Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

2016 ◽  
Vol 21 (1) ◽  
pp. 237-270 ◽  
Author(s):  
Peng Wang ◽  
Jialin Hong ◽  
Dongsheng Xu
Keyword(s):  
Case Studies ◽  
Hamiltonian Systems ◽  
Multiplicative Noise ◽  
Additive Noise ◽  
Symplectic Methods ◽  
Computational Tools ◽  
Stochastic Hamiltonian Systems ◽  
Runge Kutta ◽  
Strong Order

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.

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

2014 ◽  
Vol 16 (1) ◽  
pp. 169-200 ◽  
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.

Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

2016 ◽  
Vol 07 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Beibei Zhu ◽  
Zhenxuan Hu ◽  
Yifa Tang ◽  
Ruili Zhang
Keyword(s):  
Numerical Simulation ◽  
Hamiltonian System ◽  
Second Order ◽  
Symplectic Methods ◽  
Kutta Method ◽  
Numerical Accuracy ◽  
Runge Kutta Method ◽  
Simulation Results ◽  
Runge Kutta

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian system. The numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy conservation. Furthermore, they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

Explicit pseudo-symplectic methods for stochastic Hamiltonian systems

2017 ◽  
Vol 58 (1) ◽  
pp. 163-178 ◽  
Author(s):  
Xinyan Niu ◽  
Jianbo Cui ◽  
Jialin Hong ◽  
Zhihui Liu
Keyword(s):  
Hamiltonian Systems ◽  
Symplectic Methods ◽  
Stochastic Hamiltonian Systems
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