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The computation time required by standard finite difference methods withfixed timesteps for solving fractional diffusion equations is usually verylarge because the number of operations required to find the solution scales asthe square of the number of timesteps. Besides, the solutions of these problemsusually involve markedly different time scales, which leads to quiteinhomogeneous numerical errors. A natural way to address these difficulties isby resorting to adaptive numerical methods where the size of the timesteps ischosen according to the behaviour of the solution. A key feature of thesemethods is then the efficiency of the adaptive algorithm employed todynamically set the size of every timestep. Here we discuss two adaptivemethods based on the step-doubling technique. These methods are, in many cases,immensely faster than the corresponding standard method with fixed timestepsand they allow a tolerance level to be set for the numerical errors that turnsout to be a good indicator of the actual errors.

Fast, Accurate and Robust Adaptive Finite Difference Methods for Fractional Diffusion Equations: The Size of the Timesteps does Matter