Résolution numérique des grands systèmes différentiels linéaires

Legras, Jean

doi:10.1007/bf02165235

Cited by 9 publications

(5 citation statements)

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“…For this reason they are of potential interest to the Maxwell equations. Exponential integrators do have a long history [10,21,22,27,30,31,38,46] and have undergone a revival during the last decade, see e.g. [4,9,11,24,26,33].…”

Section: Exponential Integrationmentioning

confidence: 99%

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Unconditionally stable integration of Maxwell’s equations

Verwer

Botchev

2009

Linear Algebra and its Applications

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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:

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Section: Exponential Integrationmentioning

confidence: 99%

“…Formula (4.2) can be found there. A closely related, somewhat later contribution, is [31]. In the recent literature this approach is sometimes called exponential time differencing, see e.g.…”

Section: The Exponential Integrator Ek2mentioning

confidence: 99%

Unconditionally stable integration of Maxwell’s equations

Verwer

Botchev

2009

Linear Algebra and its Applications

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“…This method, which goes back to [54,55] can be derived by interpolating the function g under the integral in (13) linearly and evaluating the integral. This approach is sometimes referred to as exponential time differencing [56,57] and is closely related to a class of exponential Runge-Kutta-Rosenbrock methods [58].…”

Section: Exponential Time Integrationmentioning

confidence: 99%

Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling

Botchev

2016

Journal of Computational and Applied Mathematics

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MSC: 65F60 65F30 65N22 65L05 35Q61 Keywords: Finite difference time domain (FDTD) method Matrix exponential Maxwell's equations Krylov subspace methods Photonic crystals a b s t r a c tThe exponential time integration, i.e., time integration which involves the matrix exponential, is an attractive tool for time domain modeling involving Maxwell's equations. However, its application in practice often requires a substantial knowledge of numerical linear algebra algorithms, such as Krylov subspace methods.In this note we discuss exponential Krylov subspace time integration methods and provide a simple guide on how to use these methods in practice. While specifically aiming at nanophotonics applications, we intentionally keep the presentation as general as possible and consider full vector Maxwell's equations with damping (i.e., with nonzero conductivity terms).Efficient techniques such as the Krylov shift-and-invert method and residual-based stopping criteria are discussed in detail. Numerical experiments are presented to demonstrate the efficiency of the discussed methods and their mesh independent convergence. Some of the algorithms described here are available as Octave/Matlab codes from www.math.utwente.nl/~botchevma/expm/.

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“…Krylov subspace methods have been a successful tool for doing this since the end of the eighties; we list chronologically [54,65,23,46,57,24,39]. This progress in numerical linear algebra has triggered advances in numerical time integration methods (see, e.g., [31,40,18]) and a revival of the exponential time integrators [41], which have been developed since the sixties [19,49,48].…”

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confidence: 99%

A residual concept for Krylov subspace evaluation of the $φ$ matrix function

Botchev,

Knizhnerman,

Tyrtyshnikov

2020

Preprint

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An efficient Krylov subspace algorithm for computing actions of the ϕ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient restarting procedure. For matrices with numerical range in the stable complex half plane, we analyze residual convergence and prove that the restarted method is guaranteed to converge for any Krylov subspace dimension. Numerical tests demonstrate efficiency of our approach for solving large scale evolution problems resulting from discretized in space time-dependent PDEs, in particular, diffusion and convection-diffusion problems.

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Résolution numérique des grands systèmes différentiels linéaires

Cited by 9 publications

References 0 publications

Unconditionally stable integration of Maxwell’s equations

Unconditionally stable integration of Maxwell’s equations

Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling

A residual concept for Krylov subspace evaluation of the $φ$ matrix function

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