Mehlin, Michaela. Efficient explicit time integration for the simulation of acoustic and electromagnetic waves. 2015, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_11437
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Abstract
The efficient and accurate numerical simulation of timedependent wave phenomena is of fundamental importance in acoustic, electromagnetic or seismic wave propagation.
Model problems describing wave propagation include the wave equation and Maxwell's equations, which we study in this work. Both models are partial differential equations in space and time. Following the methodoflines approach we first discretize the two model problems in space using finite element methods (FEM) in their continuous or discontinuous form. FEM are increasingly popular in the presence of heterogeneous media or complex geometry due to their inherent flexibility: elements can be small precisely where small features are located, and larger elsewhere. Such a local mesh refinement, however, also imposes severe stability constraints on explicit time integration, as the maximal timestep is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small region, the use of implicit methods, or a very small timestep in the entire computational domain, are generally too high a price to pay.
Local timestepping (LTS) methods alleviate that geometry induced stability restriction by dividing the elements into two distinct regions: the "coarse region" which contains the larger elements and is integrated in time using an explicit method, and the "fine region" which contains the smaller elements and is integrated in time using either smaller timesteps or an implicit scheme.
Here we first present LTS schemes based on explicit RungeKutta (RK) methods. Starting from classical or lowstorage explicit RK methods, we derive explicit LTS methods of arbitrarily high accuracy.
We prove that the LTSRKs(p) methods yield the same rate of convergence as the underlying RKs scheme. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTSRK methods.
As a second method we propose local exponential AdamsBashforth (LexpAB) schemes. Unlike LTS schemes, LexpAB methods overcome the severe stability restrictions caused by local mesh refinement not by integrating with a smaller timestep but by using the exact matrix exponential in the fine region. Thus, they present an interesting alternative to the LTS schemes. Numerical experiments in 1D and 2D confirm the expected order of convergence and demonstrate the versatility of the approach in cases of extreme refinement.
Model problems describing wave propagation include the wave equation and Maxwell's equations, which we study in this work. Both models are partial differential equations in space and time. Following the methodoflines approach we first discretize the two model problems in space using finite element methods (FEM) in their continuous or discontinuous form. FEM are increasingly popular in the presence of heterogeneous media or complex geometry due to their inherent flexibility: elements can be small precisely where small features are located, and larger elsewhere. Such a local mesh refinement, however, also imposes severe stability constraints on explicit time integration, as the maximal timestep is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small region, the use of implicit methods, or a very small timestep in the entire computational domain, are generally too high a price to pay.
Local timestepping (LTS) methods alleviate that geometry induced stability restriction by dividing the elements into two distinct regions: the "coarse region" which contains the larger elements and is integrated in time using an explicit method, and the "fine region" which contains the smaller elements and is integrated in time using either smaller timesteps or an implicit scheme.
Here we first present LTS schemes based on explicit RungeKutta (RK) methods. Starting from classical or lowstorage explicit RK methods, we derive explicit LTS methods of arbitrarily high accuracy.
We prove that the LTSRKs(p) methods yield the same rate of convergence as the underlying RKs scheme. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTSRK methods.
As a second method we propose local exponential AdamsBashforth (LexpAB) schemes. Unlike LTS schemes, LexpAB methods overcome the severe stability restrictions caused by local mesh refinement not by integrating with a smaller timestep but by using the exact matrix exponential in the fine region. Thus, they present an interesting alternative to the LTS schemes. Numerical experiments in 1D and 2D confirm the expected order of convergence and demonstrate the versatility of the approach in cases of extreme refinement.
Advisors:  Grote, Marcus J. 

Committee Members:  Lanteri, Stéphane 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Numerik (Grote) 
UniBasel Contributors:  Mehlin, Michaela and Grote, Marcus J. 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  11437 
Thesis status:  Complete 
Number of Pages:  111 S. 
Language:  English 
Identification Number: 

edoc DOI:  
Last Modified:  22 Jan 2018 15:52 
Deposited On:  02 Nov 2015 14:47 
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