Variational integrators for stochastic dissipative Hamiltonian systems

Michael Kraus, Tomasz M. Tyranowski* (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

12 Citations (Scopus)
1 Downloads (Pure)


Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian, which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange–d’Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange–d’Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether’s theorem. Furthermore, mean-square and weak Lagrange–d’Alembert Runge–Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behaviour compared to nongeometric methods. The Vlasov–Fokker–Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.
Original languageEnglish
Pages (from-to)1318–1367
Number of pages50
JournalIMA Journal of Numerical Analysis
Issue number2
Early online date22 Jul 2020
Publication statusPublished - Apr 2021
Externally publishedYes


  • n/a OA procedure
  • stochastic forced Hamiltonian systems
  • geometric integration
  • stochastic variational integrators
  • collisional plasma kinetics


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