Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations

TL;DR

This paper develops high-order fractional-compact numerical algorithms for Riesz derivatives, improving accuracy in solving fractional reaction-dispersion equations while maintaining computational efficiency.

Contribution

It introduces generalized high-order fractional-compact formulas for Riesz derivatives and applies them to multidimensional fractional equations with proven stability and convergence.

Findings

Algorithms achieve high convergence orders

Numerical results confirm efficiency and accuracy

Stability analysis supports practical use

Abstract

It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the nonlocal properties of fractional operators. Therefore, developing some high-order numerical approximation formulas for fractional derivatives play a more important role in numerically solving fractional differential equations. This paper focuses on constructing (generalized) high-order fractional-compact numerical approximation formulas for Riesz derivatives. Then we apply the developed formulas to the one- and two-dimension Riesz spatial fractional reaction-dispersion equations. The stability and convergence of the derived numerical algorithms are strictly studied by using the energy analysis method. Finally, numerical simulations are given to demonstrate the efficiency and convergence orders of the presented numerical algorithms.