[Paper Review] High-order numerical methods for the Riesz space fractional advection-dispersion equations

Libo Feng, Pinghui Zhuang+3 more|arXiv (Cornell University)|Mar 31, 2020

Fractional Differential Equations Solutions30 references32 citations

TL;DR

The authors develop high-order finite difference schemes using weighted and shifted Grünwald difference operators for the Riesz space fractional advection-dispersion equations, proving unconditional stability and second-order convergence, and enhancing accuracy to fourth order via Richardson extrapolation; they validate with numerical examples.

ABSTRACT

In this paper, we propose high-order numerical methods for the Riesz space fractional advection-dispersion equations (RSFADE) on a {f}inite domain. The RSFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivative with the Riesz fractional derivatives of order $α\in(0,1)$ and $β\in(1,2]$, respectively. Firstly, we utilize the weighted and shifted Grünwald difference operators to approximate the Riesz fractional derivative and present the {f}inite difference method for the RSFADE. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of $\mathcal {O}(τ^2+h^2)$. Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be $\mathcal {O}(τ^4+h^4)$. Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are excellent with the theoretical analysis.

Motivation & Objective

• Motivate accurate numerical solutions for Riesz space fractional advection-dispersion equations on finite domains.
• Develop a Crank-Nicolson finite difference scheme based on weighted and shifted Grünwald difference operators.
• Prove unconditional stability and convergence of the scheme.
• Enhance convergence order via Richardson extrapolation.
• Demonstrate efficiency and accuracy through numerical experiments with RFDE/RFADE cases.

Proposed method

• Approximate the Riesz fractional derivatives using weighted and shifted Grünwald difference (WSGD) operators.
• Discretize in time with a Crank-Nicolson scheme to obtain a matrix form (I+D)U^n = (I−D)U^{n-1}.
• Show that the resulting operator D is strictly diagonally dominant and that (I+D) is invertible.
• Prove unconditional stability and convergence of the scheme with O(τ^2 + h^2) accuracy.
• Apply Richardson extrapolation to achieve O(τ^4 + h^4) accuracy.
• Support theory with lemmas on Grünwald weights and WSGD properties.

Experimental results

Research questions

• RQ1 Can a high-order finite difference scheme accurately approximate the Riesz space fractional derivatives in RSFADE on a finite domain?
• RQ2 Is the Crank-Nicolson-based scheme unconditionally stable and convergent, and can its convergence be enhanced via Richardson extrapolation?
• RQ3 What are the practical convergence rates observed in numerical experiments for RFDE/RFADE with varying alpha and beta?
• RQ4 How does the method perform in matrix form with interior-point discretization and boundary conditions?

Key findings

• The proposed CN–WSGD scheme achieves unconditional stability.
• The scheme converges with O(τ^2 + h^2) accuracy.
• Richardson extrapolation improves convergence to O(τ^4 + h^4).
• Numerical examples confirm second-order convergence and the effectiveness of REM in achieving fourth-order accuracy.
• The discrete operator D is strictly diagonally dominant and symmetric positive definite, ensuring invertibility and stability.
• Results are consistent with theoretical analysis across RFDE and RFADE cases with various α and β values.