{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T15:01:15Z","timestamp":1780066875629,"version":"3.54.0"},"reference-count":129,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,9,1]],"date-time":"2018-09-01T00:00:00Z","timestamp":1535760000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/axioms7030063","type":"journal-article","created":{"date-parts":[[2018,9,3]],"date-time":"2018-09-03T10:50:51Z","timestamp":1535971851000},"page":"63","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":49,"title":["Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8201-8372","authenticated-orcid":false,"given":"Michael","family":"Dumbser","sequence":"first","affiliation":[{"name":"Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6070-8372","authenticated-orcid":false,"given":"Francesco","family":"Fambri","sequence":"additional","affiliation":[{"name":"Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maurizio","family":"Tavelli","sequence":"additional","affiliation":[{"name":"Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Bader","sequence":"additional","affiliation":[{"name":"Department of Informatics, Technical University of Munich, D-85748 Munich, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tobias","family":"Weinzierl","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Durham, Durham DH1 3LE, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,1]]},"reference":[{"key":"ref_1","unstructured":"Riemann, B. 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