A high‐order discontinuous Galerkin method for the incompressible Navier‐Stokes equations on arbitrary grids

Abstract

Summary A high‐order discontinuous Galerkin (DG) method is proposed in this work for solving the two‐dimensional steady and unsteady incompressible Navier‐Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax‐Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.

Parallel discontinuous Galerkin finite element method for computing hyperbolic conservation law on unstructured meshes

“…Compared with the NASA data, the calculation results of the present algorithm are not very ideal, but it can be seen that the speed-up trend of the parallel algorithm is also basically linear. A comparison between the present method and the parallel method (Hicken and Zingg, 2008) has been given in Figure 14(b). Compared with the parallel algorithm in the reference, the present algorithm has higher parallel efficiency.…”

“…Xia et al (2014) propose a parallel h-adaptive reconstructed DGM for three-dimensional compressible flows. Zhang et al (2019) add an artificial compressibility term for relaxing the incompressibility constraint and apply the LLF flux to present DGM for the two-dimensional unsteady and steady incompressible Navier-Stokes (NS) equations on arbitrary grids.…”

“…Moreover, discontinuous Galerkin methods require much more storage resources on unstructured grids. To improve the efficiency, the various numerical schemes and parallel algorithms are developed in literature (Luo et al, 2011;Xia et al, 2014;Zhang et al, 2019). Xia et al (2014) propose a parallel h-adaptive reconstructed DGM for three-dimensional compressible flows.…”

High precision compact numerical approximation in exponential form for the system of 2D quasilinear elliptic BVPs on a discrete irrational region

“…A second order FDM for a system of EPDEs in a square domain has been given by Pandey and Jaboob [48] . Zhang et al [6] have proposed a highly accurate discontinuous Galerkin method for solving the 2D NS equations in conservative form on arbitrary grids. In 2020, Li and Zhang [50] have reduced the classical continuous FEM to a high order FDM applied to a 2D elliptic equation with smooth coefficients on a rectangular domain.…”

Higher order approximation in exponential form based on half-step grid-points for 2D quasilinear elliptic BVPs on a variant domain

“…Further, Zhang et al. [47] have solved 2D NS equations using a Galerkin method. Nadeem and Aziz [63] have designed a meshless and a Haar wavelet collocation method for 2D and 3D PDEs of elliptic type.…”