%0 Journal Article
%T Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion Equation
%A Yuan-Ming Wang
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/293706
%X This paper is concerned with two alternating direction implicit (ADI) finite difference methods for solving a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods is provided in the discrete maximum norm. It is shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. Numerical results are given to confirm the theoretical analysis results. 1. Introduction Fractional differential equations and fractional calculus arise in various application problems in science and engineering [1每16]. Various numerical methods have been developed for the computation of fractional differential equations [17每34]. Fractional subdiffusion equations describe a special type of anomalous diffusion [35], and it is a more difficult task to solve this kind of equation numerically. Numerical works for fractional subdiffusion equations are mostly focused on one-dimensional problems due to the memory effect in fractional derivatives; see, for example, [19, 20, 20每26, 31每33, 36每42]. A two-dimensional anomalous subdiffusion equation was numerically treated in [43, 44], where explicit and implicit finite difference schemes were proposed. Chen et al. [28] extended their work in [43] to a variable-order subdiffusion equation. Liu et al. [45] developed an implicit meshless approach based on the radial basis function for the numerical simulation of a two-dimensional subdiffusion problem. Chen and Liu [18] considered an implicit difference scheme for a three-dimensional fractional advection-diffusion equation, and a Richardson extrapolation was applied to improve the accuracy. The complexity of the fractional differential equations comes from the involving fractional derivatives that are nonlocal and have the character of history dependence and universal mutuality. This means that the computations would be costly if the implicit schemes were applied, especially for solving multidimensional problems [43, 44]. Some researchers have explored some techniques for reducing this cost. These techniques include the adaptive technique [46] and the matrix transfer technique [47, 48]. It is well known that alternating direction implicit (ADI) methods are unconditionally stable as the traditional implicit methods. On the other hand, they reduce a multidimensional problem to a series of independent one-dimensional problems, and thus the computational complexities and the computational cost can be greatly reduced. Therefore, ADI methods for fractional differential equations have the potential to
%U http://www.hindawi.com/journals/amp/2013/293706/