Scalable ADER-DG Transport Method with Polynomial Order Independent CFL Limit
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914176720437248 |
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| author | Ricardo, Kieran Duru, Kenneth |
| author_facet | Ricardo, Kieran Duru, Kenneth |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scalable ADER-DG Transport Method with Polynomial Order Independent CFL Limit Ricardo, Kieran Duru, Kenneth Numerical Analysis Computational Engineering, Finance, and Science Atmospheric and Oceanic Physics Computational Physics 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. |
| title | Scalable ADER-DG Transport Method with Polynomial Order Independent CFL Limit |
| topic | Numerical Analysis Computational Engineering, Finance, and Science Atmospheric and Oceanic Physics Computational Physics |
| url | https://arxiv.org/abs/2507.07304 |