A finite difference method with non-uniform timesteps for fractional diffusion equations

摘要

An implicit finite difference method with non-uniform timesteps for solvingthe fractional diffusion equation in the Caputo form is proposed. The methodallows one to build adaptive methods where the size of the timesteps isadjusted to the behaviour of the solution in order to keep the numerical errorssmall without the penalty of a huge computational cost. The method isunconditionally stable and convergent. In fact, it is shown that consistencyand stability implies convergence for a rather general class of fractionalfinite difference methods to which the present method belongs. The hugecomputational advantage of adaptive methods against fixed step methods forfractional diffusion equations is illustrated by solving the problem of thedispersion of a flux of subdiffusive particles stemming from a point source.