Bona–Smith-Type systems in Bounded Domains with Slip-Wall Boundary Conditions: Theoretical Justification and a Conservative Numerical Scheme

Considered herein is a class of Boussinesq systems of Bona–Smith type that describe the propagation of long surface water waves of small amplitude in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situation...

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Vydáno v:Journal of scientific computing Ročník 102; číslo 1; s. 16
Hlavní autoři: Antonopoulos, Dimitrios, Mitsotakis, Dimitrios
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.01.2025
Springer Nature B.V
Témata:
Algorithms
Approximation
Boundary conditions
Boundary value problems
Boussinesq equations
Computational Mathematics and Numerical Analysis
Energy conservation
Initial conditions
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Numerical models
Propagation
Runge-Kutta method
Slip
Surface water
Theoretical
Topography
Velocity
Vorticity
Water conservation
Water waves
Wave propagation
Finite element method
Long waves
Energy conservation
35Q35
76B25
Boussinesq systems
76M10
Slip-wall conditions
ISSN:0885-7474, 1573-7691
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Shrnutí:Considered herein is a class of Boussinesq systems of Bona–Smith type that describe the propagation of long surface water waves of small amplitude in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situations involving water waves in channels, ports, and generally in basins with solid boundaries. We prove that, given appropriate initial conditions, the corresponding initial-boundary value problems have unique solutions locally in time, which is a fundamental property of deterministic mathematical modeling. Moreover, we demonstrate that the systems under consideration adhere to three basic conservation laws for water waves: mass, vorticity, and energy conservation. The theoretical analysis of these specific Boussinesq systems leads to a conservative mixed finite element formulation. Using explicit, relaxation Runge–Kutta methods for the discretization in time, we devise a fully discrete scheme for the numerical solution of initial-boundary value problems with slip-wall conditions, preserving mass, vorticity, and energy. Finally, we present a series of challenging numerical experiments to assess the applicability of the new numerical model.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02742-8