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

Abstract

In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier–Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Although the resulting adaptive staggered grids are more complex than classical uniform Cartesian meshes, the numerical scheme can still be written in a rather compact form. Thanks to the use of a tensor-product formulation for the definition of the nodal basis in the d-dimensional space (d=2,3), all the discrete operators can be efficiently written as a combination of linear one-dimensional operators acting in the d space directions separately.Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier–Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. In order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor, an implicit discretization is also used for the diffusive terms in the momentum equation. A particular feature of our staggered DG approach is that the viscous stress tensor is discretized on the dual mesh. This corresponds to the use of a lifting operator, but on the staggered grid. Both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix-free conjugate gradient method, for which fast convergence is observed. Moreover, it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner. Due to the explicit discretization of the nonlinear convective terms, the final algorithm is stable within the classical CFL-type time step restriction for explicit DG methods based on the fluid velocity. Moreover, by using a suitable local time stepping technique based on the ADER-DG framework, each element can use its optimal local time step also within the AMR context.The new space-time adaptive staggered DG scheme has been thoroughly verified for polynomial degrees up to N=9 for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. The high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution.To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier–Stokes equations on space-time adaptive meshes in two and three space dimensions.