An Adaptive Difference Method for Variable-Order Diffusion Equations

An adaptive finite difference scheme for variable-order fractional-time subdiffusion equations in the Caputo form is studied. The fractional-time derivative is discretized by the L1 procedure but using nonhomogeneous timesteps. The size of these timesteps is chosen by an adaptive algorithm to keep the local error bounded around a preset value, a value that can be chosen at will. For some types of problems, this adaptive method is much faster than the corresponding usual method with fixed timesteps while keeping the local error of the numerical solution around the preset values. These findings turn out to be similar to those found for constant-order fractional diffusion equations.