A direct discontinuous Galerkin method for the incompressible Navier-Stokes equations on arbitrary grids

A high order direct discontinuous Galerkin (DG) method is proposed in this work for solving the two-dimensional steady and unsteady incompressible Navier-Stokes (INS) equations on arbitrary grids. The inviscid term is discretized by using a simplified artificial compressibility flux which can be obtained explicitly and straightforwardly without entailing any numerical iterative procedure for solving a Riemann problem of a nonlinear system. And the viscous term is discretized by the direct DG (DDG) method which was developed based on the weak formulation of the scalar diffusion problems on structured grids. A number of test cases are presented to assess the performance of the DDG method compared to the widely used BR2 method for solving the INS equations. The numerical results indicate that the DDG method can achieve the designed order of accuracy and is able to deliver comparable accuracy as the BR2 method. Due to its simplicity in implementation, the DDG method provides an attractive alternative for solving the INS equations on arbitrary grids. (C) 2018 Elsevier Inc. All rights reserved.