Numerical investigation of stochastic canonical Hamiltonian systems by high order stochastic partitioned Runge-Kutta methods

Abstract

In this paper, a family of arbitrary high order quadratic invariants and energy conservation parametric stochastic partitioned Runge-Kutta methods (SPRK) are constructed for stochastic canonical Hamiltonian systems where the parameters depend on some truncated random variables, step size and numerical solutions. We first apply the P-series and bi-coloured trees theory to analyze the mean-square and weak convergence order conditions of SPRK methods solving a class of single integrand stochastic differential equations. Then, a class of SPRK methods with parameters are obtained by means of W-transform and technique of truncated Wiener increments, and we prove that the methods are symplectic. Combining with order conditions, there exists a special parameter α* which enables convergence order in each iteration and can preserve the energy of the stochastic canonical Hamiltonian systems. Finally, the representative stochastic canonical Hamiltonian systems are selected to verify the good performance of the proposed parameter methods.