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Long-term behaviour of symmetric partitioned linear multistep methods I. Global error and conservation of invariants

B. Cano, A. Durán, M. Rodríguez

Abstract

In this paper an asymptotic expansion of the global error on the stepsize for partitioned linear multistep methods is proved. This provides a tool to analyse the behaviour of these integrators with respect to error growth with time and conservation of invariants. In particular, symmetric partitioned linear multistep methods with no common roots in their first characteristic polynomials, except unity, appear as efficient methods to approximate non-separable Hamiltonian systems since they can be explicit and show good long term behaviour at the same time. As a case study, a thorough analysis is given for small oscillations of the double pendulum problem, which is illustrated by numerical experiments.

Long-term behaviour of symmetric partitioned linear multistep methods I. Global error and conservation of invariants

Abstract

In this paper an asymptotic expansion of the global error on the stepsize for partitioned linear multistep methods is proved. This provides a tool to analyse the behaviour of these integrators with respect to error growth with time and conservation of invariants. In particular, symmetric partitioned linear multistep methods with no common roots in their first characteristic polynomials, except unity, appear as efficient methods to approximate non-separable Hamiltonian systems since they can be explicit and show good long term behaviour at the same time. As a case study, a thorough analysis is given for small oscillations of the double pendulum problem, which is illustrated by numerical experiments.

Paper Structure

This paper contains 20 sections, 6 theorems, 69 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Asymptotic expansion of the global error
  3. Symmetric methods
  4. Error growth analysis
  5. Coefficients associated to the root $x_1=1$
  6. Symmetric methods
  7. Coefficients associated to the roots $x_{i}\,\, (i=2,\dots,m)$
  8. Coefficients associated to the non-common roots of unitary modulus
  9. Conservation of invariants
  10. Smooth part of the numerical solution
  11. Symmetric methods
  12. Whole numerical solution when $m=1$
  13. Double pendulum problem
  14. Error associated to the smooth part of the numerical solution
  15. Symmetric PLMMs
  16. ...and 5 more sections

Key Result

Theorem 2.1

For fixed $t_{n}=nh, n=1,\ldots$, and under conditions (i)-(v), there are smooth functions $e_{j,i,\alpha}, e_{j,i,\alpha,\beta}$, with $\alpha,\beta \in \{p,q \}$, such that, as $h\rightarrow 0$, Here, the functions $e_{j,1,\alpha}$, with $\alpha \in \{p,q\}$, $j=r,\ldots,2r-1$, satisfy and the functions $e_{j,i,\alpha}$, with $\alpha \in \{p,q\}$, $i=2,\ldots,m$, $j=r,\ldots,2r-1$ are solution

Figures (4)

  • Figure 1: Error in the Hamiltonian against time measured at integer multiples of $2\pi$ when integrating the double pendulum problem with symmetric PLMM2 (\ref{['plmm2']}).
  • Figure 2: Error in the Hamiltonian against time measured at integer multiples of $2\pi$ when integrating the double pendulum problem with the symmetric non-partitioned LMM (\ref{['lmm2']}).
  • Figure 3: Error in the Hamiltonian against time measured at integer multiples of $2\pi$ when integrating the double pendulum problem with Adams method of 3rd order (\ref{['adams3']}).
  • Figure 4: Error in the Hamiltonian against time measured at integer multiples of $2\pi$ when integrating the double pendulum problem with PLMM (\ref{['sim_nosim']}).

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Corollary 4.1
  • Theorem 4.1
  • proof
  • ...and 4 more