Variational integrators for stochastic dissipative Hamiltonian systems
Michael Kraus, Tomasz M Tyranowski
Abstract
Abstract 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.
BibTeX
@article{Kraus_2020,
title={Variational integrators for stochastic dissipative Hamiltonian systems},
volume={41},
ISSN={1464-3642},
url={http://dx.doi.org/10.1093/imanum/draa022},
DOI={10.1093/imanum/draa022},
number={2},
journal={IMA Journal of Numerical Analysis},
publisher={Oxford University Press (OUP)},
author={Kraus, Michael and Tyranowski, Tomasz M},
year={2020},
month=jul,
pages={1318–1367} }