Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids

摘要

In this paper two new families of arbitrary high order accurate spectral discontinuous Galerkin (DG) finite element methods are derived on staggered Cartesian grids for the solution of the incompressible Navier-Stokes (NS) equations in two and three space dimensions. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. Thanks to the use of a nodal basis on a tensor-product domain, all discrete operators can be written efficiently as a combination of simple one-dimensional operators in a dimension-by-dimension fashion. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor θ ∈ [0.5,1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. From our numerical experiments we find that the pressure system appears to be reasonably well-conditioned, since in all test cases shown in this paper the use of a preconditioner was not necessary. This is a rather unique feature among existing implicit DG schemes for the Navier-Stokes equations. In order to avoid a stability restriction due to the viscous terms, the latter are discretized implicitly using again a staggered mesh approach, where the viscous stress tensor is also defined on the dual mesh.The second family of staggered DG schemes proposed in this paper achieves high order of accuracy also in time by expressing the numerical solution in terms of piecewise space-time polynomials. In order to circumvent the low order of accuracy of the adopted fractional stepping, a simple iterative Picard procedure is introduced, which leads to a space-time pressure-correction algorithm. In this manner, the symmetry and positive definiteness of the pressure system are not compromised. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. These features are typically not easy to obtain all at the same time for a numerical method applied to the incompressible Navier-Stokes equations. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N = 11, using a large set of non-trivial test problems in two and three space dimensions, for which erither analyticl, numerical or experimental reference solutions exist.