A New Class of Uniformly Accurate Numerical Schemes for Highly Oscillatory Evolution Equations

We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing fo...

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Published in:	Foundations of computational mathematics Vol. 20; no. 1; pp. 1 - 33
Main Authors:	Chartier, Philippe, Lemou, Mohammed, Méhats, Florian, Vilmart, Gilles
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.02.2020 Springer Nature B.V Springer Verlag
Subjects:	Applications of Mathematics Computer Science Economics Evolution Linear and Multilinear Algebras Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Stiffness Standard averaging Uniform accuracy Highly oscillatory problems Stroboscopic averaging 65L20 35K15 Pullback method 74Q10 Micro–macro method uniform accuracy AMS subject classification : 65L20 pullback method standard averaging uniform accuracy micro-macro method stroboscopic averaging highly-oscillatory problems
ISSN:	1615-3375, 1615-3383
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Abstract	We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.
AbstractList	We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime. We introduce a new methodology to design uniformly accurate methods for oscil-latory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime. We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum of regimes, from non-stiff to highly oscillatory. Thanks to an averaging transformation, the stiffness of the problem is softened, allowing for standard schemes to retain their usual orders of convergence. Overall, high-order numerical approximations are obtained with errors and at a cost independent of the regime.
Author	Vilmart, Gilles Chartier, Philippe Méhats, Florian Lemou, Mohammed
Author_xml	– sequence: 1 givenname: Philippe surname: Chartier fullname: Chartier, Philippe email: Philippe.Chartier@inria.fr organization: Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625 – sequence: 2 givenname: Mohammed surname: Lemou fullname: Lemou, Mohammed organization: Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625 – sequence: 3 givenname: Florian surname: Méhats fullname: Méhats, Florian organization: Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625 – sequence: 4 givenname: Gilles surname: Vilmart fullname: Vilmart, Gilles organization: Université de Genève, Section de mathématiques
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Keywords	Standard averaging Uniform accuracy Highly oscillatory problems Stroboscopic averaging 65L20 35K15 Pullback method 74Q10 Micro–macro method uniform accuracy AMS subject classification : 65L20 pullback method standard averaging uniform accuracy micro-macro method stroboscopic averaging highly-oscillatory problems
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Snippet	We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum... We introduce a new methodology to design uniformly accurate methods for oscillatory evolution equations. The targeted models are envisaged in a wide spectrum... We introduce a new methodology to design uniformly accurate methods for oscil-latory evolution equations. The targeted models are envisaged in a wide spectrum...
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Title	A New Class of Uniformly Accurate Numerical Schemes for Highly Oscillatory Evolution Equations
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