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Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödinger equations

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Abstract

This work is devoted to the numerical simulation of nonlinear Schrödinger and Klein–Gordon equations. We present a general strategy to construct numerical schemes which are uniformly accurate with respect to the oscillation frequency. This is a stronger feature than the usual so called “Asymptotic preserving” property, the last being also satisfied by our scheme in the highly oscillatory limit. Our strategy enables to simulate the oscillatory problem without using any mesh or time step refinement, and the orders of our schemes are preserved uniformly in all regimes. In other words, since our numerical method is not based on the derivation and the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, in the highly oscillatory limit regime, and in the intermediate regime with the same order of accuracy. In the same spirit as in Crouseilles et al. (J Comput Phys 248, 287–308, 2013), the method is based on two main ingredients. First, we embed our problem in a suitable “two-scale” reformulation with the introduction of an additional variable. Then a link is made with classical strategies based on Chapman–Enskog expansions in kinetic theory despite the dispersive context of the targeted equations, allowing to separate the fast time scale from the slow one. Uniformly accurate schemes are eventually derived from this new formulation and their properties and performances are assessed both theoretically and numerically.

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Notes

  1. In the sequel, we explicitly mention the dependence \(\mathcal{F}_0(U^\varepsilon _0)\) while \(\mathcal{F}_0\) stands for \(\mathcal{F}(0,\tau ,\underline{U}_0^\varepsilon )\) and \((\partial _t \mathcal{F})_0\) stands for \(\partial _t \mathcal{F}(0,\tau ,\underline{U}_0^\varepsilon )\).

  2. The notation \(\mathcal{O}_{X^\sigma }\) is used here for terms uniformly bounded in \(\mathbb {T}\) with the appropriate \(X^\sigma \)-norm.

  3. Notice that \(LX^\sigma \) is continuously embedded in \(X^{\sigma -2}\).

  4. For simplicity, we omit here the \(t\) dependency in \(h_1\) and \(h_2\) which are all evaluated at \(t=0\).

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Acknowledgments

We acknowledge support by the ANR-FWF Project Lodiquas (ANR-11-IS01-0003).

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Authors and Affiliations

  1. INRIA-Rennes Bretagne Atlantique, IPSO Project, Rennes, France

    Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou & Florian Méhats

  2. CNRS and IRMAR, Université de Rennes 1, Rennes, France

    Mohammed Lemou

  3. IRMAR, Université de Rennes 1, Rennes, France

    Florian Méhats

Authors
  1. Philippe Chartier
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  2. Nicolas Crouseilles
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  3. Mohammed Lemou
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  4. Florian Méhats
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Correspondence to Nicolas Crouseilles.

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Chartier, P., Crouseilles, N., Lemou, M. et al. Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödinger equations. Numer. Math. 129, 211–250 (2015). https://doi.org/10.1007/s00211-014-0638-9

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  • DOI: https://doi.org/10.1007/s00211-014-0638-9

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