Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids

摘要

In this paper a new high order semi-implicit discontinuous Galerkin method(SI-DG) is presented for the solution of the incompressible Navier-Stokesequations on staggered space-time adaptive Cartesian grids (AMR) in two andthree space-dimensions. The pressure is written in the form of piecewisepolynomials on the main grid, which is dynamically adapted within acell-by-cell AMR framework. According to the time dependent main grid,different face-based spatially staggered dual grids are defined for thepiece-wise polynomials of the respective velocity components. Arbitrary highorder of accuracy is achieved in space, while a very simple semi-implicit timediscretization is obtained via an explicit discretization of the nonlinearconvective terms, and an implicit discretization of the pressure gradient inthe momentum equation and of the divergence of the velocity field in thecontinuity equation. The real advantages of the staggered grid arise in thesolution of the Schur complement associated with the saddle point problem ofthe discretized incompressible Navier-Stokes equations, i.e. after substitutingthe discrete momentum equations into the discrete continuity equation. Thisleads to a linear system for only one unknown, the scalar pressure. Indeed, theresulting linear pressure system is shown to be symmetric andpositive-definite. The new space-time adaptive staggered DG scheme has beenthoroughly verified for a large set of non-trivial test problems in two andthree space dimensions, for which analytical, numerical or experimentalreference solutions exist. To the knowledge of the authors, this is the firststaggered semi-implicit DG scheme for the incompressible Navier-Stokesequations on space-time adaptive meshes in two and three space dimensions.