High-Accuracy Time Discretization of Stochastic Fractional Diffusion Equation

Abstract

A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying the semi-implicit Euler scheme. The solution of the equation is only Hölder continuous in time, which is disadvantageous to improve the temporal convergence rate. Firstly, the system is transformed into an equivalent form having better regularity than the original one in time. But the regularity of nonlinear term remains unchanged. Then, combining Lagrange mean value theorem and independent increments of Brownian motion leads to a higher accuracy discretization of nonlinear term which ensures the implementation of the proposed time discretization scheme without loss of convergence rate. Our scheme can improve the convergence rate from ${\min \{\frac{\gamma }{2\alpha },\frac{1}{2}\}}$ to ${\min \{\frac{\gamma }{\alpha },1\}}$ in the sense of mean-squared $L^2$-norm. The theoretical error estimates are confirmed by extensive numerical experiments.