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

翻译：我们提出了一个新的公式,用于非线性相对面数字通量,并将直接不连续的Galerkin方法与刘延(H. Liu, J. Yan, 直接不连续的Galerkin(DGG) 界面校正(DDGIC))相校(DDGIC) 的界面校正(DDGIC) 结合界面校正(H. Liu, J. Yan, 直接不连续的Galerkin(DDGGG) 方法) 与界面校正(H. Liu, J. Yan, 直接不连续的Galerkin(DG) 8(3) (2010 541) 的通信) 相匹配,用于压缩可压缩的 Navier- Stokeks 方程式。 新的DDGIC 和ROUMQA 的精确度计算方法将进行精确到显示的DGA+DGMA的精度。