Implicit Runge–Kutta and spectral Galerkin methods for Riesz space fractional/distributed-order diffusion equation

A numerical method with high accuracy both in time and in space is constructed for the Riesz space fractional diffusion equation, in which the temporal component is discretized by an s-stage implicit Runge–Kutta method and the spatial component is approximated by a spectral Galerkin method. For an algebraically stable Runge–Kutta method of order p(p≥s+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p\ge s+1)$$\end{document}, the unconditional stability of the full discretization is proven and the convergence order of s+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s+1$$\end{document} in time is obtained. The optimal error estimate in space, with convergence order only depending on the regularity of initial value and f, is also derived. Meanwhile, this kind of method is applied to the Riesz space distributed-order diffusion equation, and similar stability and convergence results are obtained. Finally, numerical experiments are provided to illustrate the theoretical results.