A novel second order finite difference discrete scheme for fractal mobile/immobile transport model based on equivalent transformative Caputo formulation

TL;DR

This paper introduces a second order finite difference scheme based on a transformative Caputo formulation for fractal mobile/immobile transport models, improving efficiency and convergence rate, with rigorous stability analysis and numerical validation.

Contribution

The paper proposes a novel second order finite difference scheme using a transformative Caputo derivative, enhancing computational efficiency and convergence for fractal transport models.

Findings

The scheme achieves unconditional stability.

Convergence rate improves from O(τ^{2-α}) to O(τ^{3-α}).

Numerical experiments confirm theoretical accuracy.

Abstract

In this article, we present a new second order finite difference discrete scheme for fractal mobile/immobile transport model based on equivalent transformative Caputo formulation. The new transformative formulation takes the singular kernel away to make the integral calculation more efficient. Furthermore, this definition is also effective where $\alpha$ is a positive integer. Besides, the T-Caputo derivative also helps to increase the convergence rate of the discretization of $\alpha$-order($0<\alpha<1$) Caputo derivative from $O(\tau^{2-\alpha})$ to $O(\tau^{3-\alpha})$, where $\tau$ is the time step. For numerical analysis, a Crank-Nicholson finite difference scheme to solve fractal mobile/immobile transport model is introduced and analyzed. The unconditional stability and a priori estimates of the scheme are given rigorously. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.