2025/07/09 by Kieran Ricardo, Ricardo, Kieran, Kenneth Duru +1
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Atmospheric and Oceanic Physics (physics.ao-ph) #Computational Engineering #Computational Physics (physics.comp-ph) #Electromagnetic Simulation and Numerical Methods #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Finance #Numerical Analysis (math.NA) #Numerical methods for differential equations #and Science (cs.CE)
paper · pdf · doi:10.48550/arxiv.2507.07304
openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 utilises the domain of dependence. As a result, our method remains stable for CFL numbers up to ≈ 1/√(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.