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

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.