High Accuracy Spectral Method for the Space-Fractional Diffusion Equations (Journal of Mathematical Study, 2014)
In this paper, a high order accurate spectral method is presentedfor the space-fractional diffusion equations. Based on Fourierspectral method in space and  Chebyshev collocation method in time,three high order accuracy schemes are proposed. The main advantagesof this method are that it yields a fully diagonal representation ofthe fractional operator, with increased accuracy and efficiencycompared with low-order counterparts, and a completelystraightforward extension to high spatial dimensions. Some numericalexamples, including Allen-Cahn equation, are conducted to verify theeffectiveness of this method.