Pseudo-time stepping methods for space–time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Abstract

The space-time discontinuous Galerkin discretization of the compressible Navier-Stokes equations results in a non-linear system of algebraic equations, which we solve with pseudo-time stepping methods. We show that explicit Runge-Kutta methods developed for the Euler equations suffer from a severe stability constraint linked to the viscous part of the equations and propose an alternative to relieve this constraint while preserving locality. To evaluate its effectiveness, we compare with an implicit-explicit Runge-Kutta method which does not suffer from the viscous stability constraint. We analyze the stability of the methods and illustrate their performance by computing the flow around a 2D airfoil and a 3D delta wing at low and moderate Reynolds numbers.

Multigrid Optimization for Space-Time Discontinuous Galerkin Discretizations of Advection Dominated Flows

“…For the 2D advection-diffusion equation we have shown that by minimizing the spectral radius of the multigrid error transformation operator, a significant improvement in the multigrid performance can be achieved. The algorithms have been tested on a 2D problem containing boundary layers, where the optimized Runge-Kutta smoothers show a significant improvement compared to the original EXI-EXV Runge-Kutta smoother discussed in [2,3]. …”

Discontinuous Galerkin finite element methods for (non)conservative partial differential equations

“…This method was applied in the early 2000's in a space-time DG context in [79] for second order accurate discretizations of the Euler equations and has been a preferred method to solve space-time DG discretizations since [4,40,41,42,61,64]. Our objective is to improve this method for higher order space-time DG discretizations since it preserves the locality of the DG scheme and is very useful in a FAS multigrid scheme for nonlinear problems.…”

“…We introduce a rescaling that better balances the different operators for all aspect ratio cells, whether the flow is inviscid or viscous. This in contrast to the pseudo-time algorithm in [40] in which a combination of two RungeKutta smoothers was used for flows with inviscid and viscous regions: the EXI Runge-Kutta smoother for inviscid flows and the EXV Runge-Kutta smoother for viscous flows. This scaling is important since it also allows to optimize the multigrid algorithm for only the unit aspect ratio.…”

“…Depending on the stability of the smoother, we use different CFL τ and VN τ numbers. For the original EXI-EXV pseudo-time stepping scheme, see [40], we define with h = ∆x 1 √ 1 + AR 2 . The EXV scheme was used when Re i ≤ 1 and the EXI scheme was used otherwise.…”

“…The EXV scheme was used when Re i ≤ 1 and the EXI scheme was used otherwise. For the EXI-EXV smoother in [40], no massmatrix was used in the pseudo-time calculation. The mass-matrix was, however, used in the computations with the optimized smoothers.…”