The basic equation of fluid dynamics -- Partial differential equations: analytic aspects -- Finite volume and finite difference discretization on nonuniform grids -- The stationary convection-diffusion equation -- The nonstationary convection-diffusion equation -- The incompressible Navier-Stokes equations -- Iterative methods -- The shallow-water equations -- Scalar conservation laws -- The Euler equations in one space dimension -- Discretization in general domains -- Numerical solution of the Euler equations in general domains -- Numerical solution of the Navier-Stokes equations in general domains -- Unified methods for computing incompressible and compressible flow.

The book is aimed at graduate students, researchers, engineers and physicists involved in flow computations. An up-to-date account is given of the present state-of-the-art of numerical methods employed in computational fluid dynamics. The underlying numerical principles are treated with a fair amount of detail, using elementary mathematical analysis. Attention is given to difficulties arising from geometric complexity of the flow domain and of nonuniform structured boundary-fitted grids. Uniform accuracy and efficiency for singular perturbation problems is studied, pointing the way to accurate computation of flows at high Reynolds number. Much attention is given to stability analysis, and useful stability conditions are provided, some of them new, for many numerical schemes used in practice. Unified methods for compressible and incompressible flows are discussed. Numerical analysis of the shallow-water equations is included. The theory of hyperbolic conservation laws is treated. Godunov's order barrier and how to overcome it by means of slope-limited schemes is discussed. An introduction is given to efficient iterative solution methods, using Krylov subspace and multigrid acceleration. Many pointers are given to current literature, to help the reader to quickly reach the current research frontier.