Barré de Saint-Venant equations' well-posedness

The Adhémar Barré de Saint-Venant equations, also known as the Shallow Water Equations (SWE), are fundamental to the study of open-channel flow and are the basis of all Fluvial Hydraulics software such as MIKE11, MIKE21, SRH-2D and TELEMAC. These non-linear partial differential equations, admitting no-analytical solutions except for simplified cases, are solvable using computers and numerical methods. All approved numerical methods can be used to solve the SWE although Preissmann’s scheme became the most widely employed in many software applications such as FLDWVE, SRH-1D, and HEC-RAS. By the end of the last century, some researchers pointed out a serious limitation of the existing hydraulics software and the numerical codes solving the one-dimensional SWE (1D SWE): their inability to simulate transcritical flows. As Preissmann’s scheme was the dominant numerical scheme used in hydraulics tools, the hydraulic community concluded that this scheme can’t simulate transcritical flows due to its numerical instability. To solve this issue, some researchers suggested modifications to Preissmann’s scheme, others used numerical techniques to eliminate the observed oscillations in the results of the simulations, while others proposed modifying slightly the original SWE by cancelling the inertia terms. This paper will prove that Preissmann scheme has nothing to do with these accusations and that the proposed improvements to this scheme are unnecessary. In fact, it will be proven that the observed instability is not a numerical one related to the numerical scheme used to solve the SWE, but rather a mathematical instability inherent to the SWE. It will be shown that, independently of the numerical method used, there will be no solution to the 1D SWE except if a specific condition is met: any problem based on the SWE is well-posed if this condition holds. In particular, in the case of very wide channels, this condition reduces to a limitation on Froude number, that is F<1.5 (F<2) if Manning (Chézy) equation is adopted for the friction slope.