A symplectic structure preserved by the trapezoidal rule

Hon Wah TAM*, Dao Liu Wang

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

3 Citations (Scopus)


The trapezoidal rule has often been referred to as being symmetric or time-reversible and is therefore good for Hamiltonian systems. However, it is well-known that the trapezoidal rule is not symplectic, but is related to the mid-point rule (which is symplectic) through a coordinate transformation. In this paper, we show that the trapezoidal rule preserves a symplectic structure different from the original one by O(h2). The ideas in this paper also motivate us to apply Richardson's extrapolation to the trapezoidal rule. Numerical results show that the extrapolated trapezoidal rule preserves the Hamiltonian up to O(h4).

Original languageEnglish
Pages (from-to)2193-2197
Number of pages5
JournalJournal of the Physical Society of Japan
Issue number9
Publication statusPublished - Sept 2003

Scopus Subject Areas

  • Physics and Astronomy(all)

User-Defined Keywords

  • Richardson's extrapolation
  • Symplectic structure
  • Trapezoidal rule


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