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

Abstract

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.

arXiv comment26 pages, 7 figures

References in corpus (4)

• Two-dimensional compressible viscous flow around a circular cylinder
• Third order Maximum-Principle-Satisfying Direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangle mesh
• A reference solution of the flow over a circular cylinder at Re=40
• A Sparse and High-Order Accurate Line-Based Discontinuous Galerkin Method for Unstructured Meshes