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 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.

[1]  E. Platen,et al.  Balanced Implicit Methods for Stiff Stochastic Systems , 1998 .

[2]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[3]  C. Scovel,et al.  Symplectic integration of Hamiltonian systems , 1990 .

[4]  Francesco Petruccione,et al.  Simulation of one-dimensional noisy Hamiltonian systems and their application to particle storage rings , 1994 .

[5]  C. Scovel,et al.  Symplectic integration of Hamiltonian systems , 1990 .

[6]  G. Kallianpur.  Stochastic differential equations and diffusion processes , 1981 .

[7]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[8]  G. Milstein.  Numerical Integration of Stochastic Differential Equations , 1994 .

[9]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[10]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[11]  J. M. Sanz-Serna,et al.  Symplectic integrators for Hamiltonian problems: an overview , 1992, Acta Numerica.

[12]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[13]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[14]  S. V. Tret'jakov,et al.  Numerical integration of Hamiltonian systems with external noise , 1994 .

[15]  G. N. Milstein,et al.  Symplectic Integration of Hamiltonian Systems with Additive Noise , 2001, SIAM J. Numer. Anal..