Explicit exactly energy-conserving methods for Hamiltonian systems

Stefan Bilbao*, Michele Ducceschi, Fabiana Zama

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as St{\"o}rmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.
Original languageEnglish
Article number111697
Number of pages21
JournalJournal of Computational Physics
Volume472
Early online date18 Oct 2022
DOIs
Publication statusPublished - 1 Jan 2023

Keywords / Materials (for Non-textual outputs)

  • Hamiltonian systems
  • geometric numerical integration
  • energy-conserving methods
  • explicit methods
  • finite difference methods

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