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Weighted average finite difference methods for fractional diffusion equations. (English) Zbl 1094.65085
Summary: A class of finite difference methods for solving fractional diffusion equations is considered. These methods are an extension of the weighted average methods for ordinary (non-fractional) diffusion equations. Their accuracy is of order ${\left({\Delta }x\right)}^{2}$ and ${\Delta }t$, except for the fractional version of the Crank-Nicolson method, where the accuracy with respect to the timestep is of order ${\left({\Delta }t\right)}^{2}$ if a second-order approximation to the fractional time-derivative is used. Their stability is analyzed by means of a recently proposed procedure akin to the standard von Neumann stability analysis [S. B. Yuste and L. Acedo, SIAM J. Numer. Anal. 42, No. 5, 1862–1874 (2005; Zbl 1119.65379)]. A simple and accurate stability criterion valid for different discretization schemes of the fractional derivative, arbitrary weight factor, and arbitrary order of the fractional derivative, is found and checked numerically. Some examples are provided in which the new methods’ numerical solutions are obtained and compared against exact solutions.
##### MSC:
 65M06 Finite difference methods (IVP of PDE)