Adaptive Finite Difference Method with Non-uniform Timesteps for Fractional Diffusion Equations

Details Adaptive Finite Difference Method with Non-uniform Timesteps for Fractional Diffusion Equations Adaptive Finite Difference Method with Non-uniform Timesteps for Fractional Diffusion Equations

2011-09-29

arxiv.org/abs/1109.6622v1

Mathematics

Numerical Analysis

19 pages, 3 figures

Scientific paper

An adaptive implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to adapt the size of the timesteps to the behaviour of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The computational advantages of the present adaptive method against fixed step methods are illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.

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