Multi-revolution composition methods for highly oscillatory differential equations

We introduce a new class of multi-revolution composition methods for the approximation of the N th-iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, the technique benefits from the properties of standard compositi...

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Bibliographic Details
Published inNumerische Mathematik Vol. 128; no. 1; pp. 167 - 192
Main Authors Chartier, Philippe, Makazaga, Joseba, Murua, Ander, Vilmart, Gilles
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2014
Springer Verlag
Subjects
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Simulation
Theoretical
35Q55
37L05
34K33
near-identity map
differential equation
highly-oscillatory
asymptotic preserving
averaging
geometric integration
composition method
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ISSN0029-599X
0945-3245
DOI10.1007/s00211-013-0602-0

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Summary:We introduce a new class of multi-revolution composition methods for the approximation of the N th-iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, the technique benefits from the properties of standard composition methods: it is intrinsically geometric and well-suited for Hamiltonian or divergence-free equations for instance. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schrödinger equation, illustrate the theoretical results, as well as the efficiency and versatility of the methods.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-013-0602-0