Séance Séminaire

Séminaire ACSIOM

An a posteriori subcell limiter for Discontinuous Galerkin methods based on robust Finite Volumes. Applications to Euler, MagnetoHydrodynamics and multi-phase flows

The purpose of this joined work is to propose a novel a posteriori finite volume subcell limiter technique for the Discontinuous Galerkin finite element method for nonlinear systems of hyperbolic conservation laws in multiple space dimensions that works well for arbitrary high order of accuracy in space and time and that does not destroy the natural subcell resolution properties of the DG method. High order time discretization is achieved via a one-step ADER approach that uses a local space?time discontinuous Galerkin predictor method to evolve the data locally in time within each cell. Our new limiting strategy is based on a paradigm, which a posteriori verifies the validity of a discrete candidate solution against physical and numerical detection criteria after each time step. For those troubled cells that need limiting, our new limiter approach recomputes the discrete solution by scattering the DG polynomials at the previous time step onto a set of N' = 2N + 1 finite volume subcells per space dimension. A robust but accurate finite volume scheme then updates the subcell averages of the conservative variables within the detected troubled cells. The recomputed subcell averages are subsequently gathered back into high order cell-centered DG polynomials on the main grid via a subgrid reconstruction operator. We illustrate the performance of the new a posteriori subcell finite volume limiter approach for (very high order) DG methods via the simulation of numerous test cases run on Cartesian and unstructured grids in two and three space dimensions, using DG schemes of up to tenth order of accuracy in space and time (N = 9). Moreover we will present the coupling of our approach with AMR techniques and for different systems of PDEs (Euler, MHD, multiphase).