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
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 \(\partial _t \mathcal{F}(0,\tau ,\underline{U}_0^\varepsilon )\).
The notation \(\mathcal{O}_{X^\sigma }\) is used here for terms uniformly bounded in \(\mathbb {T}\) with the appropriate \(X^\sigma \)-norm.
Notice that \(LX^\sigma \) is continuously embedded in \(X^{\sigma -2}\).
For simplicity, we omit here the \(t\) dependency in \(h_1\) and \(h_2\) which are all evaluated at \(t=0\).
References
Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)
Bao, W., Dong, X.: Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime. Numer. Math. 120, 189–229 (2012)
Castella, F., Chartier, P., Murua, A., Méhats, F.: Stroboscopic averaging for the nonlinear Schrödinger equation, preprint HAL-00732850. http://hal.archives-ouvertes.fr
Chartier, P., Murua, A., Sanz-Serna, J.M.: Higher-order averaging, formal series and numerical integration I: B-series. Found. Comput. Math. 10(6), 695–727 (2010)
Crouseilles, N., Lemou, M., Méhats, F.: Asymptotic preserving schemes for highly oscillatory kinetic equations. J. Comput. Phys. 248, 287–308 (2013)
Degond, P.: Macroscopic limits of the Boltzmann equation: a review in modeling and computational methods for kinetic equations. In: Degond, P., Pareschi, L., Russo, G. (eds.) Modeling and Simulation in Science, Engineering and Technology Series, pp. 3–57. Birkhauser, Basel (2003)
Faou, E., Schratz, K.: Asymptotic preserving schemes for the Klein–Gordon equation in the non-relativistic limit regime. Numer. Math. (2014, to appear)
Frénod, E., Raviart, P.-A., Sonnendrücker, E.: Two scale expansion of a singularly perturbed convection equation. J. Math. Pures Appl. 80, 815–843 (2001)
Grébert, B., Villegas-Blas, C.: On the energy exchange between resonant modes in nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 127–134 (2011)
Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454 (1999)
Machihara, S.: The nonrelativistic limit of the nonlinear Klein-Gordon equation. Funkcial. Ekvac. 44, 243–252 (2001)
Machihara, S., Nakanishi, K., Ozawa, T.: Nonrelativistic limit in the energy space for nonlinear Klein–Gordon equations. Math. Ann. 322, 603–621 (2002)
Masmoudi, N., Nakanishi, K.: From nonlinear Klein–Gordon equation to a system of coupled nonlinear Schršödinger equations. Math. Ann. 324, 359–389 (2002)
Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989)
Perko, L.M.: Higher order averaging and related methods for perturbed periodic and quasi-periodic systems. SIAM J. Appl. Math. 17, 698–724 (1969)
Sanders, J.A., Verhulst, F.: Averaging methods in nonlinear dynamical systems. Applied Mathematical Sciences, vol. 59. Springer, New York (1985)
Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150(5–7), 262–268 (1990)
Acknowledgments
We acknowledge support by the ANR-FWF Project Lodiquas (ANR-11-IS01-0003).
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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