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

摘要

In this paper we propose a novel arbitrary high order accurate semi-implicitspace-time DG method for the solution of the three-dimensional incompressibleNavier-Stokes equations on staggered unstructured curved tetrahedral meshes. Astypical for space-time DG schemes, the discrete solution is represented interms of space-time basis functions. This allows to achieve very high order ofaccuracy also in time, which is not easy to obtain for the incompressibleNavier-Stokes equations. Similar to staggered finite difference schemes, in ourapproach the discrete pressure is defined on the primary tetrahedral grid,while the discrete velocity is defined on a face-based staggered dual grid. Avery simple and efficient Picard iteration is used in order to derive aspace-time pressure correction algorithm that achieves also high order ofaccuracy in time and that avoids the direct solution of global nonlinearsystems. Formal substitution of the discrete momentum equation on the dual gridinto the discrete continuity equation on the primary grid yields a very sparsefive-point block system for the scalar pressure, which is conveniently solvedwith a matrix-free GMRES algorithm. From numerical experiments we find that thelinear system seems to be reasonably well conditioned, since all simulationsshown in this paper could be run without the use of any preconditioner. For apiecewise constant polynomial approximation in time and proper boundaryconditions, the resulting system is symmetric and positive definite. Thisallows us to use even faster iterative solvers, like the conjugate gradientmethod. The proposed method is verified for approximation polynomials of degreeup to four in space and time by solving a series of typical 3D test problemsand by comparing the obtained numerical results with available exact analyticalsolutions, or with other numerical or experimental reference data.