High-order stochastic symplectic partitioned Runge-Kutta methods for stochastic Hamiltonian systems with additive noise

Han, Minggang; Ma, Qiang; Ding, Xiaohua

doi:10.1016/j.amc.2018.10.041

Cited by 15 publications

(24 citation statements)

References 22 publications

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“…This figure again clearly illustrates the excellent long time behaviour of the proposed numerical scheme as stated in the above theorem. Furthermore, the drift-preserving scheme (3) outperforms the numerical schemes from [5,22] in terms of preserving the expected value of the energy (compare Figure 8 to [5, Table 2] and [22, Figure 6.7]).…”

Section: 4mentioning

confidence: 98%

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Drift-preserving numerical integrators for stochastic Hamiltonian systems

et al. 2020

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The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

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Section: 4mentioning

confidence: 98%

“…Hénon-Heiles problem with two additive noises. Finally, we consider the Hénon-Heiles problem with two additive noises from [5,22]. This SDE is given by the Hamiltonian and with Σ " diagpσ 1 , σ 2 q and W " pW 1 , W 2 q J in (1).…”

Section: 4mentioning

confidence: 99%

Drift-preserving numerical integrators for stochastic Hamiltonian systems

et al. 2020

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“…the Hamiltonian considered in [9] (where the matrix is diagonal), the linear stochastic oscillator from [44], and various stochastic Hamiltonian systems studied in [36,Chap. 4], see also [35], or [26,27,42,50].…”

Section: Settingmentioning

confidence: 99%

Drift-preserving numerical integrators for stochastic Poisson systems

Cohen

Vilmart

2021

International Journal of Computer Mathematics

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We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.

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“…The present setting covers, for instance, the following examples: simplified versions of the stochastic rigid bodies studied in [45,47], the stochastic Hamiltonian systems considered in [13] by taking BpXq " J " ˆ0 ´Id m Id m 0 ˙the constant canonical symplectic matrix, for which the SDE (2) yields dpptq " ´∇V pqptqq dt `Σ dW ptq, dqptq " pptq dt, the Hamiltonian considered in [9] (where the matrix Σ is diagonal), the linear stochastic oscillator from [44], and various stochastic Hamiltonian systems studied in [36,Chap. 4], see also [35], or [42,50,27,26].…”

Section: Drift-preserving Scheme For Stochastic Poisson Problemmentioning

confidence: 99%

Drift-preserving numerical integrators for stochastic Poisson systems

Cohen,

Vilmart

2020

Preprint

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We perform a numerical analysis of randomly perturbed Poisson systems. For the considered Itô perturbation of Poisson differential equations, we show the longtime behavior of the energy and quadratic Casimirs for the exact solution. We then propose and analyze a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence one, weak order of convergence two. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

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High-order stochastic symplectic partitioned Runge-Kutta methods for stochastic Hamiltonian systems with additive noise

Cited by 15 publications

References 22 publications

Drift-preserving numerical integrators for stochastic Hamiltonian systems

Drift-preserving numerical integrators for stochastic Hamiltonian systems

Drift-preserving numerical integrators for stochastic Poisson systems

Drift-preserving numerical integrators for stochastic Poisson systems

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