The influence of source terms on stability, accuracy and conservation in two‐dimensional shallow flow simulation using triangular finite volumes

TLDR

The two‑dimensional shallow water model, a hyperbolic system, is widely used to simulate unsteady surface wave propagation, and numerical scheme development typically begins with scalar equations that may include source terms. This study aims to present a complete formulation of a second‑order finite volume scheme for the shallow water equation, with particular attention to its reduction to first order. The authors analyze explicit first‑ and second‑order upwind finite volume schemes, emphasizing stability constraints, numerical conservation, positivity preservation in the presence of source terms, and defining time‑step limits at cell edges via the CFL condition. © 2007 John Wiley & Sons, Ltd.

Abstract

Abstract The two‐dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two‐dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second‐order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case. The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant–Friedrichs–Lewy (CFL) condition. Copyright © 2007 John Wiley & Sons, Ltd.