Adaptive numerical solutions of time-fractional advection-diffusion-reaction equations

In this paper, we propose an adaptive procedure, recently developed for fractional ordinary differential equations, for the solutions of time-fractional advection-diffusion-reaction equations involving the Caputo derivative. We focus on the adaptivity of the discretization in time direction defining a step size selection function that allows adapting the time step size according to the local behaviour of the solution. The new approach is easy to implement and reveals to have a low computational cost. Test problems are reported and comparisons with results found in literature confirm the accuracy and efficiency of the step-size selection procedure for the solutions of fractional partial differential equations. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

In this paper, an adaptive procedure is proposed for solving time-fractional advection-diffusion-reaction equations involving the Caputo derivative, focusing on adaptivity in the time direction by defining a step size selection function based on the local behavior of the solution. The new approach is easy to implement with low computational cost, and test problems confirm the accuracy and efficiency of the step-size selection procedure for solving fractional partial differential equations.