Patch dynamics: macroscopic simulation of multiscale systems

Samaey, Giovanni; Kevrekidis, Ioannis G.; Roose, Dirk

doi:10.1002/pamm.200700767

Cited by 4 publications

(4 citation statements)

References 7 publications

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“…The HMM method we suggest here is described in three separate steps. We follow the same strategy as in [1] for parabolic equations and in [14] for the onedimensional advection equation. See [8], [9] and [2] for additional details and proofs.…”

Section: Description Of the Methodsmentioning

confidence: 99%

Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time

Engquist,

Holst,

Runborg

2011

Preprint

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Multiscale problems are computationally costly to solve by direct simulation because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multiscale wave propagation following the framework of the heterogeneous multiscale method. The numerical methods couple simulations on macro-and microscales for problems with rapidly fluctuating material coefficients. The computational complexity of the new method is significantly lower than that of traditional techniques. We focus on HMM approximation applied to long time integration of onedimensional wave propagation problems in both periodic and non-periodic medium and show that the dispersive effect that appear after long time is fully captured.

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Section: Description Of the Methodsmentioning

confidence: 99%

Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time

Engquist,

Holst,

Runborg

2011

Preprint

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“…The HMM algorithm we use here is based on a central finite difference scheme fitted in the framework described in [1]. It is similar to the schemes in [3], for parabolic equations, and in [5] for the one dimensional advection equation. A more detailed description of the HMM algorithm follows:…”

Section: Heterogeneous Multiscale Methodsmentioning

confidence: 99%

Multiscale methods for the wave equation

Engquist

Holst

Runborg

2007

Proc Appl Math and Mech

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We consider the wave equation in a medium with a rapidly varying speed of propagation. We construct a multiscale scheme based on the heterogeneous multiscale method, which can compute the correct coarse behavior of wave pulses traveling in the medium, at a computational cost essentially independent of the size of the small scale variations. This is verified by theoretical results and numerical examples.

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“…We follow the same strategy as in [1] for parabolic equations and in [19] for the one-dimensional advection equation. See also [8].…”

Section: Hmm For the Wave Equationmentioning

confidence: 99%

“…c.f. discussion about micro solver boundary conditions in [19]. In this way we do not need to worry about the effects of boundary conditions.…”

Section: Hmm For the Wave Equationmentioning

confidence: 99%

Multi-scale methods for wave propagation in heterogeneous media

Engquist

Holst²,

Runborg³

2011

Communications in Mathematical Sciences

100

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Multi-scale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multi-scale wave propagation in the framework of heterogeneous multi-scale method. The numerical methods couples simulations on macro-and micro-scales for problems with rapidly oscillating coefficients. We show that the complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the micro-scale. A convergence proof is given and numerical results are presented for periodic problems in one, two and three dimensions. The method is also successfully applied to non-periodic problems and for long time integration where dispersive effects occur.

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Patch dynamics: macroscopic simulation of multiscale systems

Cited by 4 publications

References 7 publications

Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time

Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time

Multiscale methods for the wave equation

Multi-scale methods for wave propagation in heterogeneous media

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