Abstract
In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq″ (t)+ Bq(t) = f(q(t)), where A ∈ ℝm×m is a symmetric positive definite matrix, B ∈ ℝm×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q) = −∇qV (q) for a real-valued function V (q). The energy-preserving formulae can exactly preserve the Hamiltonian \(H(q', q) = \frac{1}{2}q'^{\rm{T}} Aq' + \frac{1}{2}q^{\rm{T}} Bq + V(q)\). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.
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Bratsos, A.: A numerical method for the one-dimentional sine-Gordon equation. Numer. Mehods Partial Differ. Equ., 24, 833–844 (2008)
Chabassier, J., Joly, P.: Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string. Comput. Methods Appl. Mech. Engrg., 199, 2779–2795 (2010)
Chen, C., Tang, Q.: Continuous finite element methods for Hamiltonian systems. Appl. Math. Mech., 28, 1071–1080 (2007)
Dehghan, M., Mohebbi, A., Asgari, Z.: Fourth-order compact solution of the nonlinear Klein–Gordon equation. Numer. Algor., 520, 523–54 (2009)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin, Heidelberg, 2006
Iserles, A., Quispel, G., Tse, P.: B-series methods cannot be volume-preserving. BIT Numer. Math., 47, 351–378 (2007)
Janssen, J., Vandewalle, S.: On SOR waveform relaxation methods. SIAM J. Numer. Anal., 34, 2456–2481 (1997)
Jiménez, S., Vázquez, L.: Analysis of four numerical schemes for a nonlinear Klein–Gordon equation. Appl. Math. Comput., 35, 61–94 (1990)
Khanamiryan, M.: Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations, Part I. BIT Numer. Math., 48, 743–762 (2008)
Liu, C., Shi, W., Wu, X.: An efficient high-order explicit scheme for solving Hamiltonian nonlinear wave equations. Appl. Math. Comput., 246, 696–710 (2014)
Liu, K., Shi, W., Wu, X.: An extended discrete gradient formula for oscillatory Hamiltonian systems. J. Phys. A., 46, 165203 (2013)
Lubich, C., Ostermann, A.: Multigrid dynamic iteration for parabolic equations. BIT Numer. Math., 27, 216–234 (1987)
Mohebbi, A., Dehghan, M.: High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods. Math. Comput. Model., 51, 537–549 (2010)
Quispel, G., McLaren, D.: A new class of energy-preserving numerical integration methods. J. Phys. A., 41, 045206 (2008)
Sun, Z.: Numerical Methods of Partial Differential Equations (2nd version, in Chinese), Science Press, Beijing, 2012
Tourigny, Y.: Product approximation for nonlinear Klein–Gordon equations. IMA J. Numer. Anal., 9, 449–462 (1990)
Vandewalle, S.: Parallel multigrid waveform relaxation for parabolic problems, in: Teubner Scripts on Numerical Mathematics, Stuttgart, 1993
Wang, B.: Triangular splitting implementation of RKN-type Fourier collocation methods for second-order differential equations. Math. Meth. Appl. Sci., 41, 1998–2011 (2018)
Wang, B., Iserles, A., Wu, X.: Arbitrary-order trigonometric Fourier collocation methods for multifrequency oscillatory systems. Found. Comput. Math., 16, 151–181 (2016)
Wang, B., Li, G.: Bounds on asymptotic-numerical solvers for ordinary differential equations with extrinsic oscillation. Appl. Math. Model., 39, 2528–2538 (2015)
Wang, B., Meng, F., Fang, Y.: Efficient implementation of RKN-type Fourier collocation methods for second-order differential equations. Appl. Numer. Math., 119, 164–178 (2017)
Wang, B., Li, T., Wu, Y.: Arbitrary-order functionally fitted energy-diminishing methods for gradient systems. Appl. Math. Lett., 83, 130–139 (2018)
Wang, B., Wu, X.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A, 376, 1185–1190 (2012)
Wang, B., Wu, X., Meng, F.: Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second order differential equations. J. Comput. Appl. Math., 313, 185–201 (2017)
Wang, B., Yang, H., Meng, F.: Sixth order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Calcolo, 54, 117–140 (2017)
Wang, B., Wu, X., Meng, F., et al.: Exponential Fourier collocation methods for solving first-order differential equations. J. Comput. Math., 35, 711–736 (2017)
Wu, X., Liu, K., Shi, W.: Structure-Preserving Algorithms for Oscillatory Differential Equations II, Springer-Verlag, Heidelberg, 2015
Wu, X., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys., 235, 587–605 (2013)
Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations, Springer-Verlag, Heidelberg, 2013
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Supported by NSFC (Grant No. 11571302), NSF of Shandong Province (Grant No. ZR2018MA024) and the foundation of Scientific Project of Shandong Universities (Grant Nos. J17KA190 and KJ2018BAI031)
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Fang, Y.L., Liu, C.Y. & Wang, B. Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System. Acta. Math. Sin.-English Ser. 34, 1863–1878 (2018). https://doi.org/10.1007/s10114-018-6300-1
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DOI: https://doi.org/10.1007/s10114-018-6300-1
Keywords
- Nonlinear Hamiltonian wave equations
- energy-preserving schemes
- Average Vector Field method
- oscillatory systems