Structure-preserving stochastic Runge–Kutta–Nyström methods for nonlinear second-order stochastic differential equations with multiplicative noise

A class of stochastic Runge–Kutta–Nyström (SRKN) methods for the strong approximation of second-order stochastic differential equations (SDEs) are proposed. The conditions for strong convergence global order 1.0 are given. The symplectic conditions for a given SRKN method to solve second-order stochastic Hamiltonian systems with multiplicative noise are derived. Meanwhile, this paper also proves that the stochastic symplectic Runge–Kutta–Nyström (SSRKN) methods conserve the quadratic invariants of underlying SDEs. Some low-stage SSRKN methods with strong global order 1.0 are obtained by using the order and symplectic conditions. Then the methods are applied to three numerical experiments to verify our theoretical analysis and show the efficiency of the SSRKN methods over long-time simulation.