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In this paper, we construct stochastic symplectic Runge--Kutta (SSRK) methodsof high strong order for Hamiltonian systems with additive noise. By means ofcolored rooted tree theory, we combine conditions of mean-square order 1.5 andsymplectic conditions to get totally derivative-free schemes. We also achievemean-square order 2.0 symplectic schemes for a class of second-orderHamiltonian systems with additive noise by similar analysis. Finally, linearand non-linear systems are solved numerically, which verifies the theoreticalanalysis on convergence order. Especially for the stochastic harmonicoscillator with additive noise, the linear growth property can be preservedexactly over long-time simulation.

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