[Paper Review] Second order difference approximation for a class of Riesz space fractional advection-dispersion equations with delay

Mahdi Saedshoar Heris, Mohammad Javidi|arXiv (Cornell University)|Nov 26, 2018

Fractional Differential Equations Solutions44 references4 citations

TL;DR

This paper proposes a second-order accurate finite difference scheme based on fractional backward differentiation formulas (FBDF) and shifted Gr"{u}nwald difference (WSGD) methods for solving Riesz space fractional advection-dispersion equations with delay (RFADED). The method achieves conditional stability and convergence with accuracy of $\mathcal{O}(\kappa^2 + h^2)$, and analytical solutions in terms of Mittag-Leffler functions are derived. Numerical results confirm high accuracy and efficiency for various fractional orders and delay parameters.

ABSTRACT

In this paper, we propose numerical scheme for the Riesz space fractional advection-dispersion equations with delay (RFADED). Firstly, analytical solution for RFADED in terms of the functions of Mittag-Leffler type is derived. Secondly, the fractional backward differential formulas (FBDF) method and shifted Grünwald method are introduced to the Riesz space fractional derivatives. Next, stability and convergency of these methods have been proved. Thirdly, Crank-Nicolson scheme for solving this problem is proposed. We prove that the scheme is conditionally stable and convergent with the order accuracy of ${ m O}({κ^2} + {h^2})$. Finally, some numerical results are given to demonstrate that presented method is a computationally efficient and accurate method for solving RFADED.

Motivation & Objective

• To develop a numerical scheme for solving Riesz space fractional advection-dispersion equations with delay (RFADED), which are complex due to the combination of fractional derivatives and time delays.
• To derive an analytical solution for RFADED in terms of Mittag-Leffler functions, enabling theoretical validation of numerical results.
• To establish a second-order accurate finite difference method using fractional backward differentiation formulas (FBDF) and shifted Gr"{u}nwald difference (WSGD) for spatial Riesz fractional derivatives.
• To prove the conditional stability and convergence of the proposed Crank-Nicolson-type scheme with $\mathcal{O}(\kappa^2 + h^2)$ accuracy.
• To demonstrate the method's computational efficiency and accuracy through numerical experiments with varying fractional orders and delay parameters.

Proposed method

• The Riesz space fractional derivative is approximated using the shifted Gr"{u}nwald difference (WSGD) method, which provides second-order accuracy in space.
• The time-fractional derivative of order $\gamma \in (0,1)$ is approximated using the fractional backward differentiation formula (FBDF) method, ensuring second-order accuracy in time.
• A Crank-Nicolson-type scheme is constructed by combining FBDF for time and WSGD for space, resulting in a linear system solvable at each time step.
• The stability and convergence of the scheme are rigorously proven using a discrete energy method, showing conditional stability under a time-step restriction.
• The method handles the delay term by treating the initial history function $g(x,t)$ explicitly during the initial time interval $t \in [0, \tau]$, reducing the problem to a standard RFADE.
• Analytical solutions are derived using the Laplace transform and expressed in terms of Mittag-Leffler functions, providing a benchmark for numerical validation.

Experimental results

Research questions

• RQ1 Can a second-order accurate finite difference scheme be constructed for Riesz space fractional advection-dispersion equations with delay (RFADED) using FBDF and WSGD methods?
• RQ2 What is the stability and convergence behavior of the proposed Crank-Nicolson-type scheme for RFADED, and what conditions ensure its validity?
• RQ3 How does the proposed method perform in terms of accuracy and computational efficiency for different values of fractional orders $\alpha$, $\beta$, and $\gamma$, and delay $\tau$?
• RQ4 Can analytical solutions for RFADED be expressed in terms of Mittag-Leffler functions, and how do they compare with numerical results?
• RQ5 What is the convergence rate of the method in both space and time, and how does it compare with standard L1/Grünwald approximations?

Key findings

• The proposed Crank-Nicolson scheme is conditionally stable and convergent with second-order accuracy in both time ($\kappa^2$) and space ($h^2$), achieving $\mathcal{O}(\kappa^2 + h^2)$ convergence rate.
• Numerical experiments show that the convergence order in space is approximately 2.0 for $\alpha = 0.5$, $\beta = 1.6$, and $\gamma = 0.2$, with errors decreasing from $1.4195 \times 10^{-4}$ to $9.0898 \times 10^{-6}$ as $h$ is halved.
• For $\alpha = 0.2$, $\beta = 1.9$, and $\gamma = 0.5$, the spatial convergence order reaches 2.00 at $h = 1/128$, confirming second-order accuracy.
• The method maintains second-order convergence in time, with convergence orders of 1.85–2.02 across different parameter sets, indicating robust temporal accuracy.
• The absolute errors in Example 2 are consistently below $6.1 \times 10^{-4}$ at $h = \kappa = 1/16$, decreasing to $1.03 \times 10^{-5}$ at $h = \kappa = 1/128$, demonstrating high accuracy.
• The analytical solution for RFADED is successfully derived in terms of Mittag-Leffler functions, providing an exact benchmark for validating the numerical scheme.