Discontinuous Galerkin method for incompressible viscous flow based on the entropically damped artificial compressibility

Abstract

The entropically damped artificial compressibility (EDAC) method was integrated into the Runge–Kutta discontinuous Galerkin (RKDG) framework to solve incompressible Navier–Stokes equations. The discontinuous Galerkin method (DG) is known for its robustness and compactness, making it an optimal choice for developing high-order numerical methods on unstructured meshes. The EDAC method adds an entropic damping term to the continuity equation of the traditional artificial compressibility method to alleviate pressure fluctuations during computation. This combination enhances numerical stability and improves computational accuracy, especially in capturing smooth and physically accurate pressure fields. The viscous terms of the momentum equations and the entropic damping term of the continuity equation are discretized together using the symmetric interior penalty Galerkin (SIPG) method by introducing the penalty term to ensure the continuity of the interface solution, which allows the framework to maintain numerical stability on unstructured grids and locally refined grids. Next, quadtree adaptive mesh refinement (AMR) technology was incorporated, and guardcells were used to realize data communication among leaf blocks, thereby enhancing the computational efficiency of the framework. Finally, several test cases were conducted to validate the framework’s precision, accuracy, boundary conditions, and effectiveness of the adaptive technology.