Scalable ADER-DG Transport Method with Polynomial Order Independent CFL Limit

TL;DR

This paper introduces a locally implicit, globally explicit ADER-DG scheme for transport problems that maintains stability at high polynomial orders with an element-width based CFL condition independent of polynomial degree, enhancing efficiency.

Contribution

The paper presents a novel ADER-DG method that achieves polynomial order independent CFL stability by solving local implicit problems, improving high-order transport simulations.

Findings

Stable for CFL numbers up to ~1/√d in d dimensions

Proven stability in 1D and extended to higher dimensions

Demonstrated accuracy on linear and nonlinear test cases

Abstract

Discontinuous Galerkin (DG) methods are known to suffer from increasingly restrictive explicit time-step constraints as the polynomial order increases, limiting their efficiency at high orders for explicit time-stepping schemes. In this paper, we introduce a novel \emph{locally implicit}, but \emph{globally explicit} ADER-DG scheme designed for transport-dominated problems. The method achieves a maximum stable time step governed by an element-width based CFL condition that is independent of the polynomial degree. By solving a set of element-local implicit problems at each time step, our approach more effectively utilises the domain of dependence. As a result, our method remains stable for CFL numbers up to $\approx 1/\sqrt{d}$ in $d$ spatial dimensions. We provide a rigorous stability proof in one dimension, and extend the analysis to two and three dimensions using a semi-analytical von Neumann stability analysis. The accuracy and convergence of the method are demonstrated through numerical experiments for both linear and nonlinear test cases, including numerical simulations of a transport problem on a cubed sphere 2D manifold embedded in 3D.