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Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Research paper by Weien Zhou, Jingjing Zhang, Jialin Hong, Songhe Song

Indexed on: 16 Jan '16Published on: 16 Jan '16Published in: Mathematics - Numerical Analysis



Abstract

In this paper, we construct stochastic symplectic Runge-Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. By means of colored rooted tree theory, we combine conditions of mean-square order 1.5 and symplectic conditions to get totally derivative-free schemes. We also achieve mean-square order 2.0 symplectic schemes for a class of second-order Hamiltonian systems with additive noise by similar analysis. Finally, linear and non-linear systems are solved numerically, which verifies the theoretical analysis on convergence order. Especially for the stochastic harmonic oscillator with additive noise, the linear growth property can be preserved exactly over long-time simulation.