Composite symmetric general linear methods (COSY-GLMs) for the long-time integration of reversible Hamiltonian systems

Terence J. T. Norton

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Abstract

In choosing a numerical method for the long-time integration of reversible Hamiltonian systems one must take into consideration several key factors: order of the method, ability to preserve invariants of the system, and efficiency of the computation. In this paper, 6th-order composite symmetric general linear methods (COSY-GLMs) are constructed using a generalisation of the composition theory associated with Runge–Kutta methods (RKMs). A novel aspect of this approach involves a nonlinear transformation which is used to convert the GLM to a canonical form in which its starting and finishing methods are trivial. Numerical experiments include efficiency comparisons to symmetric diagonally-implicit RKMs, where it is shown that COSY-GLMs of the same order typically require half the number of function evaluations, as well as long-time computations of both separable and non-separable Hamiltonian systems which demonstrate the preservation properties of the new methods.
Original languageEnglish
Pages (from-to)397-421
Number of pages25
JournalBIT Numerical Mathematics
Volume58
Issue number2
Early online date10 Jan 2018
DOIs
Publication statusPublished - 1 Jun 2018

Keywords

  • General linear methods
  • Reversible Hamiltonian systems
  • Symmetric composition

ASJC Scopus subject areas

  • Software
  • Computer Networks and Communications
  • Computational Mathematics
  • Applied Mathematics

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