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 DGfinite element methods are derived on staggered Cartesian grids for thesolution of the inc.NS equations in two and three space dimensions. Pressureand velocity are expressed in the form of piecewise polynomials along differentmeshes. 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 firstfamily, h.o. of accuracy is achieved only in space, while a simplesemi-implicit time discretization is derived for the pressure gradient in themomentum equation. The resulting linear system for the pressure is symmetricand 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-freeconjugate gradient method. The use of a preconditioner was not necessary. Thisis a rather unique feature among existing implicit DG schemes for the NSequations. In order to avoid a stability restriction due to the viscous terms,the latter are discretized implicitly. The second family of staggered DGschemes achieves h.o. of accuracy also in time by expressing the numericalsolution in terms of piecewise space-time polynomials. In order to circumventthe low order of accuracy of the adopted fractional stepping, a simpleiterative Picard procedure is introduced. In this manner, the symmetry andpositive definiteness of the pressure system are not compromised. The resultingalgorithm is stable, computationally very efficient, and at the same timearbitrary h.o. accurate in both space and time. The new numerical method hasbeen thoroughly validated for approximation polynomials of degree up to N=11,using a large set of non-trivial test problems in two and three spacedimensions, for which either analytical, numerical or experimental referencesolutions exist.