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Local Time-Stepping Methods for Wave Propagation

English title Local Time-Stepping Methods for Wave Propagation
Applicant Grote Marcus
Number 169243
Funding scheme Project funding
Research institution Fachbereich Mathematik Departement Mathematik und Informatik Universität Basel
Institution of higher education University of Basel - BS
Main discipline Mathematics
Start/End 01.02.2017 - 31.01.2020
Approved amount 358'296.00
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Keywords (5)

numerical analysis; adaptive mesh refinement; finite element methods; local time-stepping; wave propagation

Lay Summary (German)

Lead
Von den kleinsten Wellen des sichtbaren Lichts im Nanometerbereich bis zu den kilometerlangen seismischen Wellen gewaltiger Erdbeben durchlaufen Wellen verschiedenster Art allgegenwärtig alle Bereiche unseres täglichen Lebens. Mathematisch werden alle diese verschiedensten Phänomene im Grunde durch die klassische Wellengleichung beschrieben, die schon im XVIII. Jahrhundert von D. Bernoulli und J.-B. R. d’Alembert hergeleitet wurde. Da sich deren Lösung nur in den einfachsten “akademischen” Fällen mit Bleistift und Papier bestimmen lässt, greift man in der Praxis auf Methoden der Numerischen Mathematik zurück, mit denen sich näherungsweise Wellenphänomene auch in kompliziertesten Situationen auf dem Computer simulieren lassen.
Lay summary

Numerische Verfahren zur Simulation von Wellenphänomenen basieren auf einer Raum- und Zeitdiskretisierung der Wellengleichung. Für die räumliche Diskretisierung sind finite Elemente Verfahren bestens geeignet, besonders weil sie auch in beliebig komplizierten
Geometrien einsetzbar sind. Für die zeitliche Diskretisierung sind explizite Zeitschrittverfahren besonders effizient
und lassen sich auch gut auf parallelen Hochleistungsrechnern einsetzen. Dabei benutzen Standardverfahren den gleichen Zeitschritt
im gesamten Rechengebiet, der wegen der CFL Stabilitätsbedingung durch die kleinste Maschenweite bestimmt ist. 
Dies ist bei lokal verfeinerten Gittern höchst ineffizient. Um diese CFL Stabilitätsschranke zu umgehen, wurden in den letzten Jahren
verschiedene lokale Zeitschrittverfahren entwickelt, die in den kleineren Elementen kleinere Zeitschritte und in den grösseren Elementen
grössere Zeitschritte erlauben.

Auch wenn lokale Zeitschrittverfahren sich in der Praxis schon bewährt haben, gibt es bisher noch keine
allgemeine Konvergenztheorie. Ziel dieses Projekts ist es deshalb, einerseits eine rigorose Konvergenztheorie für explizite lokale Zeitschritte zu etablieren
und andererseits deren Nutzen sowohl für Langzeitsimulationen wie auch zeitharmonische Simulationen zu demonstrieren.  
Wichtige Anwendungen sind die medizinische Bildgebung, die Seismik, der Mobilfunk oder auch die zerstörungsfreie Materialprüfung.  In all diesen Anwendungen bilden effiziente numerische Verfahren die Grundlage für effiziente und detailtreue Simulationen. 

Direct link to Lay Summary Last update: 05.04.2017

Responsible applicant and co-applicants

Employees

Publications

Publication
Parallel controllability methods for the Helmholtz equation
Grote Marcus J., Nataf Frédéric, Tang Jet Hoe, Tournier Pierre-Henri (2020), Parallel controllability methods for the Helmholtz equation, in Computer Methods in Applied Mechanics and Engineering, 362, 112846-112846.
On controllability methods for the Helmholtz equation
Grote Marcus J., Tang Jet Hoe (2019), On controllability methods for the Helmholtz equation, in Journal of Computational and Applied Mathematics, 358, 306-326.
Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation
Grote Marcus J., Mehlin Michaela, Sauter Stefan A. (2018), Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation, in SIAM Journal on Numerical Analysis, 56(2), 994-1021.

