  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.
