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Axioms 2018, 7(3), 63; https://doi.org/10.3390/axioms7030063

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

1
Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy
2
Department of Informatics, Technical University of Munich, D-85748 Munich, Germany
3
Department of Computer Science, University of Durham, Durham DH1 3LE, UK
*
Author to whom correspondence should be addressed.
Received: 21 May 2018 / Revised: 21 August 2018 / Accepted: 22 August 2018 / Published: 1 September 2018
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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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. View Full-Text
Keywords: hyperbolic partial differential equations; high order discontinuous Galerkin finite element schemes; shock waves and discontinuities; vectorization and parallelization; high performance computing hyperbolic partial differential equations; high order discontinuous Galerkin finite element schemes; shock waves and discontinuities; vectorization and parallelization; high performance computing
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Dumbser, M.; Fambri, F.; Tavelli, M.; Bader, M.; Weinzierl, T. Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine. Axioms 2018, 7, 63.

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