Browsing by Author "Gassner, Gregor"

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Discontinuous Galerkin methods for the unsteady compressible Navier-Stokes equations
(2009) Gassner, Gregor; Munz, Claus-Dieter
In this work a new explicit arbitrary high order accurate discontinuous Galerkin finite element solver for the unsteady compressible Navier-Stokes equations is developed. Although the focus is on the compressible Navier-Stokes equations, the developed framework can directly be applied to other pure hyperbolic, pure parabolic or mixed hyperbolic/parabolic time dependent conservation laws. Discontinuous Galerkin finite element based methods have several important properties. They are locally conservative schemes, despite their affiliation to the class of finite element methods. They allow to use arbitrary unstructured non-conforming meshes, while remaining their formal (high) order of accuracy, even with skewed and anisotropic grid cells. The resolution can be adapted locally by increasing or decreasing the local polynomial degree, without the difficulties of a conforming finite element approach. The (formal) order of accuracy is essential a parameter, as one only has to choose the polynomial degree of the approximation. The most important property of discontinuous Galerkin schemes is that the solution in a grid cell depends only on data from grid cells sharing a face, independent of the approximation order. This compactness yields an excellent parallelizability of the method, which is essential for large scale computations. Based on the standard spatial discontinuous Galerkin framework several modifications to increase the computational efficiency are proposed. In a first step a novel construction guideline for modal and nodal basis functions on arbitrary shaped grid cells is introduced. For the discretization of problems with high order derivatives a novel weak formulation is introduced and applied to the second order compressible Navier-Stokes equations. For the approximation of the viscous fluxes a new approximation based on local Riemann solutions is used. This spatial discretization is combined with a new time discretization, which allows time consistent local time stepping. In order to validate the high accuracy and efficiency of the developed method, several test cases including the linearized Euler equations, the non-linear Euler equations and the compressible Navier-Stokes equations are considered. Finally, this method is applied to solve two and three dimensional compressible unsteady flow problems.