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Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine.

Dumbser, Michael and Fambri, Francesco and Tavelli, Maruizio and Bader, Michael and Weinzierl, Tobias (2018) 'Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine.', Axioms., 7 (3). p. 63.


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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Publisher statement:© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (
Date accepted:22 August 2018
Date deposited:20 September 2018
Date of first online publication:01 September 2018
Date first made open access:20 September 2018

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