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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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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