Abstract

We study ordinary nonlinear differential equations which arise from steady nonlinear conservation laws with source terms. Two examples of conservation laws which lead to these equations are the Saint-Venant and the Euler equations. In each case there is a reduction to a scalar equation and we use the ideas of upwinding and discretisation of source terms to devise methods for the solution. Numerical results are presented with both the Engquist-Osher and the Roe scheme with different strategies for discretising the source terms based on balance ideas. Acknowledgements Firstly, I would like to express my gratitude to Professor Mike Baines. His supervision, support and patience were constant throughout this work and encouraged me to go on. My thanks go also to Professor Nancy Nichols. Her supervision and advice were very helpful. As a team, their supervision complemented each other and I benefited from their knowledge and teaching. I am grateful to my sponsors in Portugal, Funda ção para a Ciência e a Tecnologia (grant PRAXIS XXI/BD/15905/98 from the Subprograma Ciência e Tecnologia do 2o Quadro Comunitário de Apoio, andtheEscola Superior de Tecnologia e Gestão from the Instituto Politécnico de Leiria, who made this project viable. I wish to thank the help of staff and colleagues in the Mathematics Department in Reading who always made me feel welcome. Studying in Department of Mathematics of the University of Reading afforded the opportunity to learn with very good teachers and to meet colleagues and fellow researchers. In Reading, I met new friends and their friendship and support were very important in making me feel less lonely. I would like to thank especially Jessica, Ana Teresa, Hussain and Giovanni. We shared very happy moments that I will cherish forever. I extend my thanks to Helena, who made my stay in the University of Reading possible, and to Fernando, Cacilda, Teresa Mota and Cristine and other members of the Brazilian and Portuguese Speakers Society. Among the friends I met in in the Mathematics Department, I will remember with

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