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.
Axioms 2018, 7(3), 63; https://doi.org/10.3390/axioms7030063
Received: 21 May 2018
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Revised: 21 August 2018
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Accepted: 22 August 2018
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Published: 1 September 2018
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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.
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Keywords:
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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MDPI and ACS Style
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. https://doi.org/10.3390/axioms7030063
AMA Style
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(3):63. https://doi.org/10.3390/axioms7030063
Chicago/Turabian StyleDumbser, Michael, Francesco Fambri, Maurizio Tavelli, Michael Bader, and Tobias Weinzierl. 2018. "Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine" Axioms 7, no. 3: 63. https://doi.org/10.3390/axioms7030063
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