WITHDRAWN: High-order numerical methods for the Riesz space fractional advection–dispersion equations

2016 ◽  
Author(s):  
L.B. Feng ◽  
P. Zhuang ◽  
F. Liu ◽  
I. Turner ◽  
J. Li
Keyword(s):  
Numerical Methods ◽  
High Order ◽  
Riesz Space ◽  
Dispersion Equations ◽  
Advection Dispersion

scholarly journals Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
X. Wang ◽  
F. Liu ◽  
X. Chen
Keyword(s):  
Numerical Methods ◽  
Quadrature Rule ◽  
Second Order ◽  
Riesz Space ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Alternating Direction ◽  
The Stability ◽  
Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

Legendre Spectral Collocation Technique for Advection Dispersion Equations Included Riesz Fractional

2021 ◽  
Vol 6 (1) ◽  
pp. 9
Author(s):  
Mohamed M. Al-Shomrani ◽  
Mohamed A. Abdelkawy
Keyword(s):  
Numerical Methods ◽  
Numerical Algorithm ◽  
Spectral Collocation ◽  
Theoretical Formulation ◽  
Numerical Examples ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Collocation Technique ◽  
Collocation Approach

The advection–dispersion equations have gotten a lot of theoretical attention. The difficulty in dealing with these problems stems from the fact that there is no perfect answer and that tackling them using local numerical methods is tough. The Riesz fractional advection–dispersion equations are quantitatively studied in this research. The numerical methodology is based on the collocation approach and a simple numerical algorithm. To show the technique’s performance and competency, a comprehensive theoretical formulation is provided, along with numerical examples.

scholarly journals An advanced numerical modeling for Riesz space fractional advection–dispersion equations by a meshfree approach

2016 ◽  
Vol 40 (17-18) ◽  
pp. 7816-7829 ◽  
Author(s):  
Z.B. Yuan ◽  
Y.F. Nie ◽  
F. Liu ◽  
I. Turner ◽  
G.Y. Zhang ◽  
...  
Keyword(s):  
Numerical Modeling ◽  
Riesz Space ◽  
Dispersion Equations ◽  
Advection Dispersion

scholarly journals High-Order Algorithms for Riesz Derivative and their Applications (III)

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Hengfei Ding ◽  
Changpin Li
Keyword(s):  
Numerical Methods ◽  
Theoretical Analysis ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Turbulent Diffusion ◽  
Numerical Experiments ◽  
High Order ◽  
Riesz Space ◽  
Sixth Order ◽  
Convergence Orders

AbstractNumerical methods for fractional calculus attract increasing interest due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz type turbulent diffusion equation (or, Riesz space fractional turbulent diffusion equation). Numerical experiments are displayed which support the theoretical analysis.

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