Collaboration

Group / person Country
Types of collaboration
Prof. Martin Gander, Université de Genève Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
Prof. Assyr Abdulle, EPFL Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Prof. Stefan Sauter, Universität Zürich Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Dr. Michaela Mehlin, KIT Karlsruhe Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Guest Lectures in Numerical Analysis and Applied Mathematics Individual talk High-order local time-stepping methods 18.11.2019 Umea Universitet, Sweden Grote Marcus;
European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) Talk given at a conference Parallel Controllability Methods for the Helmholtz equation 30.09.2019 Egmond aan Zee, Netherlands Tang Jet Hoe;
International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2019) Talk given at a conference Scalable HPC Solution of the Helmholtz Equation via Exact Controllability 25.08.2019 Vienna, Austria Tang Jet Hoe;
International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2019) Talk given at a conference Efficient Uncertainty Quantification for Wave Propagation in Complex Geometry 25.08.2019 Vienna, Austria Michel Simon;
Swiss Numerics Colloquium Poster Uncertainty Quantification for Wave Propagation in Complex Geometry 10.05.2019 USI, Lugano, Switzerland Michel Simon;
Swiss Numerics Colloquium Poster Scalable HPC Solution of the Helmholtz Equation via Exact Controllability 10.05.2019 Lugano, Switzerland Tang Jet Hoe;
Conference on Mathematics of Wave Phenomena Talk given at a conference Parallel HPC Solution of the Helmholtz Equation with Controllability Methods 23.07.2018 KIT, Karlsruhe, Germany Tang Jet Hoe;
Swiss Numerics Colloquium Poster Parallel HPC Solution of the Helmholtz Equation with Controllability Methods 20.04.2018 ETH, Zurich, Switzerland Tang Jet Hoe;
Zurich Colloquium ACM Individual talk High-Order Explicit Local Time-Stepping Methods For Wave Propagation 29.11.2017 ETH Zurich, Switzerland Grote Marcus;
6th Workshop on Parallel-in-Time Methods Talk given at a conference On exact controllability methods for the Helmholtz equation 23.10.2017 Monte Verità, Switzerland Tang Jet Hoe;
PDE afternoon talk series Individual talk High-Order Explicit Local Time-Stepping Methods For Wave Propagation 17.10.2017 University of Vienna, Austria Grote Marcus;
Numerical methods for wave propagation Talk given at a conference on Controllability Methods for the Helmholtz Equation 31.08.2017 LJLL, Université Pierre et Marie Curie, Paris, France Grote Marcus;
Colloquium in honor of P. Joly's 60th birthday Talk given at a conference High-Order Explicit Explicit Local Time-Stepping Methods For Wave Propagation 28.08.2017 ENSTA ParisTech, France Grote Marcus;
International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2017) Talk given at a conference Convergence Analysis of Energy Conserving Explicit Local Time-stepping Methods for the Wave Equation 15.05.2017 Minneapolis, United States of America Grote Marcus;
International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2017) Talk given at a conference Local Time-Stepping for the Solution of the Helmholtz Equation via Controllability Methods 15.05.2017 Minneapolis, United States of America Tang Jet Hoe;
Swiss Numerics Colloquium Poster Controllability Methods for the Helmholtz Equation in Bounded or Unbounded Domains 28.04.2017 Basel, Switzerland Tang Jet Hoe;
HONOM 2017 Talk given at a conference High-Order Explicit Local Time-Stepping Methods For Wave Propagation 27.03.2017 Stuttgart, Germany Grote Marcus;
Space-time methods for time-dependent PDEs Talk given at a conference High-Order Explicit Local Time-Stepping Methods For Wave Propagation 13.03.2017 Oberwolfach, Germany Grote Marcus;


Associated projects

Number Title Start Funding scheme
188583 Local Time-Stepping Methods: Stability, Convergence and Advanced Applications 01.02.2020 Project funding
188583 Local Time-Stepping Methods: Stability, Convergence and Advanced Applications 01.02.2020 Project funding

Abstract

Wave phenomena are ubiquitous across science, technology and medicine. Typical applications include ultrasound imaging, wireless communications and seismic tomography. In this proposal we shall analyze and further develop time integration methods for the numerical simulation of acoustic, electromagnetic or elastic wave phenomena. For the spatial discretization, we use either conforming finite elements or discontinuous Galerkin methods, which accomodate arbitrary meshes and geometry. For the time discretization, we consider recently derived local time-stepping (LTS) methods, which overcome the bottleneck due to local mesh refinement by taking smaller time-steps precisely where the smallest elements are located.Explicit LTS methods have already proved useful in many applications and shown nearly optimal speed-up on HPC architectures. Convergence (in the ODE sense) to the semi-discrete solution on a fixed mesh is fairly standard. However, a general convergence theory in the PDE sense, which establishes convergence to the (true) continuous solution as both the time-step and the mesh-size tend to zero, is still lacking.This proposal therefore aims at establishing a rigourous convergence theory for explicit LTS methods, which fall into two separate categories. Hence this proposal consists of two separate projects. In the first project, we shall prove optimal space-time convergence of LTS methods based on energy conserving leap-frog (LF) methods. Moreover, we shall compare the accuracy of different fourth-order LTS methods in particular for long-time simulations. In the second project, we shall derive a complete space-time convergence theory for Runge-Kutta (RK) based LTS methods. Moreover, we shall demonstrate the usefulness of LTS methods also in time-harmonic regimes, when the controllability method is used for the solution of the Helmholtz equation.
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