A family of stable numerical solvers for the shallow water equations with source terms q Tom as Chac on Rebollo a, * , Antonio Dom  ınguez Delgado b , Enrique D. Fern andez Nieto b a Departamento de Ecuaciones Diferenciales y An alisis Num erico, Universidad de Sevilla, C/Tarfia, s/n. 41080 Sevilla, Spain b Departamento de Matem atica Aplicada I, Universidad de Sevilla, ETS Arquitectura Avda, Reina Mercedes, N. 2, 41012 Sevilla, Spain Received 23 November 2001; received in revised form 3 June 2002 Abstract In this work we introduce a multiparametric family of stable and accurate numerical schemes for 1D shallow water equations. These schemes are based upon the splitting of the discretization of the source term into centered and de- centered parts. These schemes are specifically designed to fulfill the enhanced consistency condition of Berm udez and V azquez, necessary to obtain accurate solutions when source terms arise. Our general family of schemes contains as particular cases the extensions already known of Roe and Van Leer schemes, and as new contributions, extensions of Steger–Warming, Vijayasundaram, Lax–Friedrichs and Lax–Wendroff schemes with and without flux-limiters. We include some meaningful numerical tests, which show the good stability and consistency properties of several of the new methods proposed. We also include a linear stability analysis that sets natural sufficient conditions of stability for our general methods. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Finite volume method; Upwinding; Shallow water; Source terms 1. Introduction and motivation This paper deals with the numerical solution of 1D shallow water equations for channels with variable depth and width. These equations are a couple of conservation laws linking the depth h and the discharge q, which in condensated form read as follows: q This research was partially supported by Spanish Government Research Projects REN2000-1162-C02-01 and REN2000-1168-C02- 01. * Corresponding author. Tel.: +34-54-557989; fax: +34-54-552898. E-mail addresses: chacon@numer.us.es (T. Chaco ´n Rebollo), domdel@us.es (A. Dom  ınguez Delgado), edofer@us.es (E.D. Ferna ´ndez Nieto). 0045-7825/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0045-7825(02)00551-0 Comput. Methods Appl. Mech. Engrg. 192 (2003) 203–225 www.elsevier.com/locate/cma