A novel way constructing symplectic stochastic partitioned runge-kutta methods for stochastic hamiltonian systems

In this paper, a novel way of constructing symplectic stochastic partitioned Runge-Kutta methods for stochastic Hamiltonian systems is presented. First, a new class of continuous-stage stochastic partitioned Runge-Kutta methods for partitioned stochastic differential equations are proposed. The order conditions of the continuous-stage stochastic partitioned Runge-Kutta methods are derived via the stochastic B-series theory. The symplectic conditions of the continuous-stage stochastic partitioned Runge-Kutta methods when applied to stochastic Hamiltonian systems are analyzed. Then we prove applying any quadrature formula to a symplectic continuous-stage stochastic partitioned Runge-Kutta method will result in a classical symplectic stochastic partitioned Runge-Kutta method. In this way, various symplectic stochastic partitioned Runge-Kutta methods can be easily constructed by using different quadrature formulas. A concrete symplectic continuous-stage stochastic partitioned Runge-Kutta method of order 1 is constructed and two retrieved stochastic partitioned Runge-Kutta methods are obtained. Numerical experiments are presented to verify the theoretical results and show the effectiveness of the derived methods.