Stochastic discrete Hamiltonian variational integrators

• Published in: BIT Vol. 58; no. 4; pp. 1009 - 1048
• Main Authors: Holm, Darryl D., Tyranowski, Tomasz M.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2018; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Computer simulation; Hamiltonian functions; Integrators; Lie groups; Mathematics; Mathematics and Statistics; Numeric Computing; Numerical stability; Runge-Kutta method; Stochastic differential equations; 65C30; Variational integrators; Geometric mechanics; Geometric numerical integration methods; Stochastic Hamiltonian systems
• ISSN: 0006-3835; 1572-9125; 1572-9125
• DOI: 10.1007/s10543-018-0720-2

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. 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 its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.