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Local Time-Stepping Methods for Wave Propagation

Research Project
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01.02.2017
 - 31.01.2020

Wave phenomena are ubiquitous across science, technology and medicine. Typical applications in- clude 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 meth- ods, 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.

Collaborations & Cooperations

2021 - Participation or Organization of Collaborations on a national level
Sauter, Stefan, Prof., Universität Zürich, Research cooperation

Funding

Local Time-Stepping Methods for Wave Propagation

SNF Projekt (GrantsTool), 10.2016-09.2019 (36)
PI : Grote, Marcus J..

Members (3)

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Marcus J. Grote

Principal Investigator
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Jet Hoe Tang

Project Member
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Simon René Jonas Michel

Project Member