An efficient conservative splitting characteristic difference method for solving 2-d space-fractional advection–diffusion equations

In this paper, we develop an efficient splitting characteristic difference method for solving 2-dimensional two-sided space-fractional advection–diffusion equation. The intermediate numerical solutions are first computed by the piecewise parabolic method (PPM) where x¯i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{x}_i$$\end{document} is solved by the explicit second-order Runge–Kutta scheme. Then, the interior solutions are computed by the splitting σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-implicit characteristic difference method. By some auxiliary lemmas, our scheme is proved stable in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm. The error estimate is given and we prove our schemes are of second-order convergence in space. Numerical experiments are used to verify our theoretical analysis.