On some numerical methods for solving the 1-D Saint-Venant equations of general flow regime. Part 1: Numerical methods

Abstract

Development of methods for numerical simulation of dike- or dam-break flood is one of essential problems of Fluid Mechanics at the present time. Many numerical methods for solving the 1-D Saint-Venant equations have been proposed. However, the analysis, the evaluation and the selection of appropriate and efficient methods are interest of many research groups and institutions in the world.

The purpose of this paper is to introduce and to evaluate four numerical methods for solving the 1-D homogenous Saint-Venant equations in combination with three approaches of processing source terms. The evaluation is based on the test problems, proposed by European Hydraulic Research Laboratories.

The Part 1of the paper presents some modern numerical methods for solving the 1-D Saint-Venant equations of general Bow regime, where the Bow may be mixed between sub­ critical and supercritical. The homogenous part of the system of equations is numerically solved by "shock capturing methods" for conservation laws: the Lax-Friedrichs, the self adjusting hybrid, the Roe's approximation and the Nessyahu-Tedmor methods. The source terms play an important role and are discretized by the pointwise, upwind or mixed approaches. In the second part of this paper the above methods are verified by a set of test problems, covering all of three flow regimes: subcritical, supercritical, transcritical. The re­sults show that the mixed approach of processing source terms is better than the pointwise one. The Roe approximation method with the mixed discretization of source terms is then applied for a preliminary evaluation of the Son La - Hoa Binh dam-break problem.