Research On Structure-Preserving Methods For Some Stochastic Differential Equations

In recent years,the theory of stochastic differential equations has been applied to many fields.The construction and analysis of their numerical algorithms are one of the core problems of solving random problems.In this research,we study the stucturepreserving methods for some stochastic differential equations.The main contents are as follows;1.For a class of stochastic differential equations driven by multiplicative noise with conserved quantities,we construct projection methods which preserve single ormultiple conserved quantities of original system.Based on the projection technique, these methods are able to reach high order of strong convergence by supporting methods.Theoretical results and numerical experiments show that the propersed methods not only have high mean-square order but also preserve multiple conserved quantities simultaneously.2.For a class of stochastic differential equations driven by multiplicative noise with conserved quantities,we couple the parareal algorithm with projection methods of the trajectory on a specific manifold,defined by the preservation of some conserved quantities of stochastic differential equations.First,projection methods are intro- duced as the coarse and fine propagators.Second,we also apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm.3.We construct stochastic symplectic Runge–Kutta(SSRK)methods of high strong order for Hamiltonian systems with additive noise.By means of colored rooted tree theory,we combine conditions of mean-square order 1.5 and symplectic conditions to get totally derivative-free schemes.We also achieve mean-square order 2.0 sym- plectic schemes for a class of second-order Hamiltonian systems with additive noise by similar analysis.