Weakly singular shock profiles for a non-dispersive regularization of shallow-water equations
For researchers in hyperbolic conservation laws and shallow-water modeling, this provides a theoretical justification for a regularization that avoids oscillations while preserving shock structure.
The authors show that a non-dispersive regularization of the Saint-Venant equations admits continuous, piecewise smooth traveling wave solutions for every classical shock, with energy dissipation at a single weak singularity, and also cusped solitary waves.
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.