Head reconstruction method to balance flux and source terms in shallow water equations

This paper presents the head reconstruction method (HRM), a new technique that can be used within the finite volume framework to make shallow water models well balanced, i.e., to correct the imbalance that exists between flux and source terms in the equation discretization in the case of irregular bathymetry thus providing unphysical solutions. This technique, based on considering, within each computational cell, the total head of the flow (i.e., the sum of the elevation, pressure and kinetic energies per unit weight of the fluid) as an equilibrium variable, enables the preservation of dynamic equilibria under subcritical, transcritical, and supercritical flow conditions. The new technique is applied to the one-dimensional total variation diminishing (TVD) MUSCL-Hancock scheme and the conservation property is then proven mathematically for this scheme under static equilibrium conditions. Furthermore, the effectiveness of the HRM is tested and compared with two other well balancing techniques based on considering the water elevation as an equilibrium variable in various steady flow case studies. In the end the robustness of the HRM is tested in the simulation of dam-break flow over irregular bathymetry.