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Numerical Investigation of Parallel-In-Time Methods for Dominantly Hyperbolic Equations

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Numerical Investigation of Parallel-In-Time Methods for Dominantly Hyperbolic Equations Numerical Investigation of Parallel-in-Time Methods for Dominantly Hyperbolic Equations Vom Fachbereich Maschinenbau an der Technischen Universität Darmstadt zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Dissertation vorgelegt von Andreas Schmitt, M.Sc. aus Düsseldorf Berichterstatter: Prof. Dr. rer. nat. Michael Schäfer Mitberichterstatter: Prof. Dr. rer. nat. Sebastian Schöps Tag der Einreichung: 23.07.2018 Tag der mündlichen Prüfung: 10.10.2018 Darmstadt 2019 D 17 Schmitt, Andreas Numerical Investigation of Parallel-in-Time Methods for Dominantly Hyperbolic Equations Technische Universität Darmstadt Tag der mündlichen Prüfung: 10.10.2018 Jahr der Veröffentlichung auf TUprints: 2019 URN: urn:nbn:de:tuda-tuprints-83286 Veröffentlicht unter CC BY-NC-ND 4.0 International https://creativecommons.org/licences/by-nc-nd/4.0/ Vorwort Die vorliegende Arbeit “Numerical Investigation of Parallel-in-Time Methods for Dominantly Hyperbolic Equations” ist mein Beitrag zum Forschungsgebiet der zeitparallelen Methoden. Diese Dissertation wurde als Abschlussarbeit meines Promotionsvorhabens an der Technischen Universität Darmstadt am Fachgebiet für Numerische Berechnungsverfahren im Maschinenbau und der Graduiertenschule Computational Engineering verfasst. Mit den Inhalten der Dissertation habe ich mich von Mai 2015 bis Juli 2018 beschäftigt. Ich möchte mich bei Prof. Michael Schäfer für die Möglichkeit zur Pro- motion, den Vorschlag des spannenden Themengebietes und die Betreuung bedanken. Insbesondere danke ich Ihm auch für die gegebenen Freiheiten, so dass ich das Thema nach meinen Interessen erforschen konnte. Des Weiteren danke ich Prof. Sebastian Schöps für die Übernahme der Mitberichterstatter- schaft und die konstruktiven Gespräche. Für viele konstruktive Gespräche möchte ich mich auch bei den Mitentwick- lern von SWEET bedanken. Ohne den regen fachlichen Austausch mit Martin Schreiber wäre die Arbeit nicht dieselbe. Prof. Pedro Peixoto danke ich unter Anderem für klärende Antworten zu der semi-Lagrangian Methode. Zudem gilt mein Dank allen Kollegen am FNB und der GSC, die mich über die Jahre begleitet haben, sowohl für den fachlichen Austausch, als auch den Spaß bei verschiedenen Freizeitaktivitäten. Die angenehme Arbeitsumgebung werde ich in Erinnerung behalten. Namentlich erwähnt seien hier Jakob, dessen Bürotüre nie für Disskussionen jeder Art verschlossen war, und Do- minic, der in der Phase des Zusammenschreibens ein Leidensgenosse zum gegenseitigen Motivieren war. Christian und Michael bin ich für die Unter- stützung in allen technischen Fragen dankbar. Ebenfalls dankbar bin ich Monika für ihre Hilfe bei jeglichen bürokratischen Angelegenheiten. III Abschließend danke ich noch meiner Familie und meinen Freunden, die mich während der Promotion begleitet haben. Ein besonderer Dank gilt meinen Eltern, die mich durch alle Höhen und Tiefen uneingeschränkt unterstützt haben. Darmstadt, Juli 2018 Andreas Schmitt IV Abstract Simulations aid in many scientific and industrial applications. A general am- bition for these simulations is to keep the time-to-solution as small as possible while maintaining a desired accuracy. Besides with high computational power, this can be achieved by employing multiple processing units with paralleliza- tion. Today’s state of the art is the spatial parallelization which provides a very good parallel efficiency. However, such a parallelization introduces communication and synchroniza- tion overheads leading to a maximal number of processing units which can be used efficiently. Applying a parallelization in time on top of the paralleliza- tion in space makes using more processing units possible. An issue of the parallel-in-time methods is their problem dependent efficiency. It tends to be generally bad for dominantly hyperbolic problems. The viscous Burgers equa- tion, which for small viscosities falls into that category, is used to investigate two methods of parallelization in time. First, a look is taken at the Adomian decomposition method (ADM) and possibilities of exploiting additional degrees of parallelism within this method. Its viability is questioned by comparing its discrete version (DADM) to the explicit Runge-Kutta method (ERK). The comparison shows similar restric- tions regarding their maximal time step size for both methods. Furthermore, the DADM leads to larger errors with increasing order of accuracy compared to the ERK. However, discussing the parallelization within the DADM shows a reduction of the runtime complexity from quadratic to linear is possible. With this reduction in the runtime DADM seems to be a viable competitor to the ERK. This is especially