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

Abstract: Discontinuous Galerkin (DG) methods are known to suffer from increasingly restrictive time step constraints as the polynomial order increases, limiting their efficiency at high orders. In this paper, we introduce a novel locally implicit, but 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 captures the domain of dependence. As a result, our method remains stable for CFL numbers up to $1/\sqrt{d}$ 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 on both linear and nonlinear test cases.