A staggered space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations on two-dimensional triangular meshes

TL;DR

This paper introduces a high-order semi-implicit space-time discontinuous Galerkin method on staggered unstructured triangular meshes for solving 2D incompressible Navier-Stokes equations, enabling accurate simulations on complex domains with coarse grids.

Contribution

It presents a novel high-order DG scheme on staggered meshes for incompressible flows, combining isoparametric elements, a sparse pressure system, and efficient iterative solvers.

Findings

Validated for polynomial degrees up to 4 in space and time.

Achieved high accuracy with coarse meshes on complex geometries.

Reduced computational stencil and avoided Riemann solvers.

Abstract

In this paper we propose a novel arbitrary high order accurate semi-implicit space-time discontinuous Galerkin method for the solution of the two dimensional incompressible Navier-Stokes equations on staggered unstructured triangular meshes. Isoparametric finite elements are used to take into account curved domain boundaries. The discrete pressure is defined on the primal triangular grid and the discrete velocity field is defined on an edge-based staggered dual grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier-Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. Formal substitution of the discrete momentum equation into the discrete continuity equation yields a sparse four-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. A very simple and efficient Picard iteration is then used in order to achieve high order of accuracy also in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier-Stokes equations. The flexibility and accuracy of high order space-time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The use of a staggered grid allows to avoid the use of Riemann solvers in several terms of the discrete equations and significantly reduces the total stencil size of the linear system that needs to be solved for the pressure. The proposed method is validated for approximation polynomials of degree up to $p=4$ in space and time by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data.