A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations
Departamento de Física, Universidad de Extremadura, 06071 Badajoz, Spain
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Received: 3 June 2013
Revised: 6 August 2013
Published online: 7 October 2013
An implicit finite difference method with non-uniform timesteps for solving fractional diffusion and diffusion-wave equations in the Caputo form is presented. The non-uniformity of the timesteps allows one to adapt their size to the behaviour of the solution, which leads to large reductions in the computational time required to obtain the numerical solution without loss of accuracy. The stability of the method has been proved recently for the case of diffusion equations; for diffusion-wave equations its stability, although not proven, has been checked through extensive numerical calculations.
© EDP Sciences, Springer-Verlag, 2013