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Numerical Algorithms
Published in: Numerical Algorithms 4/2022

07-05-2022 | Original Paper

On nested Picard iterative integrators for highly oscillatory second-order differential equations

Author: Yan Wang

Published in: Numerical Algorithms | Issue 4/2022

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Abstract

This paper is devoted to the construction and analysis of uniformly accurate (UA) nested Picard iterative integrators (NPI) for highly oscillatory second-order differential equations. The equations involve a dimensionless parameter ε ∈ (0,1], and their solutions are highly oscillatory in time with wavelength at \(\boldsymbol {\mathcal {O}}(\varepsilon ^{2})\), which brings severe burdens in numerical computation when ε ≪ 1. In this work, we first propose two NPI schemes for solving a differential equation. The schemes are uniformly first- and second-order accurate for all ε ∈ (0,1]. Moreover, they are super convergent when the time-step size is smaller than ε2. Then, the schemes are generalized to a system of differential equations with the same uniform accuracies. Error bounds are rigorously established and numerical results are reported to confirm the error estimates.
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Appendix
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Literature
1.
Ariel, G., Engquist, B, Tsai, R.: A multiscale method for highly oscillatory ordinary differential equations with resonance. Math. Comput. 78, 929–956 (2009)CrossRefMATHMathSciNet
2.
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)CrossRefMATHMathSciNet
3.
Bao, W., Zhao, X.: Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime. J. Comput. Phys. 398, 108886 (2019)CrossRefMATHMathSciNet
4.
Bao, W., Cai, Y., Zhao, X.: A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon equation in the nonrelativistic limit regime. SIAM J. Numer. Anal. 52, 2488–2511 (2014)CrossRefMATHMathSciNet
5.
Bao, W., Dong, X., Zhao, X.: Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations. J. Math. Study 47, 111–150 (2014)CrossRefMATHMathSciNet
6.
Baumstark, S., Faou, E., Schratz, K.: Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to the classical NLS splitting. Math. Comp. 87, 1227–1254 (2018)CrossRefMATHMathSciNet
7.
Cai, Y., Wang, Y.: Uniformly accurate nested Picard iterative integrators for the Dirac equation in the nonrelativistic limit regime. SIAM J. Numer. Anal. 57, 1602–1624 (2019)CrossRefMATHMathSciNet
8.
Castella, F., Chartier, P., Méhats, F., Murua, A.: Stroboscopic averaging for the nonlinear schrödinger equation. Found. Comput. Math. 15, 519–559 (2015)CrossRefMATHMathSciNet
9.
Chartier, P., Crouseilles, N., Lemou, M., Méhats, F.: Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear schrödinger equations. Numer. Math. 129, 211–250 (2015)CrossRefMATHMathSciNet
10.
Cohen, D.: Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems. IMA J. Numer. Anal. 26, 34–59 (2005)CrossRefMATHMathSciNet
11.
Cohen, D., Hairer, E., Lubich, C.: Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math. 3, 327–345 (2003)CrossRefMATHMathSciNet
12.
Cohen, D., Hairer, E., Lubich, C.: Numerical energy conservation for multi-frequency oscillatory differential equations. BIT Numer. Math. 45, 287–305 (2005)CrossRefMATHMathSciNet
13.
Cohen, D., Hairer, E., Lubich, C.: Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations. Numer. Math. 110, 113–143 (2008)CrossRefMATHMathSciNet
14.
Condon, M., Deaño, A., Iserles, A: On highly oscillatory problems arising in electronic engineering. ESAIM Math. Model. Numer. Anal. 43, 785–804 (2009)CrossRefMATHMathSciNet
15.
Condon, M., Deaño, A., Iserles, A: On second order differential equations with highly oscillatory forcing terms. Proc. R. Soc. A 466, 1809–1828 (2010)CrossRefMATHMathSciNet
16.
Duncan, D. B.: Symplectic finite difference approximations of the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal. 34, 1742–1760 (1997)CrossRefMATHMathSciNet
17.
Engquist, B., Tsai, Y.: Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comp. 74, 1707–1742 (2005)CrossRefMATHMathSciNet
18.
Faou, E., Schratz, K.: Asymptotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime. Numer. Math. 126, 441–469 (2014)CrossRefMATHMathSciNet
19.
Garcia-Archilla, B., Sanz-Serna, J.M., Skeel, R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20, 930–963 (1998)CrossRefMATHMathSciNet
20.
Gauckler, L.: Error analysis of trigonometric integrators for semilinear wave equations. SIAM J. Numer. Anal. 53, 1082–1106 (2015)CrossRefMATHMathSciNet
21.
Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)CrossRefMATHMathSciNet
22.
Grimm, V., Hochbruck, M.: Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A: Math. Gen. 39, 5495–5507 (2006)CrossRefMATHMathSciNet
23.
Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)CrossRefMATHMathSciNet
24.
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)MATH
25.
Hochbruck, M., Ostermann, A.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)CrossRefMATHMathSciNet
26.
27.
Iserles, A: On the numerical quadrature of highly-oscillating integrals I: Fourier transforms. IMA J. Numer. Anal. 24, 365–391 (2004)CrossRefMATHMathSciNet
28.
Lorenz, K., Jahnke, T., Lubich, C.: Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition. BIT Numer. Math. 45, 91–115 (2005)CrossRefMATHMathSciNet
29.
Machihara, S., Nakanishi, K., Ozawa, T.: Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. Math. Ann. 322, 603–621 (2002)CrossRefMATHMathSciNet
30.
Masmoudi, N., Nakanishi, K.: From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrödinger equations. Math. Ann. 324, 359–389 (2002)CrossRefMATHMathSciNet
31.
32.
Ostermann, A., Schratz, K.: Low regularity exponential-type integrators for semilinear Schrödinger equations. Found. Comput. Math. 18, 731–755 (2018)CrossRefMATHMathSciNet
33.
Sanz-Serna, J. M.: Mollified impulse methods for highly oscillatory differential equations. SIAM J. Numer. Anal. 46, 1040–1059 (2008)CrossRefMATHMathSciNet
34.
35.
36.
37.
Tao, T.: Local and Global Analysis of Nonlinear Dispersive and Wave Equations. CBMS Regi. Cmf. Ser. Math. AMS, Providence (2006)
38.
Tsutsumi, M.: Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions. Nonlinear Anal. 8, 637–643 (1984)CrossRefMATHMathSciNet
39.
Wang, B., Iserles, A., Wu, X.: Arbitrary-order trigonometric Fourier collocation methods for multi-frequency oscillatory systems. Found. Comput. Math. 16, 151–181 (2016)CrossRefMATHMathSciNet
40.
Weinan, E.: Analysis of the heterogeneous multiscale method for ordinary differential equations. Commun. Math. Sci. 1, 423–436 (2003)CrossRefMATHMathSciNet
41.
Weinan, E., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: The heterogeneous multiscale method: a review. Commun. Comput. Phys. 2, 367–450 (2007)MATHMathSciNet
42.
Zhao, X.: Uniformly accurate (UA) multiscale time integrators for second order oscillatory differential equations with large initial data. BIT Numer. Math. 57, 649–683 (2017)CrossRefMATH
Metadata
Title
On nested Picard iterative integrators for highly oscillatory second-order differential equations
Author
Yan Wang
Publication date
07-05-2022
Publisher
Springer US
Published in
Numerical Algorithms / Issue 4/2022
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-022-01317-8

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