Discretization of the divergence formulation of the bed slope term in the shallow-water equations and consequences in terms of energy balance

Abstract

In this research, the influence on energy balance of the discretization scheme of the divergence formulation of the bed slope term in the shallow-water equations is analyzed theoretically (for a single topographic step) and based on two numerical tests. Different values of the main parameter controlling the discretization scheme of the divergence formulation are analyzed to identify the formulation which minimizes the energy variation resulting from the discretization. For a wide range of ambient Froude numbers and relative step heights, the theoretical value of the control parameter minimizing the energy variation falls within a very narrow range, which can reasonably be approximated by a single “optimal” value. This is a result of high practical relevance for the design of accurate numerical schemes, as confirmed by the results of the numerical tests.