A Stable and High-Order Accurate Discontinuous Galerkin Based Splitting Method for the Incompressible Navier-Stokes Equations

TL;DR

This paper introduces a high-order accurate, stable discontinuous Galerkin splitting method for incompressible Navier-Stokes equations, featuring a novel pressure postprocessing technique and demonstrated through 2D and 3D numerical tests.

Contribution

It presents a new DG-based splitting scheme with a modified upwind flux and a local Helmholtz projection postprocessing for improved stability and mass conservation.

Findings

Achieves second-order convergence in time.

Ensures local mass conservation of the velocity.

Demonstrates stability and accuracy in 2D and 3D simulations.

Abstract

In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on $H(\text{div})$ reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.