Mathematical modeling and numerical analysis for the higher order Boussinesq system

Khorbatly, Bashar; Lteif, Ralph; Israwi, Samer; Gerbi, Stéphane

doi:10.1051/m2an/2022015

Cited by 4 publications

(2 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Intermediate-term well-posedness of the higher order Boussinesq model for almost flat bottom topography. In this section, as for the flat bottom case [18], we conclude that the higher-order Boussinesq model prolonged with medium and small topography variation (β ∼ √ µ and β = µ respectively) admits a unique solution in Y s>3/2 (see Definition 2) but on longer time scales of order ε −1/2 . It should be noted that the rest-state condition (63) is not required 4.…”

Section: 32supporting

confidence: 51%

Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

Khorbatly

2023

DCDS-B

Self Cite

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In the mathematical theory of water waves, this paper focuses on the hierarchy of higher order asymptotic models. The well-posedness of the medium amplitude extended Green-Naghdi model, as well as higher-ordered Boussinesq-Peregrine and Boussinesq models, is first demonstrated. Introducing a regularization term and various physical topography variations, we show that these models admit unique solutions by a standard energy estimate method in the "hyperbolic" space <inline-formula><tex-math id="M1">\begin{document}$ H^{s+2}( \mathbb R)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ s>3/2 $\end{document}</tex-math></inline-formula>, but on short/intermediate time scales with respect to amplitude and topography parameters of order <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon^{-1/4} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \max(\sqrt{ \varepsilon}, \beta)^{-1} = \beta^{-1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon^{-1/2} $\end{document}</tex-math></inline-formula> respectively. Furthermore, we show that the extended Green-Naghdi system's long-term well-posedness is reachable on time scales of order <inline-formula><tex-math id="M6">\begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}</tex-math></inline-formula>. The above three specified models, in particular, admit longer time existence of order <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon^{-1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \max( \varepsilon, \beta)^{-1} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ \varepsilon^{-1} $\end{document}</tex-math></inline-formula>, respectively.</p>

show abstract

Section: 32supporting

confidence: 51%

Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

Khorbatly

2023

DCDS-B

Self Cite

View full text Add to dashboard Cite

show abstract

“…(2.24)). Although in some cases their Cauchy problems have been proved well-posed, [11][12][13], they do not preserve any meaningful approximation of the total energy of the Euler equations. While the particular property is not restrictive, it is rather desirable from physical as well as numerical point of view.…”

Section: Introductionmentioning

confidence: 99%

Equations for small amplitude shallow water waves over small bathymetric variations

Israwi,

Khalifeh,

Mitsotakis

2023

Proc. R. Soc. A.

View full text Add to dashboard Cite

A generalized version of the a b c d -Boussinesq class of systems is derived to accommodate variable bottom topography in two-dimensional space. This extension allows for the conservation of suitable energy functionals in some cases and enables the description of water waves in closed basins with well-justified slip–wall boundary conditions. The derived systems possess a form that ensures their solutions adhere to important principles of physics and mathematics. By demonstrating their consistency with the Euler equations and estimating their approximation error, we establish the validity of these new systems. Their derivation is based on the assumption of small bathymetric variations. With practical applications in mind, we assess the effectiveness of some of these new systems through comparisons with standard benchmarks. The results indicate that the assumptions made during the derivation are not overly restrictive. The applications of the new systems encompass a wide range of scenarios, including the study of tsunamis, tidal waves and waves in ports and lakes.

show abstract

For the Shallow Water Waves: Bilinear-Form and Similarity-Reduction Studies on a Boussinesq-Burgers System

Gao,

Tian,

Zhou

et al. 2024

Int J Theor Phys

View full text Add to dashboard Cite

Mathematical modeling and numerical analysis for the higher order Boussinesq system

Cited by 4 publications

References 44 publications

Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

Long, intermediate and short-term well-posedness of high precision shallow-water models with topography variations

Equations for small amplitude shallow water waves over small bathymetric variations

For the Shallow Water Waves: Bilinear-Form and Similarity-Reduction Studies on a Boussinesq-Burgers System

Contact Info

Product

Resources

About