A quadrature simplification method for fast implicit discontinuous Galerkin schemes

This paper describes a methodology for fast implicit time integrations with high-order discontinuous Galerkin (DG) methods. The proposed approach, named quadrature simplification by orthogonality (QSO), assumes constant-flux Jacobians in each cell and utilizes the orthogonal properties of basis functions to simplify the quadrature involved in implicit time integration schemes. QSO enables substantially faster implicit time integration without any major deterioration in the computed results. In terms of the computational cost, numerical stability, and convergence property, the performance of QSO is first assessed through shock wave and boundary layer problems using 2D structured meshes. Effects of QSO on time evolutions are also examined. The application of the proposed QSO to 3D unstructured meshes is then investigated through boundary layer computations. Finally, an illustrative application to the more complex problem of the flowfield over a delta wing is used to demonstrate the capability of high-order DG methods with QSO.