Computational Algorithms for Shallow Water Equations

This book is a thoroughly revised and enlarged version of “Shock-capturing methods for free-surface shallow flows", first published by Wiley and Sons, 2001. The book describes mathematically free-surface flows through partial differential equations and includes modern shock-capturing methods to solve them, with strong emphasis on finite volume upwind and centred methods. Such equations and methods are fundamental in simulating shallow water flows but also atmospheric flows, dispersion of dense gases and the dynamics of mixtures of materials.

The book is accompanied by numerical software in the form of sample computer programs as supplementary material. In this new edition, additional sections have been introduced to existing chapters. Also, new chapters have been included; one contains a review of the mathematics of hyperbolic partial differential equations, another introduces the numerical analysis of partial differential equations and another one deals with advanced very-high order numerical methods in the finite volume and discontinuous Galerkin frameworks. Furthermore, comprehensive modifications and corrections have been made throughout various sections of the text, and numerous figures depicting numerical results have been enhanced.

This book is primarily intended for environmental scientists, applied mathematicians and engineers in academia, research laboratories, industry and consultancy organisations. Senior undergraduate and postgraduate students involved with mathematical modelling and computational methods for environmental problems will benefit from studying this book. Lecturers could use most of the material for courses on numerical methods for wave propagation problems in hydraulics, oceanography, atmospherics and other geophysical fluid dynamics contexts.

The Shallow Water Equations.- Notions on Hyperbolic Equations.- Linear Shallow Water Equations.- Properties of the Nonlinear Equations.- Elementary Waves in Shallow Water.- Exact Riemann Solver: Wet Bed.- Exact Riemann Solver: Dry Bed.- Tests with Exact Solution.- Notions on Numerical Methods.- First-Order Methods for Systems.- Approximate Riemann Solvers.- Second-Order Non-Linear Methods.- Sources and Multidimensions.- Ader High-Order Methods.- Dam-Break Modelling.- Mach Reflection in Tsunamis.- Concluding Remarks.

Eleuterio F. Toro is currently Emeritus Professor of Mathematics at the University of Trento, Italy, and formerly Professor of Numerical Analysis (Professore Chiara Fama) at the same University. He holds an Honours B.Sc. in pure mathematics, an M.Sc. in functional analysis and differential equations, and a Ph.D. in computational mathematics. Since 1982, and prior to his appointment in Italy (2002), he held several academic positions in the UK, including a full professorship; he has also held many visiting appointments in various countries, such as Japan, Russia, China, and USA; he is Recipient of several honours and distinctions in various countries. His research has for many years focused on the construction of computational methods for solving time-dependent evolutionary partial differential equations, with emphasis on hyperbolic balance laws and applications to industrial, aerospace, environmental, and medical problems. He is Author/Editor of several books and Author of more than 300 publications.