Numerical methods and analysis for a class of fractional advection–dispersion models

In this paper, a class of fractional advection-dispersion models (FADM) is investigated. These models include five fractional advection-dispersion models: the immobile, mobile/immobile time FADM with a temporal fractional derivative 0 < γ < 1, the space FADM with skewness, both the time and space FADM and the time fractional advection-diffusion-wave model with damping with index 1 < γ < 2. They describe nonlocal dependence on either time or space, or both, to explain the development of anomalous dispersion. These equations can be used to simulate regional-scale anomalous dispersion with heavy tails, for example, the solute transport in watershed catchments and rivers. We propose computationally effective implicit numerical methods for these FADM. The stability and convergence of the implicit numerical methods are analyzed and compared systematically. Finally, some results are given to demonstrate the effectiveness of our theoretical analysis.

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Legendre Spectral Collocation Technique for Advection Dispersion Equations Included Riesz Fractional

Fractal and Fractional ◽

10.3390/fractalfract6010009 ◽

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.

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Evaluation of distribution coefficients for the prediction of strontium and cesium migration in a uniform sand

Canadian Geotechnical Journal ◽

10.1139/t82-008 ◽

1982 ◽

Vol 19 (1) ◽

pp. 92-103 ◽

Cited By ~ 45

Author(s):

W. D. Reynolds ◽

R. W. Gillham ◽

J. A. Cherry

Keyword(s):

Distribution Coefficient ◽

Breakthrough Curves ◽

Distribution Coefficients ◽

Column Experiments ◽

Advection Equation ◽

Dispersion Models ◽

Advection Dispersion ◽

Liquid Phases ◽

Long Tails ◽

The Mathematical Model

The validity of using a distribution coefficient (Kd) in the mathematical prediction of strontium and cesium transport through uniform saturated sand was investigated by comparing measured breakthrough curves with curves of simulations using the advection-dispersion and the advection equations. Values for Kd were determined by batch equilibration tests and, indirectly, by fitting the mathematical model to breakthrough data from column experiments. Although the advection-dispersion equation accurately represented the breakthrough curves for two nonreactive solutes (chloride and tritium), neither it nor the advection equation provided close representations of the strontium and cesium curves. The simulated breakthrough curves for strontium and cesium were nearly symmetrical, whereas the data curves were very asymmetrical, with long tails. Column experiments with different pore-water velocities indicated that the shape of the normalized breakthrough curves was not sensitive to velocity. This suggests that the asymmetry of the measured curves was the result of nonlinear partitioning of the cations between the solid and liquid phases, rather than nonequilibrium effects. The results indicate that the distribution coefficient, when used in advection-dispersion models for prediction of the migration of strontium and cesium in field situations, can result in significant error.

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Eulerian-Lagrangian Numerical Scheme for Simulating Advection, Dispersion, and Transient Storage in Streams and a Comparison of Numerical Methods

Journal of Environmental Engineering ◽

10.1061/(asce)0733-9372(2008)134:12(996) ◽

2008 ◽

Vol 134 (12) ◽

pp. 996-1005 ◽

Cited By ~ 5

Author(s):

T. J. Cox ◽

R. L. Runkel

Keyword(s):

Numerical Methods ◽

Numerical Scheme ◽

Transient Storage ◽

Advection Dispersion

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

Advances in Mathematical Physics ◽

10.1155/2015/590435 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 27

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.

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iCFD: Interpreted Computational Fluid Dynamics – Degeneration of CFD to one-dimensional advection-dispersion models using statistical experimental design – The secondary clarifier

Water Research ◽

10.1016/j.watres.2015.06.012 ◽

2015 ◽

Vol 83 ◽

pp. 396-411 ◽

Cited By ~ 14

Author(s):

Estelle Guyonvarch ◽

Elham Ramin ◽

Murat Kulahci ◽

Benedek Gy Plósz

Keyword(s):

Fluid Dynamics ◽

Computational Fluid Dynamics ◽

Experimental Design ◽

Statistical Experimental Design ◽

Dispersion Models ◽

One Dimensional ◽

Advection Dispersion ◽

Secondary Clarifier

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A comparison of some numerical methods in solving 1-D steady-state advection dispersion reaction equation

Civil Engineering and Environmental Systems ◽

10.1080/10286600902849968 ◽

2010 ◽

Vol 27 (2) ◽

pp. 155-172 ◽

Cited By ~ 2

Author(s):

D. Yudianto ◽

Xie Yuebo

Keyword(s):

Steady State ◽

Numerical Methods ◽

Reaction Equation ◽

Advection Dispersion

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Numerical Modeling of Phenol Adsorption on Granular Activated Carbon Fixed Bed: Comparison of Two Numerical Methods to Solve the Advection-dispersion Equation

Chemical Product and Process Modeling ◽

10.1515/cppm-2019-0130 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mounira Kolli ◽

Safia Semra ◽

Fatiha Benmahdi ◽

Mohamed Bouhelassa ◽

Michel Sardin

Keyword(s):

Experimental Data ◽

Activated Carbon ◽

Numerical Methods ◽

Dispersion Equation ◽

Fixed Bed ◽

Granular Activated Carbon ◽

Picard Iteration ◽

Advection Dispersion ◽

Newton Raphson ◽

Runge Kutta

AbstractThis paper presents a comparison between some numerical methods and techniques for solving the nonlinear advection-dispersion equation, which may be used to describe the adsorption of phenol into a granular activated carbon fixed bed under local equilibrium conditions. The adsorption is described by the Langmuir isotherm, which makes the advection-dispersion equation nonlinear. This equation is solved successively by using the approximation and linearization techniques. For each technique, two types of numerical algorithms are used. Concerning the first one, the Implicit and the Runge Kutta schemes are used. As for the second one, the Modified Picard iteration and the Newton Raphson scheme are applied. Simulation results have been compared to each other and to the experimental data as well. Both of the Implicit and the Runge Kutta algorithms have led to superimposed simulated breakthrough curves. Both of the modified Picard and Newton Raphson schemes have given identical results too. However, comparing to the experimental data, the obtained solution, using the approximation technique, has underestimated the retardation of solute and failed in fitting the experimental breakthrough. The Obtained solution, using the linearization technique, has correctly fitted the experimental results under all the conditions of: feed flow rate, activated carbon bed height and the inlet phenol concentration.

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