Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

AbstractWe study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

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Explicit pseudo-symplectic methods based on generating functions for stochastic Hamiltonian systems

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2019.112433 ◽

2020 ◽

Vol 373 ◽

pp. 112433

Author(s):

Cristina Anton

Keyword(s):

Hamiltonian Systems ◽

Generating Functions ◽

Symplectic Methods ◽

Stochastic Hamiltonian Systems

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High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

Communications in Computational Physics ◽

10.4208/cicp.311012.191113a ◽

2014 ◽

Vol 16 (1) ◽

pp. 169-200 ◽

Cited By ~ 11

Author(s):

Jian Deng ◽

Cristina Anton ◽

Yau Shu Wong

Keyword(s):

Hamiltonian Systems ◽

High Order ◽

Symplectic Integrators ◽

Symplectic Methods ◽

Computational Tools ◽

Fully Implicit ◽

Stochastic Hamiltonian Systems ◽

Hamiltonian Functions ◽

Symplectic Schemes

AbstractThe construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

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