High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes (Communications in Computational Physics, 2021)
In this paper, the second-order and third-order Runge-Kutta discontinuousGalerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory(WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENOlimiter is an extension of a finite volume multi-resolution WENO scheme developedin [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This newWENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCFindicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order andthird-order RKDG methods with the associated multi-resolution WENO limiters aredeveloped as examples for general high-order RKDG methods, which could maintainthe original order of accuracy in smooth regions and keep essentially non-oscillatoryproperty near strong discontinuities by gradually degrading from the optimal orderto the first order. The linear weights inside the procedure of the new multi-resolutionWENO limiters can be set as any positive numbers on the condition that they sumto one. A series of polynomials of different degrees within the troubled cell itselfare applied in a WENO fashion to modify the DG solutions in the troubled cell ontetrahedral meshes. These new WENO limiters are very simple to construct, and canbe easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Suchspatial reconstruction methodology improves the robustness in the simulation on thesame compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes)and three-dimensional tests are performed to demonstrate the good performance ofthe RKDG methods with new multi-resolution WENO limiters.