicit time scheme procedure combined with an automatic h-adaptive hierarchical mesh refinement allowing non-conforming meshes. The time scheme is the implicit Second Order Backward Difference Formula (BDF2) with varying time step. The numerical test cases including subsonic, transsonic and supersonic flows show that the current slope limiting with quadrilateral meshes process together with the implicit time scheme is able to remove overshoots and undershoots around high gradient regions while preserving the high accuracy of the DG method. While combining this procedure with the automatic h-adaptive mesh refinement, one can improve the accuracy of the solutions and be able to capture quite precisely the features of the flows under consideration. The AHMR automatic procedure presented can easily be implemented in the numerical resolution of any physical models. Furthermore the limiting method used in this paper can be generalized to any type of mesh in two dimensions.