Second-order explicit difference schemes for the space fractional advection diffusion equation

In this paper, two kinds of explicit second order difference schemes are developed to solve the space fractional advection diffusion equation. The discretizations of fractional derivatives are based on the weighted and shifted Grünwald difference operators developed in [Meerschaert and Tadjeran, J.Comput.Appl.Math. 172 (2004) 65-77; Tian et al., arXiv:1201.5949; Li and Deng, arXiv:1310.7671]. The stability of the presented difference schemes are discussed by means of von Neumann analysis. The analysis shows that the presented numerical schemes are both conditionally stable. The necessary conditions of stability is discussed. Finally, the results of numerical experiments are given to illustrate the performance of the presented numerical methods.