Quadrature-free forms of discontinuous Galerkin methods in solving compressible flows on triangular and tetrahedral grids

Abstract

The nonlinear stability would affect the accuracy and robustness of the discontinuous Galerkin (DG) method in solving the inviscid and viscous compressible flows. This issue is studied by proposing a new framework to incorporate the quadrature-based and quadrature-free discontinuous Galerkin methods on triangular and tetrahedral grids. Four nodal DG schemes are derived by choosing special test functions and different collocation points for the approximate solution and flux function. The new framework can be more efficient in computational cost and easier in code implementation especially for the quadrature-free scheme. These four schemes are the same for the linear system but different for the nonlinear system due to aliasing errors. Numerical examples are provided to validate the performance in accuracy and nonlinear stability.