A Hybridized Multiscale Discontinuous Galerkin Method for Compressible Flows

Abstract

We introduce a hybridized multiscale discontinuous Galerkin (HMDG) method for the numerical solution of compressible flows. The HMDG method is developed upon extending the hybridizable discontinuous Galerkin (HDG) method presented in [30]. The extension is carried out by modifying the local approximation spaces on elements. Our local approximation spaces are characterized by two integers (n*, k*), where n* is the number of subcells within an element and k* is the polynomial degree of shape functions defined on the subcells. The selection of the value of (n*, k*) on a particular element depends on the smoothness of the solution on that element. More specifically, for elements on which the solution is smooth, we choose the smallest value n* = 1 and the highest degree k* = k. For elements containing shocks in the solution, we use the largest value n* = n and the lowest degree k* = 0 so as to capture shocks without using artificial viscosity and limiting slopes/fluxes. The proposed method thus combines the accuracy and efficiency of higli-order approximations with the robustness of low-order approximations. Numerical results are presented to demonstrate the performance of the proposed method.