A Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids

摘要

A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible Navier-Stokes equations on arbitrary grids in the framework of DG methods. The DDG method, originally introduced for scalar diffusion problems on structured grids, is extended to discretize viscous and heat fluxes in the Navier-Stokes equations. Two approaches of implementing the DDG method to compute numerical diffusive fluxes for the Navier-Stokes equations are presented: one is based on the conservative variables, and the other is based on the primitive variables. The importance of the characteristic cell size used in the DDG formulation on unstructured grids is examined. The numerical fluxes on the boundary by the DDG method are discussed. A number of test cases are presented to assess the performance of the DDG method for solving the compressible Navier-Stokes equations. The numerical results obtained indicate that the DDG method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used BR2 method at a significantly reduced cost, clearly demonstrating that the DDG method provides an attractive alternative for solving the compressible Navier-Stokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost.