Analytical Solution of the Burgers Equation for Simulating Translatory Waves in Conveyance Channels

Odai, Samuel Nii; Kubo, Naritaka; Onizuka, Kotaro; Osato, Koji

doi:10.1061/(asce)0733-9429(2006)132:2(194)

Cited by 12 publications

(12 citation statements)

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“…By using a concept based on the wavelength, the BEM was tested analytically and the results confirmed those obtained numerically. The numerical and analytical approaches presented in this paper confirm that the BEM developed by Onizuka and Odai (1998) and modified by Odai et al (2006) is suitable for describing the shallow water equations because it captures the monoclinal-rising-wave phenomenon very well.…”

Section: Introductionsupporting

confidence: 61%

“…For further discussion on the analytical solution of the BEM see Odai et al (2006). A translation property of the Burgers equation is of interest.…”

Section: Transformation Of the Bemmentioning

confidence: 99%

“…The primary advantage of the Burgers' equation is that the general closed form solutions are known for the initial-value problem. The analytical solutions of the Burgers equation model (BEM) for application to openchannel flows have been fully described by Odai et al (2006).…”

Section: Introductionmentioning

confidence: 99%

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The Burgers equation model and the monoclinal wave phenomenon in conveyance channels

Odai

Kubo

2007

Journal of Hydraulic Research

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In this paper, the ability of the Burgers equation model (BEM) to describe the monoclinal-rising-wave phenomenon is presented. The BEM is transformed by standardizing the initial and final normal flow depths; and by introducing a moving coordinate system into the equation. The resulting equation is tested numerically and analytically by introducing an abrupt flow increase between initial and final steady flow causing a positive translatory wave at the upstream end of the channel. The wave attained a stable form after a long time has elapsed, and it finally approached a monoclinal-rising-wave. In the case of a negative translatory wave the wave never attained a stable form, but changed continually with the passage of time. Comparison made between the numerical solutions presented graphically and the analytical deductions confirm the ability of the BEM to describe the monoclinal-wave phenomenon in conveyance channels. RÉSUMÉDans cet article, on présente la capacité du modèle de l'équation de Burgers (BEM) à décrire le phénomène de formation de l'onde monoclinale. Le BEM est transformé en normalisant les tirants d'eau normaux initiaux et finaux, et en introduisant un système de coordonnées mobiles dans l'équation. L'équation résultante a été testée numériquement et analytiquement en imposant une augmentation brusque de débit, entre l'état stationnaire initial et l'état final, ce qui provoque une vague en translation positive à l'extrémité amont du canal. La vague a atteint une forme stable au bout d'un temps assez long, et a approché finalement une onde monoclinale. Dans le cas d'une vague en translation négative, celle-ci n'a jamais pu atteindre une forme stable, mais a changé continuellement au fil du temps. La comparaison faite entre les solutions numériques présentées en forme de graphiques et les déductions analytiques confirment la capacité du BEM à décrire le phénomène d'onde monoclinale dans les canaux de transport.

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Section: Introductionsupporting

confidence: 61%

“…For further discussion on the analytical solution of the BEM see Odai et al (2006). A translation property of the Burgers equation is of interest.…”

Section: Transformation Of the Bemmentioning

confidence: 99%

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The Burgers equation model and the monoclinal wave phenomenon in conveyance channels

Odai

Kubo

2007

Journal of Hydraulic Research

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“…To collocate the integral Equation (14) at the GG mesh grid ) also be the first-and second-order barycentric Gegenbauer integration vectors required for approximating single-and double-definite integrals over the interval [ −1, 1] and the region  2 , respectively, as described by Algorithms 6 and 7 and Equation (3.9) in the work of Elgindy. 51 Using these essential numerical quadrature tools, we are able to write the discrete forms of Equations (12) and (14) in the following linear algebraic system form…”

Section: The Chbgpmmentioning

confidence: 99%

“…Benton and Platzman 13 used the transformation to classify several distinct exact solutions of Burgers' equation in tabular forms. Odai et al 14 solved the reduced linear diffusion equation produced by Cole-Hopf transformation via means of Green's function assuming an infinite domain. Schiffner et al 15 derived an explicit analytical solution to Burgers' equation based on Volterra series.…”

Section: Introductionmentioning

confidence: 99%

High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method

Elgindy

Dahy

2018

Math Methods in App Sciences

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We present a novel high-order numerical method to solve viscous Burgers' equation with smooth initial and boundary data. The proposed method combines Cole-Hopf transformation with well-conditioned integral reformulations to reduce the problem into either a single easy-to-solve integral equation with no constraints or an integral equation provided by a single integral boundary condition. Fully exponential convergence rates are established in both spatial and temporal directions by embracing a full Gegenbauer collocation scheme based on Gegenbauer-Gauss mesh grids using apt Gegenbauer parameter values and the latest technology of barycentric Gegenbauer differentiation and integration matrices. The global collocation matrices of the reduced algebraic linear systems were derived allowing for direct linear system solvers to be used. Rigorous error and convergence analyses are presented in addition to two easy-to-implement pseudocodes of the proposed computational algorithms. We further show three numerical tests to support the theoretical investigations and demonstrate the superior accuracy of the method even when the viscosity parameter → 0, in the absence of any adaptive strategies typically required for adaptive refinements.

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2006

Hydrological Processes

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Analytical Solution of the Burgers Equation for Simulating Translatory Waves in Conveyance Channels

Cited by 12 publications

References 12 publications

The Burgers equation model and the monoclinal wave phenomenon in conveyance channels

The Burgers equation model and the monoclinal wave phenomenon in conveyance channels

High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method

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