Numerical Methods for Stochastic Systems Preserving Symplectic Structure

Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2002-01, Vol.40 (4), p.1583-1604
Main Authors: Milstein, G. N., Repin, Yu. M., Tretyakov, M. V.
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Methods
Noise
Numerical analysis
Random variables
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Summary:Stochastic Hamiltonian systems with multiplicative noise, phase flows of which preserve symplectic structure, are considered. To construct symplectic methods for such systems, sufficiently general fully implicit schemes, i.e., schemes with implicitness both in deterministic and stochastic terms, are needed. A new class of fully implicit methods for stochastic systems is proposed. Increments of Wiener processes in these fully implicit schemes are substituted by some truncated random variables. A number of symplectic integrators is constructed. Special attention is paid to systems with separable Hamiltonians. Some results of numerical experiments are presented. They demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones.
ISSN:0036-1429
1095-7170
DOI:10.1137/S0036142901395588