In this talk, we present stochastic symplectic Runge-Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. Based on colored rooted tree theory,   we combine conditions of mean-square order up to 1.5 and symplecticity to obtain 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. Numerical experiments on linear and non-linear systems are provided to verify the theoretical analysis on convergence order. Especially for the stochastic harmonic oscillator with additive noise, two schemes are provided which exactly preserve the linear growth property over long-time simulation.
['Weien Zhou', 'Jingjing Zhang', 'Jialin Hong', 'Songhe Song']
Stochastic symplectic Runge-Kutta methods for the strong approximation of Hamiltonian systems with additive noise