Geometric-integration tools for the simulation of musical sounds

Ai Ishikawa*, Dominik L. Michels, Takaharu Yaguchi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


During the last decade, much attention has been given to sound rendering and the simulation of acoustic phenomena by solving appropriate models described by Hamiltonian partial differential equations. In this contribution, we introduce a procedure to develop appropriate tools inspired from geometric integration in order to simulate musical sounds. Geometric integrators are numerical integrators of excellent quality that are designed exclusively for Hamiltonian ordinary differential equations. The introduced procedure is a combination of two techniques in geometric integration: the semi-discretization method by Celledoni et al. (J Comput Phys 231:6770–6789, 2012) and symplectic partitioned Runge–Kutta methods. This combination turns out to be a right procedure that derives numerical schemes that are effective and suitable for computation of musical sounds. By using this procedure we derive a series of explicit integration algorithms for a simple model describing piano sounds as a representative example for virtual instruments. We demonstrate the advantage of the numerical methods by evaluating a variety of numerical test cases.

Original languageEnglish (US)
Pages (from-to)511-540
Number of pages30
JournalJapan Journal of Industrial and Applied Mathematics
Issue number2
StatePublished - Jul 1 2018

Bibliographical note

Publisher Copyright:
© 2018, The Author(s).


  • Acoustic phenomena
  • Acoustic simulation
  • Geometric integration
  • Musical sounds
  • Partitioned Runge–Kutta methods
  • Separable Hamiltonian system
  • Sound rendering
  • Sound simulation
  • Symplectic integration
  • Virtual instruments
  • Virtual piano

ASJC Scopus subject areas

  • General Engineering
  • Applied Mathematics


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