A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line

Barros, Felipe P. J. de; Colbrook, Matthew J.; Fokas, A. S.

doi:10.1016/j.ijheatmasstransfer.2019.05.018

Cited by 30 publications

(20 citation statements)

References 37 publications

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“…Through parameterization of the contours, these explicit solutions can be numerically evaluated. [15][16][17] The application of the UTM is systematic, regardless of the types of boundary data, e.g., nonhomogeneous Dirichlet, Neumann, and Robin conditions. This is one reason the UTM is more general and effective than standard methods for evolution IBVPs.…”

Section: The Continuous Utmmentioning

confidence: 99%

“…The exact (continuous) solution is given by 𝑞(𝑥, 𝑇) = 𝜙(𝑥 + 𝑇), while the semidiscrete solution is obtained from the representation (17) with the standard forward discretization stencil. Figure 3 shows the semidiscrete solution 𝑞 𝑛 (𝑇) (left panel) and a log-log error plot (right panel) of the ∞-norm of 𝑞 𝑛 (0.5) − 𝑞(𝑥 𝑛 , 0.5), as a function of ℎ.…”

Section: Forward Discretization Of 𝒒 𝒕 = 𝒄 𝒒 𝒙mentioning

confidence: 99%

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The numerical solutions of linear semidiscrete evolution problems on the half‐line using the Unified Transform Method

Cisneros

Deconinck

2021

Stud Appl Math

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We discuss a semidiscrete analogue of the Unified Transform Method (UTM), introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations of constant coefficients. The semidiscrete method is applied to various spacial discretizations of several first-and second-order linear equations on the half-line 𝑥 ≥ 0, producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. We additionally show how the UTM treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and propose the notion of "natural" discretizations. We consider the continuum limit of the semidiscrete solutions and provide several numerical examples.

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Section: The Continuous Utmmentioning

confidence: 99%

Section: Forward Discretization Of 𝒒 𝒕 = 𝒄 𝒒 𝒙mentioning

confidence: 99%

The numerical solutions of linear semidiscrete evolution problems on the half‐line using the Unified Transform Method

Cisneros

Deconinck

2021

Stud Appl Math

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“…However, little work has been conducted to analytically solve the above complicated model for the transient solute dispersion in submerged vegetated flows, due to the difficulty resulted from the spatial variation of the vertical turbulent diffusion coefficient. Although the hybrid numerical-analytical method of Generalized Integral Transform Technique (Cotta, 1993) has been widely used to model solute transport in river and channel flows with nonuniform velocity profile and any forms of turbulent diffusivity (de Barros & Cotta, 2007;de Barros et al, 2006de Barros et al, , 2019Cotta et al, 2001Cotta et al, , 2013, its application to the submerged vegetated flows for 10.1029/2019WR025586 an instantaneous release has not been illustrated and the basic characteristics of the dispersion process for solute transport in the submerged vegetated flow has not ever discussed. Although numerical methods can give good solutions for solute transport in submerged vegetated flows, they are generally computationally expensive.…”

Section: 1029/2019wr025586mentioning

confidence: 99%

Transient Solute Dispersion in Wetland Flows With Submerged Vegetation: An Analytical Study in Terms of Time‐Dependent Properties

Guo

Jiang

Chen

2020

Water Resources Research

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Predicting the concentration distribution of solute transport in vegetated flows is significant for associated environmental applications in water resources. While a semianalytical study has been made for the steady dispersion with a continuous release (Rubol et al., 2016, https://doi.org/10.1002/2016WR018907), an in‐depth analytical investigation has yet to be performed for the more complicated case of transient dispersion due to an instantaneous release. In regard to the pioneering benchmark observation (Murphy et al., 2007, https://doi.org/10.1029/2006WR005229) of transient solute dispersion in a submerged vegetated flow for an instantaneous source release at the top of the canopy, an extensive analytical effort is paid in this work to derive the time‐dependent basic properties and to investigate the temporal evolution of concentration distribution. Obtained large‐time asymptotic dispersivities agree well with available experimental results (Murphy et al., 2007, https://doi.org/10.1029/2006WR005229), with higher significance ( R2=0.87, n=24) for correlation between analytical results and experimental data. Analytical solutions covering effects of dispersivity, skewness, and kurtosis of detailed concentration distribution are compared satisfactorily with numerical results obtained by the random displacement method. The findings illuminate the applicability of this analytical approach in environmental risk assessment and water quality management.

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“…Therefore, many researchers are more prefer to the numerical method to solve. Special for the advectiondispersion model, some numerical methods were developed such as random lattice Boltzmann method [10], Haar wavelets coupled with finite differences [11], unified transform/Fokas method [12], a numerical method based on Legendre scaling functions [13] and many more. Each of these methods had some superiority and weakness.…”

Section: Introductionmentioning

confidence: 99%

Parameter Estimation on Two-Dimensional Advection-Dispersion Model of Biological Oxygen Demand in Facultative Waste Water Stabilization Pond: Case Study at Sewon Wastewater Treatment Facility

Sunarsih¹,

Sasongko²,

Sutrisno³

2020

Adv. sci. technol. eng. syst. j.

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To build a precise mathematical model describing a natural phenomenon, parameter estimation is needed to achieve the best parameter value. In this paper, we have calculated the best parameter value for a two-dimensional advection-dispersion differential equation of the biological oxygen demand degradation process in a facultative wastewater stabilization pond. This research was conducted with case study data collected from Sewon, Bantul facultative wastewater treatment facility located in Yogyakarta, Indonesia. The method employed in this research is based on the least square value by minimizing the difference between the observed data and the simulated data via quadratic programming using the interior point algorithm. This method gave the best value for the parameters observed in the model i.e. dispersion constant and the flow rate velocity. From the results, we have achieved that the best value for dispersion constant is 0.25, the velocity of the flow rate in the x-direction is 0.1, and the velocity of the flow rate in the y-direction is 0.15 whereas the relative error of this parameter estimation was 11.5% that is acceptable.

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A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line

Cited by 30 publications

References 37 publications

The numerical solutions of linear semidiscrete evolution problems on the half‐line using the Unified Transform Method

The numerical solutions of linear semidiscrete evolution problems on the half‐line using the Unified Transform Method

Transient Solute Dispersion in Wetland Flows With Submerged Vegetation: An Analytical Study in Terms of Time‐Dependent Properties

Parameter Estimation on Two-Dimensional Advection-Dispersion Model of Biological Oxygen Demand in Facultative Waste Water Stabilization Pond: Case Study at Sewon Wastewater Treatment Facility

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