High-Order Symplectic and Symmetric Composition Integrators for Multi-frequency Oscillatory Hamiltonian Systems

Xinyuan Wu () and Bin Wang
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Xinyuan Wu: Nanjing University, Department of Mathematics
Bin Wang: Qufu Normal University, School of Mathematical Sciences

Chapter Chapter 5 in Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations, 2018, pp 107-133 from Springer

Abstract: Abstract This chapter presents symplectic and symmetric composition methods based onExtended discrete gradient formula Adapted Runge–Kutta–Nyström (ARKN) and extended Runge–Kutta–Nyström (ERKN) integrators for solving multi-frequency and multi-dimensional oscillatory Hamiltonian systems with the Hamiltonian $$H(p,q)=\dfrac{1}{2}p^{\intercal }p+\dfrac{1}{2}q^{\intercal }Kq+U(q)$$ , where $$p=q'$$ and K is a symmetricSymplectic and symmetric composition integrators and positive semi-definite matrix. We first consider the symplecticity conditions for multi-frequency and multi-dimensional ARKN integrators. We then analyse the symplecticity of the adjoint integrators of the multi-frequency and multi-dimensional symplectic ARKN and ERKN integrators, respectively. On the basis of the theoretical analysis, and using the idea of composition methods, we derive four new high-order symplectic and symmetric integrators. The numerical results quantitatively show the advantage and efficiency of the high-order symplectic and symmetric integrators.

Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-9004-2_5

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DOI: 10.1007/978-981-10-9004-2_5

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