Abstract
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.
Highlights
Stochastic differential equations (SDEs) play an important role in modeling dynamical systems subject to internal or external random fluctuations
The stochastic variational integrators proposed in [7,8] were formulated for dynamical systems which are described by a Lagrangian and which are subject to noise whose magnitude depends only on the position q
In this paper we have presented a general framework for constructing a new class of stochastic symplectic integrators for stochastic Hamiltonian systems
Summary
Stochastic differential equations (SDEs) play an important role in modeling dynamical systems subject to internal or external random fluctuations. Geometric integration of deterministic Hamiltonian systems has been thoroughly studied (see [18,41,55] and the references therein) and symplectic integrators have been shown to demonstrate superior performance in long-time simulations of Hamiltonian systems, compared to non-symplectic methods; so it is natural to pursue a similar approach for stochastic Hamiltonian systems. An important class of geometric integrators are variational integrators This type of numerical schemes is based on discrete variational principles and provides a natural framework for the discretization of Lagrangian systems, including forced, dissipative, or constrained ones.
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