High Accuracy Spectral Method for the Space-Fractional Diffusion Equations

In this paper, a high order accurate spectral method is presented for the space-fractional diffusion equations. Based on Fourier spectral method in space and Chebyshev collocation method in time, three high order accuracy schemes are proposed. Themain advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency comparedwith low-order counterparts, and a completely straightforward extension to high spatial dimensions. Some numerical examples, including Allen-Cahn equation, are conducted to verify the effectiveness of this method.