Prof. Dr. Michael Dumbser
In this talk we present a unified family of high order accurate finite volume and discontinuous Galerkin finite element schemes on moving unstructured and adaptive Cartesian meshes for the solution of conservative and non-conservative hyperbolic partial differential equations. The PNPM approach adopted here uses piecewise polynomials uh of degree N to represent the data in each cell. For the computation of fluxes and source terms, another set of piecewise polynomials wh of degree M ≥ N is used, which is computed from the underlying polynomials uh using a reconstruction or recovery operator. The PNPM method contains classical high order finite volume schemes (N = 0) and high order discontinuous Galerkin (DG) finite element methods (N = M) as two special cases of a more general class of numerical schemes. The schemes are derived in general ALE form so that Eulerian schemes on fixed meshes and Lagrangian schemes on moving meshes can be recovered as special cases of the ALE formulation. Furthermore, the method can also be naturally implemented on space-time adaptive Cartesian grids (AMR), together with time-accurate local time stepping (LTS). To assure the robustness of the method at discontinuities, a nonlinear WENO reconstruction is performed. The time integration is carried out in one single step using a high order accurate local space-time Galerkin predictor that is also able to deal with stiff source terms.
Applications are shown for the compressible Euler and Navier-Stokes equations, for the MHD equations and for the Baer-Nunziato model of compressible multi-phase flows.
Datum: 19.06.2014, Raum: G03-106, Zeit: 17:00