An Adaptive Difference Method for Variable-Order Diffusion Equations

TL;DR

This paper introduces an adaptive finite difference scheme for variable-order fractional diffusion equations that adjusts timesteps to maintain error bounds, resulting in faster computations while preserving accuracy.

Contribution

It presents a novel adaptive algorithm for variable-order fractional diffusion equations that improves efficiency over fixed-timestep methods.

Findings

Adaptive method is significantly faster for certain problems.

Maintains local error around preset bounds.

Performance similar to constant-order fractional equations.

Abstract

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 in order 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 turns out to be similar to those found for constant-order fractional diffusion equations.