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A shallow water equation on the circle
Communications on Pure and Applied Mathematics, 1999The purpose of this paper is to study the spatially periodic case of the shallow water equation \[ \partial v/\partial t+ v\partial t/\partial x+\partial p/\partial x= 0\tag{1} \] in which the ``pressure'' \(p\) is \((1- d^2/dx^2)^{-1}(v^2+{1\over 2} v^{\prime 2})\). The equation has been much studied in recent years, starting with \textit{R.
Constantin, A., McKean, H. P.
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On ?The shallow water equations? By M. Shinbrot
Journal of Engineering Mathematics, 1972The paper [1] above describes two steady shallow water flows which differ from the known flows of this type. It is suggested that the conditions imposed in obtaining these solutions are not consistent.
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SLOW EIGENMODES OF THE SHALLOW-WATER EQUATIONS
Waves and Stability in Continuous Media, 2002We present a survey of recent work on the lowest end of the eigenmode spectrum of the shallow water equations in a rotating reference frame. The results are complemented with numerical simulations of the fully nonlinear equations. Having care of using physically correct, mass-conserving boundary conditions, a description in terms of slow eigenmodes ...
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Shallow Water equations for Non-Newtonian fluids
Journal of Non-Newtonian Fluid Mechanics, 2010zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Fernández-Nieto, Enrique D. +2 more
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Abstract This chapter first presents a simple derivation of the shallow water equation. It then proposes an implicit (time) and centered (space) discretization and tests it in one dimension. The chapter then examines solutions to look at front steepening and testing the dependence of the wave velocity on the water column depth expected ...
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Derivation of the Shallow Water Equations
2020This chapter presents a derivation of the shallow water equations, which describe flows where horizontal length-scales are much larger than the vertical. This occurs when hydrodynamic processes are small in comparison to gravity effects, and is therefore appropriate for tidal flows in channels where the tidal wavelength \(\lambda _w\) is greater than ...
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On a Shallow Water Equation Perturbed in Schwartz Class
Mathematical Physics, Analysis and Geometry, 2012zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Some numerical solutions of the shallow water equations
SUT Journal of Mathematics, 1994The Taylor-Galerkin approximation of the second order is presented for numerical solutions to the shallow water equations. Two-step scheme is used in the time integration, and linear triangular finite elements are used in spatial discretization. The method is applied to the simulation of tidal current in the Osaka Bay and in the Ariake Bay.
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