true for high-order schemes, as fewer function evaluations have to be run serial. Increasing the order of accuracy is also embarrassingly easy with the DADM compared to the ERK. V The second method investigated in this thesis is the Parareal algorithm. Here, the focus lies on the potential of the implicit Runge-Kutta method with semi-Lagrangian advection (SLIRK) as the coarse solver for the Parareal al- gorithm. Its potential compared to using the explicit Runge-Kutta method (ERK) and the implicit-explicit Runge-Kutta method (IMEX) is tested with three different benchmarks. The comparison shows the ERK is in contrast to the other two methods not able to provide speedup potential with the chosen benchmarks. For advection dominated problems SLIRK performs better than IMEX due to its stability. The stability of SLIRK leads to speedup poten- tial for a larger range of viscosities with the Parareal algorithm. Still, the instability of Parareal itself causes a decreasing potential with a decreasing viscosity. With an inviscid case the number of iterations to convergence for Parareal is too large to yield a reasonable speedup. An additional result worth mentioning is it was possible to show the importance of predicting the phase of the solution correctly for the convergence of Parareal. VI Kurzfassung Viele wissenschaftliche und industrielle Anwendungen werden von Simula- tionen unterstützt. Bei diesen Simulationen ist eine so gering wie mögliche Laufzeit bei einer vorgegebenen Genauigkeit gewünscht. Außer durch hohe Rechenleistung kann dies durch die Nutzung mehrerer Recheneinheiten mit Hilfe von Parallelisierung verringert werden. Derzeitiger Stand der Technik ist eine Parallelisierung im Raum, welche eine gute Effizienz aufweist. Eine Parallelisierung erfordert jedoch unweigerlich Kommunikations- und Synchronisierungs-Overheads, welche zu einer maximalen Anzahl an Rech- eneinheiten führen, die effizient genutzt werden können. Ein Anwenden der Zeitparallelisierung zusätzlich zur räumlichen Parallelisierung kann die Ef- fizienzgrenze allerdings weiter nach oben verschieben. Die Effizienz der Zeit- parallelisierung ist jedoch abhängig von der Anwendung. Insbesondere Prob- leme, die vorwiegend hyperbolischer Natur sind, führen zu schlechter Effizienz. Anhand der viskosen Burgersgleichung, welche für kleine Viskositäten solch ein Problem ist, werden zwei Methoden der Zeitparallelisierung untersucht. Zunächst wird die Möglichkeit einer Parallelisierung der Adomian decom- position method (ADM) untersucht. Da es sich bei der diskreten Version (DADM) der ADM um ein explizites Zeitschrittverfahren handelt, wird die Brauchbarkeit der DADM in einem Vergleich zur etablierten expliziten Runge- Kutta Methode (ERK) untersucht. Der Vergleich zeigt, dass beide Methoden einen sehr ähnlichen maximalen Zeitschritt zulassen. Bei ansteigender Ver- fahrensordnung führt die DADM zu einem größeren Fehler verglichen mit der ERK. Mit der gezeigten Parallelisierung der DADM ist es möglich die quadratische Laufzeitkomplexität auf eine lineare zu verringern. Dies führt dazu, dass die DADM mit der ERK konkurrieren kann. Insbesondere in Fällen mit hoher Verfahrensordnung ist die geringere Anzahl von Funktionsauswer- VII tungen die seriell ablaufen ein Vorteil der DADM. Verglichen mit der ERK ist es sehr einfach die Verfahrensordnung der DADM zu erhöhen. Die zweite untersuchte Methode ist der Parareal Algorithmus. Hierbei wird ein besonderes Augenmerk darauf gelegt, welches Potential die implizite Runge-Kutta Methode mit einer semi-Lagrangian Advektion (SLIRK) als Groblöser des Parareal Algorithmus besitzt. Hierfür wird ein Vergleich mit der expliziten Runge-Kutta Methode (ERK) und der impliziten-expliziten Runge- Kutta Methode (IMEX) auf Basis von drei Benchmarks durchgeführt. Dieser Vergleich zeigt, dass die ERK in den getesteten Fällen kein Speedup Poten- tial besitzt. Bei kleinen Viskositäten, welche der Gleichung einen vorwiegend hyperbolischen Charakter geben, ist die SLIRK klar im Vorteil gegenüber der IMEX, da die SLIRK stabiler ist. Auf Grund dieser Stabilität führen mehr Viskositäten zu Speedup Potential mit dem Parareal Algorithmus. Ein Verringern der Viskosität resultiert jedoch in einem Anstieg der Zahl der It- erationen zur Konvergenz von Parareal durch die Instabilität von Parareal. Im Fall der Burgersgleichung ohne Viskosität kann kein Potential für Speedup gefunden werden. Zusätzlich ist zu Erwähnen, dass die korrekte Vorhersage der Phase der Lösung wichtig ist für eine schnelle Konvergenz von Parareal. VIII Contents 1 Introduction 1 1.1 Motivation . .1 1.2 State of the Art . .3 1.3 Research Question . .6 1.4 Structure of the Thesis . .7 2 Mathematical and Physical Basics 9 2.1 Classification of Partial Differential Equations . .9 2.2 Navier-Stokes and Burgers’ Equations . 11 2.2.1 Compressible Navier-Stokes Equations . 11 2.2.2 Incompressible
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