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

In: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

Author

Listed:
  • Xinyuan Wu

    (Nanjing University, Department of Mathematics
    Qufu Normal University, School of Mathematical Sciences)

  • Bin Wang

    (Qufu Normal University, School of Mathematical Sciences)

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.

Suggested Citation

  • Xinyuan Wu & Bin Wang, 2018. "High-Order Symplectic and Symmetric Composition Integrators for Multi-frequency Oscillatory Hamiltonian Systems," Springer Books, in: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations, chapter 0, pages 107-133, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-9004-2_5
    DOI: 10.1007/978-981-10-9004-2_5
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