Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine

Dumbser, Michael; Fambri, Francesco; Tavelli, Maruizio; Bader, Michael; Weinzierl, Tobias

Abstract

In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication-avoiding and cache-blocking, and are furthermore very well-suited for vectorization, and so they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing levels of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes

Citation

Dumbser, M., Fambri, F., Tavelli, M., Bader, M., & Weinzierl, T. (2018). Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine. Axioms, 7(3), Article 63. https://doi.org/10.3390/axioms7030063

• Journal Article Type: Article
• Acceptance Date: Aug 22, 2018
• Online Publication Date: Sep 1, 2018
• Publication Date: Sep 1, 2018
• Deposit Date: May 21, 2018
• Publicly Available Date: Sep 20, 2018
• Journal: Axioms
• Electronic ISSN: 2075-1680
• Publisher: MDPI
• Peer Reviewed: Peer Reviewed
• Volume: 7
• Issue: 3
• Article Number: 63
• DOI: https://doi.org/10.3390/axioms7030063
• Public URL: https://durham-repository.worktribe.com/output/1330959
• Related Public URLs: https://arxiv.org/abs/1808.03788