A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes

TL;DR

This paper introduces a high-order semi-implicit space-time discontinuous Galerkin method for 3D incompressible Navier-Stokes equations on unstructured tetrahedral meshes, achieving high accuracy and efficiency through a staggered grid and iterative pressure correction.

Contribution

It presents a novel high-order semi-implicit DG method with a staggered grid approach and efficient pressure correction for 3D incompressible flows on unstructured meshes.

Findings

High-order accuracy in space and time achieved.

Efficient pressure correction algorithm without global nonlinear solves.

Numerical experiments confirm method's accuracy and stability.

Abstract

In this paper we propose a novel arbitrary high order accurate semi-implicit space-time DG method for the solution of the three-dimensional incompressible Navier-Stokes equations on staggered unstructured curved tetrahedral meshes. As typical for space-time DG schemes, the discrete solution is represented in terms of space-time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier-Stokes equations. Similar to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. A very simple and efficient Picard iteration is used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner. For a piecewise constant polynomial approximation in time and proper boundary conditions, the resulting system is symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method. The proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3D test problems and by comparing the obtained numerical results with available exact analytical solutions, or with other numerical or experimental reference data.