A MacCormack Method for Complete Shallow Water Equations with Source Terms

TL;DR

This paper develops and analyzes an enhanced MacCormack numerical scheme for solving the 1D complete shallow water equations with source terms, focusing on stability, convergence, and comparison with analytical solutions.

Contribution

It extends the MacCormack method to include source terms in the shallow water equations and provides stability and convergence analysis for this improved scheme.

Findings

The method achieves stable solutions with controlled oscillations.

Convergence rate of the scheme is established and validated.

Numerical solutions closely match analytical solutions.

Abstract

In the last decades, more or less complex physically-based hydrological models, have been developed to solve the shallow water equations or their approximations using various numerical methods. The MacCormack method was developed for simulating overland flow with spatially variable infiltration and microtopography using the hydrodynamic flow equations. The basic MacCormack scheme is enhanced when it uses the method of fractional steps to treat the friction slope or a stiff source term and to upwind the convection term in order to control the numerical oscillations and stability. In this paper we describe, the MacCormack scheme for 1D complete shallow water equations with source terms, analyze the stability condition of the method and we provide the convergence rate of the algorithm. This work improves some well known results deeply studied in the literature which concern the Saint-Venant problem and it represents an extension of the time dependent shallow water equations without source terms. The numerical evidences consider the rate of convergence of the method and compares the numerical solution respect to the analytical one.