Abstract
In this paper, a simple class of stochastic partitioned Runge–Kutta (SPRK) methods is proposed for solving stochastic Hamiltonian systems with additive noise. Firstly, the order conditions and symplectic condictions are analysised by using colored rooted tree theory. Then a family of mean-square order 1.5 diagonally implicit stochastic symplectic partitioned Runge–Kutta (SSPRK) methods is presented. Moreover, several explicit SSPRK methods are constructed for systems with a separable Hamiltonian H0=V(p)+U(t,q). Furthermore, these methods are proved to converge with mean-square order 2.0 to the solution when they are applied to second-order stochastic Hamiltonian systems with a separable Hamiltonian and additive noise. Finally, several numerical examples are performed to demonstrate efficiency of those SSPRK methods.
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