Investigation Of Numerical Methods For Stochastic Evolution Equations

This doctoral dissertation is devoted to numerical methods for stochastic evolution equa-tions and contains two parts. The first part deals with numerical methods for stochastic par-tial differential equations (SPDEs), including stochastic exponential integrators for stochastic elastic equation driven by multiplicative Wiener process, Parareal algorithms for nonlinear s-tochastic parabolic partial differential equations and Galerkin spectral method for SPDEs driv-en by infinite-dimensional fractional Brownian motion. The second part focuses on numerical methods for stochastic ordinary differential equations (SODEs). Two classes of two-step Mil-stein methods are constructed, and their convergence and exponential mean square stability properties are studied. This thesis consists of the following six chapters:In Chapter 1, the history and current state of numerical analysis of SPDEs and SODEs are briefly reviewed. Moreover, the outline of this thesis is also given.In Chapter 2, we give a concise introduction to the infinite-dimensional Wiener process and the basics of infinite-dimensional stochastic integral in separable Hilbert space.Chapter 3 is concerned with the stochastic exponential integrator for stochastic elastic equations driven by multiplicative noise process. For space discretization, we use finite ele-ment method. By analyzing space and temporal error separately, we derive an error estimate of the fully-discrete method and obtain an optimal strong convergence order in time, namely, the convergence order in time equals the temporal regularity of the analytic solution.Chapter 4 deals with the Parareal algorithms for the nonlinear stochastic parabolic partial differential equations. It is shown that by using the stochastic exponential integrator on the coarse grid and the exact solution propagator on the fine grid, for SPDEs driven by additive space-time noise, the convergence rate is superlinear and the convergence factor is independent of the regularity of noise. Numerical experiments are performed to show that for SPDEs driven by multiplicative noise and SPDEs with locally Lipschitz drift term, the Parareal algorithms are still efficient.In Chapter 5, we consider the SPDEs driven by infinite-dimensional fractional Brownian motion. Using Galerkin spectral method in space and linear implicit Euler method in time and combining the regularity analysis of the analytic solution, we present strong convergence analysis for the fully discrete method. Our results show that the convergence order of our method are consistent with the regularities of the analytic solution both in time and in space.In Chapter 6, we propose two classes of two-step Milstein methods for Ito autonomous SODEs and analyze the strong convergence of these two schemes in LP-norm. Our results reveal that the two new schemes are strong convergent of order one. Moreover, with a restriction on stepsize, these two schemes can preserve the exponential mean square stability of the original SODEs, and the decay rate of numerical solution will converge to the decay rate of the analytic solution.