On the Efficiency of Staggered C-Grid Discretization for the Inviscid Shallow Water Equations from the Perspective of Nonstandard Calculus

Abstract

This paper provides a rationale for the commonly observed numerical efficiency of staggered C-grid discretizations for solving the inviscid shallow water equations. In particular, using the key concepts of nonstandard calculus, we aim to show that the grid staggering of the primitive variables (surface elevation and normal velocity components) is capable of dealing with flow discontinuities. After a brief introduction of hyperreals through the notion of infinitesimal increments, a nonstandard rendition of the governing equations is derived that essentially turns into a finite procedure and permits a convenient way of modeling the hydraulic jumps in open channel flow. A central result of this paper is that the discrete formulations thus obtained are distinguished by the topological structures of the solution fields and subsequently provide a natural framework for the staggered discretization of the governing equations. Another key of the present study is to demonstrate that the discretization naturally regularizes the solution of the inviscid flow passing through the hydraulic jump without the need of non-physical dissipation. The underlying justification is provided by analytically studying the distributions of the flow variables across an infinitesimal thin hydraulic jump along with the use of hyperreal Heaviside step functions. This main finding is shown to be useful to comprehend the importance of the application of staggered finite difference schemes to accurately predict rapidly varying free-surface flows. A numerical experiment is provided to confirm this result.