Abstract
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.
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Notes
Notice that \(d_1(y) = \varTheta _0(y)\).
By convention, \(|\emptyset | = \Vert \emptyset \Vert = 0\).
Notice that \(e_\infty (\tau )=0\) if at least one of its labels is different from \(1\).
References
Abdulle, A., Weinan, E., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)
Calvo, M., Jay, L.O., Montijano, J.I., Rández, L.: Approximate compositions of a near identity map by multi-revolution Runge–Kutta methods. Numer. Math. 97(4), 635–666 (2004)
Calvo, M., Montijano, J.I., Rández, L.: A family of explicit multirevolution Runge–Kutta methods of order five. In: Analytic and Numerical Techniques in Orbital Dynamics (Spanish) (Albarracín, 2002), Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, vol. 22, pp. 45–54. Acad. Cienc. Exact. Fís. Quím. Nat. Zaragoza, Zaragoza (2003)
Calvo, M., Montijano, J.I., Rández, L.: On explicit multi-revolution Runge–Kutta schemes. Adv. Comput. Math. 26(1–3), 105–120 (2007)
Calvo, M.P., Chartier, P., Murua, A., Sanz-Serna, J.M.: Numerical stroboscopic averaging for ODEs and DAEs. Appl. Numer. Math. 61(10), 1077–1095 (2011)
Castella, F., Chartier, P., Descombes, S., Vilmart, G.: Splitting methods with complex times for parabolic equations. BIT 49(3), 487–508 (2009)
Castella, F., Chartier, P., Méhats, F., Murua, A.: Stroboscopic averaging for the nonlinear Schrödinger equation. Technical report, Sept (2012)
Chartier, P., Mehats, F., Thalhammer, M.: Convergence analysis of multi-revolution composition time-splitting pseudo-spectral methods for Schrödinger equations. Part I. the linear case (2013, in preparation)
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)
Chartier, P., Murua, A., Sanz-Serna, J.M.: A formal series approach to averaging: exponentially small error estimates. Discrete Contin. Dyn. Syst. 32(9), 3009–3027 (2012)
E, W., Engquist, B.: The heterogeneous multiscale methods. Commun. Math. Sci., 1(1):87–132 (2003)
E, W., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)
Fermi, E., Pasta, J., Ulam, S.: Studies of non linear problems. Technical Report LA-1940, Los Alamos, 1955. Later published in E. Fermi: Collected Papers (Chicago 1965), and Lect. Appl. Math. 15, 143 (1974)
Filbet, F., Jin, S.: A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229(20), 7625–7648 (2010)
Grébert, B., Villegas-Blas, C.: On the energy exchange between resonant modes in nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(1), 127–134 (2011)
Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Structure-preserving algorithms for ordinary differential equations. Springer Series in Computational Mathematics, vol. 31. Springer, Heidelberg (2010). Reprint of the second edition (2006)
Hansen, E., Ostermann, A.: High order splitting methods for analytic semigroups exist. BIT 49(3), 527–542 (2009)
Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21(2), 441–454 (1999)
Kirchgraber, U.: An ODE-solver based on the method of averaging. Numer. Math. 53(6), 621–652 (1988)
Lemou, M., Mieussens, L.: A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 31(1), 334–368 (2008)
McLachlan, R.I.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16, 151–168 (1995)
McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer. 11, 341–434 (2002)
Melendo, B., Palacios, M.: A new approach to the construction of multirevolution methods and their implementation. Appl. Numer. Math. 23(2), 259–274 (1997)
Murua, A., Sanz-Serna, J.M.: Order conditions for numerical integrators obtained by composing simpler integrators. Philos. Trans. R. Soc. Lond. Ser. A 357, 1079–1100 (1999)
Petzold, L.R., Jay, L.O., Yen, J.: Numerical solution of highly oscillatory ordinary differential equations. Acta Numer. 6, 437–483 (1997)
Sanders, J., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, 2nd edn. Springer, New York (2007)
Vilmart, G.: Weak second order multi-revolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise (2013, submitted)
Acknowledgments
JM and AM were partially supported by Ministerio de Ciencia e Innovación (Spain) under project MTM2010-18246-C03-03 (co-financed by FEDER Funds of the European Union).
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Chartier, P., Makazaga, J., Murua, A. et al. Multi-revolution composition methods for highly oscillatory differential equations. Numer. Math. 128, 167–192 (2014). https://doi.org/10.1007/s00211-013-0602-0
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DOI: https://doi.org/10.1007/s00211-013-0602-0