CIRM - Videos & books Library - Highly-oscillatory evolution equations: averaging and numerics

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Authors : Lemou, Mohammed (Author of the conference)
CIRM (Publisher )

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Abstract : Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to capture low frequencies and transition regimes. We present (very briefly) two strategies allowing to remove this obstacle for a large class of evolution problems : a 2-scale method and a micro/macro method. Two different frameworks will be considered : constant frequency, and variable - possibly vanishing - frequency. The result of these approaches is the construction of numerical schemes whose order of accuracy no longer depends on the frequency of oscillation, one then speaks of uniform accuracy (UA) for these schemes. Finally, a new technique for systematizing these two methods will be presented. Its purpose is to reduce the number of inputs that the user must provide to apply the method in practice. In other words, only the values of the field defining the evolution equation (and not its derivatives) are used.These methods have been successfully applied to solve a number of evolution models: non-linear Schrödinger and Klein-Gordon equations, Vlasov-Poisson kinetic equation with strong magnetic field, quantum transport in graphene.

Keywords : multi-scale numerical methods; highly oscillatory equations; averaging; varying frequency

MSC Codes :
35Q55 - NLS-like equations (nonlinear Schrödinger)
37L05 - General theory, nonlinear semigroups, evolution equations
65L05 - Initial value problems for ODE (numerical method)

Additional resources :
https://www.cirm-math.fr/RepOrga/2355/Slides/slide_Mohammed_LEMOU.pdf

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2021-03-22_Lemou.mp4

Information on the Event

Event Title : Jean Morlet Chair 2021- Conference: Kinetic Equations: From Modeling Computation to Analysis / Chaire Jean-Morlet 2021 - Conférence : Equations cinétiques : Modélisation, Simulation et Analyse
Event Organizers : Bostan, Mihaï ; Jin, Shi ; Mehrenberger, Michel ; Montibeller, Celine
Dates : 22/03/2021 - 26/03/2021
Event Year : 2021
Event URL : https://www.chairejeanmorlet.com/2355.html

Citation Data

DOI : 10.24350/CIRM.V.19735003
Cite this video as: Lemou, Mohammed (2021). Highly-oscillatory evolution equations: averaging and numerics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19735003
URI : http://dx.doi.org/10.24350/CIRM.V.19735003

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Bibliography

CHARTIER, Philippe, LEMOU, Mohammed, MÉHATS, Florian, et al. A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations. Foundations of Computational Mathematics, 2020, vol. 20, no 1, p. 1-33. - https://doi.org/10.1007/s10208-019-09413-3

CHARTIER, Ph, LEMOU, Mohammed, MÉHATS, Florian, et al. Highly oscillatory problems with time-dependent vanishing frequency. SIAM Journal on Numerical Analysis, 2019, vol. 57, no 2, p. 925-944. - https://doi.org/10.1137/18M1203456

CROUSEILLES, Nicolas, JIN, Shi, LEMOU, Mohammed, et al. A micro-macro method for a kinetic graphene model in one space dimension. Multiscale Modeling & Simulation, 2020, vol. 18, no 1, p. 444-474. - https://doi.org/10.1137/18M1173770

CHARTIER, Philippe, LEMOU, Mohammed, MÉHATS, Florian, et al. Derivative-free high-order uniformly accurate schemes for highly-oscillatory systems. To appear 2021. - https://hal.inria.fr/hal-03141156

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