2018
Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations
CComm. Math. Sci.
Y. Pu, R.L. Pego, D. Dutykh and D. Clamond
Comm. Math. Sci. 16(5): 1361-1378 (2018)
Open Access
Abstract
We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh (Commun. Nonl. Sci. Numer. Simulat. 55 (2018) 237-247). This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.
Keywords
momentum equationcusponsGreen-Naghdi equationsconservation lawsenergy losspeakonsshallow-water equationsshooting methodshallow waterfourth-order derivativenon-dispersive regularizationweak solutionsshock stabilityweakly singular shock profilesSerre equationshyperbolic-elliptic systemlong waves
Bibliographic record
BibTeX Citation
@article{Pu2018weaklysingular,
author = {Pu, Y. and Pego, R.L. and Dutykh, D. and Clamond, D.},
title = {Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations},
journal = {Comm. Math. Sci.},
year = {2018},
volume = {16},
number = {5},
pages = {1361--1378},
doi = {10.4310/CMS.2018.v16.n7.a10},
abstract = {We study a regularization of the classical Saint-Venant (shallow-water) equations, recently introduced by D. Clamond and D. Dutykh (Commun. Nonl. Sci. Numer. Simulat. 55 (2018) 237-247). This regularization is non-dispersive and formally conserves mass, momentum and energy. We show that for every classical shock wave, the system admits a corresponding non-oscillatory traveling wave solution which is continuous and piecewise smooth, having a weak singularity at a single point where energy is dissipated as it is for the classical shock. The system also admits cusped solitary waves of both elevation and depression.},
keywords = {momentum equation, cuspons, Green-Naghdi equations, conservation laws, energy loss, peakons, shallow-water equations, shooting method, shallow water, fourth-order derivative, non-dispersive regularization, weak solutions, shock stability, weakly singular shock profiles, Serre equations, hyperbolic-elliptic system, long waves}
}