The correct modeling of convection processes, without introducing excessive artificial damping while retaining reasonable accuracy, stability, boundedness, and algorithmic simplicity, plays a key role in reproducing complex flow physics. In this book, we present the development and testing of high-resolution upwind schemes for controlling the numerical diffusion in general conservation laws and fluid dynamics problems. In particular, the ALUS (Adaptive Linear Upwind Scheme) and TOPUS (Third-Order Polynomial Upwind Scheme) are developed. These schemes are tested in 1D scalar transport, 1D convection-diffusion equations, 1D hyperbolic systems, 2D Euler equations, and in 2D/3D Navier-Stokes equations. The schemes are then combined with a Reynolds stress algebraic equation model for the simulation of 2D incompressible turbulent flows with and without free surfaces. The book is divided into several reasonably self-contained sections. Successive sections deal with increasing levels of details. This should help the reader to obtain a clear understanding of the text presented which includes both the basic concepts and many different ways of applying these concepts.

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