A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier-Stokes equations

TL;DR

This paper introduces a high-order semi-implicit discontinuous Galerkin method on staggered unstructured meshes for efficiently solving 2D incompressible Navier-Stokes equations, with a focus on computational efficiency and accuracy.

Contribution

It presents a novel high-order DG scheme on staggered meshes that simplifies pressure computation to a single sparse system, enhancing efficiency and flexibility on complex domains.

Findings

The method achieves high-order accuracy up to polynomial degree 3.

The pressure system is sparse, symmetric, and positive definite under suitable conditions.

Numerical tests validate the method's effectiveness and accuracy.

Abstract

In this paper we propose a new spatially high order accurate semi-implicit discontinuous Galerkin (DG) method for the solution of the two dimensional incompressible Navier-Stokes equations on staggered unstructured curved meshes. While the discrete pressure is defined on the primal grid, the discrete velocity vector field is defined on an edge-based dual grid. The flexibility of high order DG methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids. Formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse block four-diagonal linear equation system for only one scalar unknown, namely the pressure. The method is computationally efficient, since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions. Furthermore, all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage, which leads to savings in terms of computational effort for the time evolution part. In this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step. The method is validated for polynomial degrees up to $p=3$ by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data.