An 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 build adaptive methods where the size of the timesteps is adjusted 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 huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.

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An explicit numerical method to solve a fractional cable equation which involves two temporal Riemann-Liouville derivatives is studied. The numerical difference scheme is obtained by approximating the first-order derivative by a forward difference formula, the Riemann-Liouville derivatives by the Grünwald-Letnikov formula, and the spatial derivative by a three-point centered formula. The accuracy, stability, and convergence of the method are considered. The stability analysis is carried out by means of a kind of von Neumann method adapted to fractional equations. The convergence analysis is accomplished with a similar procedure. The von-Neumann stability analysis predicted very accurately the conditions under which the present explicit method is stable. This was thoroughly checked by means of extensive numerical integrations.

The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors.

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