A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

We propose a new formula for the nonlinear viscous numerical flux and extend direct discontinuous Galerkin method with interface correction (DDGIC) of Liu Yan (H. Liu, J. Yan, The (DDG) diffusion corrections, Communications in Computational Physics 8 (3) (2010) 541) to compressible Navier-Stokes equations. DDGIC framework is based on observation that can be represented as sum multiple individual processes corresponding each conserved variable. A set direction vectors process defined approximated by average value solution at cell interfaces. only requires computation variables' gradient, which linear original DG formula. proposed greatly simplifies implementation, thus, easily extended general equations turbulence models. Numerical experiments $P_1$, $P_2$, $P_3$ $P_4$ polynomial approximations are performed verify optimal $(k+1)^{th}$ high-order accuracy method. shown able accurately calculate physical quantities such lift, drag, friction coefficients well separation angle Strouhal number.