Abstract
Chapter 7 is concerned with the energy-preserving numerical integration for the system of oscillatory second-order differential equations \(\ddot{q}+Mq=f(q)\), where M is a symmetric positive semi-definite matrix and f(q)=−∇U(q). Based on the traditional average-vector-field (AVF) methods, adapted average-vector-field (AAVF) methods are developed. A discretization with a quadrature formula leads to a highly accurate energy-preserving ERKN-type AAVF integrator. This integrator is symmetric and is shown to preserve the Hamiltonian H if U(q) is a polynomial of degree s≤6. In the long-term integration of the well-known Fermi–Pasta–Ulam problem, the integrator is shown to preserve the energy more accurately than some existing methods in the literature. Resonance instabilities and energy exchange between stiff components are also illustrated.
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References
Brugnano, L., Iavernaro, F., Trigiante, D.: Hamiltonian boundary value methods (energy preserving discrete line integral methods). J. Numer. Anal. Ind. Appl. Math. 5, 17–37 (2010)
Castella, F., Chartier, P., Faou, E.: An averaging technique for highly oscillatory Hamiltonian problems. SIAM J. Numer. Anal. 47, 2808–2837 (2009)
Celledoni, E., McLachlan, R.I., Owren, B., Quispel, G.R.W.: Energy-preserving integrators and the structure of B-series. Found. Comput. Math. 10, 673–693 (2010)
Cieslinski, J.L., Ratkiewicz, B.: Energy-preserving numerical schemes of high accuracy for one-dimensional Hamiltonian systems. J. Phys. A, Math. Theor. 44, 155–206 (2011)
Cieslinski, J.L., Ratkiewicz, B.: Improving the accuracy of the discrete gradient method in the one dimensional case. Phys. Rev. E 81, 016704 (2010)
Cohen, D., Hairer, E.: Linear energy-preserving integrators for Poisson systems. BIT 51, 91–101 (2011)
Cohen, D., Hairer, E., Lubich, C.: Numerical energy conservation for multi-frequency oscillatory differential equations. BIT 45, 287–305 (2005)
Dahlby, M., Owren, B., Yaguchi, T.: Preserving multiple first integrals by discrete gradients. J. Phys. A, Math. Theor. 44, 305205 (2011)
Hairer, E.: Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)
Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)
Iavernaro, F., Trigiante, D.: High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. J. Numer. Anal. Ind. Appl. Math. 4, 787–787101 (2009)
Marciniak, A.: Arbitrary order numerical methods conserving integrals for solving dynamic equations. Comput. Math. Appl. 28, 33–43 (1994)
McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. R. Soc. A 357, 1021–1046 (1999)
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)
Wu, X., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587–605 (2013)
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Wu, X., You, X., Wang, B. (2013). Energy-Preserving ERKN Methods. In: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35338-3_7
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DOI: https://doi.org/10.1007/978-3-642-35338-3_7
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