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

In this paper two new families of arbitrary high order accurate spectral DG finite element methods are derived on staggered Cartesian grids for the solution of the inc.NS equations in two and three space dimensions. 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 a spatially staggered mesh. In the first family, h.o. of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived for the pressure gradient in the momentum equation. The resulting linear system for the pressure is symmetric and positive definite and either block 5-diagonal (2D) or block 7-diagonal (3D) and can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The use of a preconditioner was not necessary. This is a rather unique feature among existing implicit DG schemes for the NS equations. In order to avoid a stability restriction due to the viscous terms, the latter are discretized implicitly. The second family of staggered DG schemes achieves h.o. 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. 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 h.o. accurate in both space and time. 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 either analytical, numerical or experimental reference solutions exist.