Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

TL;DR

This paper introduces a high-order space-time adaptive DG finite element method with a posteriori sub-cell limiting for hyperbolic conservation laws, enhancing resolution and stability in complex fluid dynamics simulations.

Contribution

It presents the first implementation of a sub-cell finite volume limiter within an adaptive AMR framework for high-order DG schemes in multiple dimensions.

Findings

Enhanced resolution of fluid dynamics features.

Robustness in handling discontinuities.

Successful application to Euler and MHD equations.

Abstract

In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order \aposteriori sub-cell ADER-WENO finite volume \emph{limiter}. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in \cite{Dumbser2014}, the discrete solution within the troubled cells is \textit{recomputed} by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.