Development of a three-dimensional high-order discontinuous galerkin method flow solver on unstructured meshes

Abstract

Recent decades, the Discontinuous Galerkin method (DGM) has gained a resurgence of interest by its high-order accuracy over the conventional second-order accurate finite volume method (FVM) with combination of both advantages of FVM and Finite Element Method (FEM). In respect of DGM, the high-order accuracy is obtained by increasing the order of approximate polynomials rather than relying on extended stencils in classical FVM. Besides, the numerical flux at the interface is developed by arising from solutions of the Riemann problems in the case of FVM. Other than that, DGM maintains the compactness because only data from neighbors of element boundaries is required, which results in that the inter-element communication is minimal and boundary conditions are implemented in a straightforward way. Also, due to the locality of the DGM, refining or coursing the meshes is easily handled without the restrictions on the continuity with neighbor elements. In addition, DGM is suited for complex geometries based on the unstructured meshes. As a result, it owns the attractive features of being easily extended to high-order approximation, well suited for complex geometries, highly parallelizable and easily handling adaptive strategies. The DGM was originally introduced to solve the neutron transport problems. Then it is developed by Cockburn and Shu for solving the nonlinear systems of hyperbolic conservation problems. They presented the Runge-Kutta DG method, which used DGM for spatial discretization and the TVD Runge-Kutta method for time integration. However, the convergence rate is extremely slow for large-scale simulations because of poor CFL stability condition. To solve this problem, an implicit time integration method is adopted. In the current research, a high-order implicit DGM flow solver for the three-dimensional problems has been developed on tetrahedral grids. A fully implicit method based on Euler backward differencing and linearization of the residuals is a...