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An $O\left(N{log}^{2}N\right)$ alternating-direction finite difference method for two-dimensional fractional diffusion equations. (English) Zbl 1229.65165

Summary: Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of $O\left({N}^{3}\right)$ per time step and memory of $O\left({N}^{2}\right)$ for where $N$ is the number of grid points.

In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of $O\left(N{log}^{2}N\right)$ per time step and memory of $O\left(N\right)$, while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination.

Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5 h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.

##### MSC:
 65M06 Finite difference methods (IVP of PDE) 35K20 Second order parabolic equations, initial boundary value problems 35R11 Fractional partial differential equations 65F10 Iterative methods for linear systems 65T50 Discrete and fast Fourier transforms (numerical methods)