Two novel classes of Discontinuous Galerkin Finite Element Methods for Time-Dependent PDE

Michael Dumbser
University of Trento
Thursday 19th January 2017
AULA CONSIGLIO VII PIANO - Edificio 14 La Nave - Dipartimento di Matemcatica MOX, Politecnico di Milano
The topic of this talk are two new classes of high order discontinuous Galerkin (DG) finite element methods for the numerical solution of time-dependent partial differential equations (PDE) in two and three space dimensions. The first class of schemes (semi-implicit DG methods on staggered grids) has been developed for the incompressible and the compressible Navier-Stokes equations. The key novelty is the use of staggered grids within a DG scheme. The special location of the data (pressure and velocity) on the staggered grid allows a significant simplification of the computation of the numerical fluxes across elements. Compared to standard collocated DG schemes, the linear system to be solved for the pressure has optimal properties: it is symmetric positive definite, has the minimum number of unknowns and uses the smallest possible stencil. The second class of methods (explicit ADER-DG schemes on collocated grids) is a fully-discrete predictor-corrector scheme with piecewise polynomial basis functions in space and time. Via a weak formulation of the PDE in spacetime, the predictor constructs an approximate solution of an element-local Cauchy problem in the small, i.e. based only on element-local initial data. For that reason, time consuming MPI communication can be avoided during the arithmetically intensive predictor step. The numerical fluxes across element interfaces are then taken into account in a subsequent corrector step. The new method is therefore particularly suitable for implementation on modern supercomputers. Furthermore, it allows the use of local time steps in a very natural and easy way, while remaining fully conservative and high order accurate in both space and time. Shock waves and other discontinuities are treated via a new a posteriori limiter, which does not rely on nonlinear data reconstruction or filtering, as usual DG limiters, but which solves the PDE again, using a more robust finite volume scheme on a locally refined subgrid. We show some examples from the fields of computational fluid dynamics, magneto-hydrodynamics (MHD) and continuum mechanics. References: M. Dumbser, O. Zanotti, R. Loubère and S. Diot. A Posteriori Subcell Limiting of the Discontinuous Galerkin Finite Element Method for Hyperbolic Conservation Laws. Journal of Computational Physics, 278:47–75, 2014 O. Zanotti, F. Fambri and M. Dumbser. Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement. Monthly Notices of the Royal Astronomical Society (MNRAS), 452:3010-3029, 2015 M. Dumbser, I. Peshkov, E. Romenski and O. Zanotti. High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. Journal of Computational Physics 314:824–862, 2016 M. Dumbser and R. Loubère. A simple robust and accurate a posteriori subcell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. Journal of Computational Physics, 319:163–199, 2016 M. Tavelli and M. Dumbser. A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes. Journal of Computational Physics, 319:294–323, 2016 contact: