High-Order Local Discontinuous Galerkin Method with Multi-Resolution WENO Limiter for Navier-Stokes Equations on Triangular Meshes (Communications in Computational Physics, 2023)
In this paper, a new multi-resolution weighted essentially non-oscillatory(MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENOlimiter is a new extension of the finite volume MR-WENO schemes. Such new limiteruses information of the LDG solution essentially only within the troubled cell itself, tobuild a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to thehighest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which couldmaintain the original order of accuracy in smooth regions and could simultaneouslysuppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on conditionthat their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify thefreedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiteris very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstructionmethodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classical viscous examples are given to show the good performance of this third-order LDGmethod with associated MR-WENO limiter.
Keywords: Local discontinuous Galerkin method; multi-resolution WENO limiter; triangular meshes; Navier-Stokes equations.