### For REFEREES

 A high order difference method for fractional sub-diffusion equations with the spatially variable coefficients under periodic boundary conditions Huiqin Zhang,Yuxiang Liang,Zhibo Wang Keywords:Fractional diffusion equation, Fourier method, Variable coefficients, Stability Abstract: In this paper, we propose a difference scheme with global convergence order $O(\tau^{2}+h^4)$ for a class of the Caputo fractional equation. The difficulty caused by the spatially variable coefficients is successfully handled. The unique solvability, stability and convergence of the finite difference scheme are proved by use of the Fourier method. The obtained theoretical results are supported by numerical experiments.