Arbitrary Order Numerical Methods For Preserving Symplectic Structure And Energy Of Stochastic Hamiltonian Systems

With the development of stochastic mechanics in recent years,many scholars have paid attention to the stochastic Hamiltonian systems.As a generalization of deterministic Hamiltonian systems,the stochastic Hamiltonian system describes the motion process of conservative systems under the influence of white noise.The solutions of stochastic Hamiltonian systems are rich in physical and geometric properties,such as symplectic structure,energy conservation and momentum conservation properties.Naturally,when we construct numerical methods to simulate these systems,on the one hand,the accuracy and computational efficiency of the methods should be paid attention to.On the other hand,the numerical solutions are required to preserve the special structure of stochastic differential equations.In view of this,we study the stochastic partitioned Runge-Kutta methods for solving stochastic Hamiltonian systems,and construct a class of stochastic partitioned Runge-Kutta methods with symplectic properties,energy conservation properties and high convergence order.First of all,we study a class of single integrand Stratonovich partitioned stochastic differential equations,and apply the P-series theory to analyze the order conditions of the stochastic partitioned Runge-Kutta methods with single random variable.By analyzing the P-series expansion of exact solutions and numerical solutions,arbitrary order conditions with bi-coloured trees form are given,which describe the partitioned Runge-Kutta methods in mean square convergence and weak convergence.The study of this part shows that a single random variable can be used to construct the arbitrary higher order methods for solving the single integrand stochastic differential equations.Then,on the basis of the previous theoretical research results,we investigate the preservation of symplectic properties and energy conservation of conservative stochastic Hamiltonian systems by the stochastic partitioned Runge-Kutta methods.We construct a class of stochastic partitioned Runge-Kutta methods with parameters by means of W-transform,and prove that the methods are symplectic.And there exists a parameter ?(9)which enables convergence order in each iteration and can preserve the energy of the conservative stochastic Hamiltonian system.Finally,the representative nonlinear conservative stochastic Hamiltonian systems are selected to verify the energy conservation of the parameter method.