C.M. Klaij

Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations

An overview is given of a space-time discontinuous Galerkin finite element method for the compressible Navier-Stokes equations. This method is well suited for problems with moving (free) boundaries which require the use of deforming elements. In addition, due to the local discretization, the space-time discontinuous Galerkin method is well suited for mesh adaptation and parallel computing. The algorithm is demonstrated with computations of the unsteady flow field about a delta wing and a NACA0012 airfoil in rapid pitch up motion.

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier-Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge-Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection-diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier-Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.