Convergence Analysis Of Numerical Methods For Several Kinds Of Stochastic Partial Differential Equations

As an important model,stochastic partial differential equations have a wide range of applications in quantum field theory,statistical mechanics,financial mathematics and so on.This thesis is devoted to the convergence analysis of numerical methods for stochastic Allen-Cahn equations,fractional noise driven stochastic partial differential equations(SPDEs)and stochastic Cahn-Hilliard equations.The fully discrete scheme is composed of a spectral Galerkin method in space and an explicit scheme in time and we derive the convergence rates.The main contents are as follows:Firstly,we briefly review the development on numerics of SPDEs and in-troduce the basic theory of infinite-dimensional stochastic analysis and Malli-avin calculus.Then,we study the weak convergence of an explicit full-discretization for stochastic Allen-Cahn equation.Since the fully discrete exponential Euler and the fully discrete linear implicit Euler approximations diverge strongly and numerically weakly when used to solve the stochastic Allen-Cahn equations,a tamed exponential Euler method is applied in the temporal direction for a spatial spectral Galerkin semi-discretization and the weak convergence rates are obtained.The new approach for the weak error analysis is direct,which does not rely on the use of the associated Kolmogorov equations or It(?)’s for-mula.In the sequel,we apply the approach to the parabolic SPDEs driven by fractional Brownian motions with the Hurst parameter H∈(1/2,1)(c.f.(4-1)).Such equation is approximated by a spectral Galerkin method in space and an exponential Euler method in time.By using the improved temporal H(?)lder regularity results in negative Sobolev space,we reveal the weak convergence rates of the factional noise driven SPDE for the first time.Additionally,based on the spatial spectral Galerkin semi-discretization,we approximate the stochastic Cahn-Hilliard equation by a tamed exponential Euler method in time.By virtue of the Gagliardo-Nirenberg inequality and the energy estimate,we prove the a priori moment bounds of a full discretization and derive the strong convergence rates.In contrast to the implicit scheme in the existing literature,the explicit scheme does not need to solve the additional nonlinear equations and is easily implementable.Finally,we make a summary of the main results and introduce the further work.