A Symmetric Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations (Communications in Computational Physics, 2018)
In this work, we investigate the numerical approximation of the compressibleNavier-Stokes equations under the framework of discontinuous Galerkin methods.For discretization of the viscous and heat fluxes, we extend and apply the symmetricdirect discontinuous Galerkin (SDDG) method which is originally introducedfor scalar diffusion problems. The original compressible Navier-Stokes equations arerewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusivefluxes are constructed from the weak formulation of primal equations directlywithout converting the second-order equations to a first-order system. Additional numericalflux functions involving the jump of second order derivative of test functionsare added to the original direct discontinuous Galerkin (DDG) discretization. A numberof numerical tests are carried out to assess the practical performance of the SDDGmethod for the two dimensional compressible Navier-Stokes equations. These numericalresults obtained demonstrate that the SDDG method can achieve the optimalorder of accuracy. Especially, compared with the well-established symmetric interiorpenalty (SIP) method [18], the SDDG method can maintain the expected optimal orderof convergence with a smaller penalty coefficient.