Numerical simulations for stochastic differential equations on manifolds by stochastic symmetric projection method

Stochastic standard projection technique, as an efficient approach to simulate stochastic differential equations on manifolds, is widely used in practical applications. However, stochastic standard projection methods usually destroy the geometric properties (such as symplecticity or reversibility), even though the underlying methods are symplectic or symmetric, which seriously affect long-time behavior of the numerical solutions. In this paper, a modification of stochastic standard projection methods for stochastic differential equations on manifolds is presented. The modified methods, called the stochastic symmetric projection methods, remain the symmetry and the ρ-reversibility of the underlying methods and maintain the numerical solutions on the correct manifolds. The mean square convergence order of these methods are proved to be the same as the underlying methods’. Numerical experiments are implemented to verify the theoretical results and show the superiority of the stochastic symmetric projection methods over the stochastic standard projection methods.