High Accuracy Spectral Method for the Space-Fractional Diffusion Equations

Abstract

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. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency compared with 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.