Space-time discontinuous Galerkin method for compressible flow
Author Klaij, C.M.
Title Space-time discontinuous Galerkin method for compressible flow
Conference/Journal PhD-thesis, Twente University
Month September
Year 2006

In this thesis, a space-time discontinuous Galerkin (DG) method for the compressible Navier-Stokes equations is presented. We explain the space-time setting, derive the weak formulation and discuss our choices for the numerical fluxes. The resulting numerical method allows local grid adaptation as well as moving and deforming boundaries, which we illustrate by computing the flow around a 3D delta wing on an adapted mesh and by simulating the dynamic stall phenomenon of a 2D airfoil in rapid pitch-up maneuver.

The space-time DG discretization 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 show their performance for 2D and 3D simulations.

To improve the efficiency of the method, we apply fully explicit multigrid pseudo-time integration with Runge-Kutta smoothing. We analyze the convergence of multigrid iteration for solving the algebraic equations arising from a space-time DG discretization of the advection-diffusion equation. Depending on the P├ęclet number, we find multigrid convergence factors between 0.50 and 0.74 with Fourier two-level analysis. We illustrate the analysis with a numerical example.

For the Navier-Stokes equations, we consider multigrid with linear basis functions on the fine grid and constant basis functions on the coarse grids. This facilitates the definition of inter-grid transfer operators on non-uniform grids. Two-level Fourier analysis shows that this multigrid iteration converges twice as fast as single-grid iteration for steady-state cases. This prediction is confirmed by the 2D and 3D simulations.

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