One of the major achievements in computational fluid dynamics has been the development of numerical methods for simulating compressible flows, combining higher-order accuracy in smooth regions with a sharp, oscillation-free representation of embedded shocks methods and now known as "high-resolution schemes". Together with introductions from the editors written from the modern vantage point this volume collects in one place many of the most significant papers in the development of high-resolution schemes as occured at ICASE.

High resolution upwind and centered methods are today a mature generation of computational techniques applicable to a wide range of engineering and scientific disciplines, Computational Fluid Dynamics (CFD) being the most prominent up to now. This textbook gives a comprehensive, coherent and practical presentation of this class of techniques. The book is designed to provide readers with an understanding of the basic concepts, some of the underlying theory, the ability to critically use the current research papers on the subject, and, above all, with the required information for the practical implementation of the methods. Applications include: compressible, steady, unsteady, reactive, viscous, non-viscous and free surface flows.

This book presents the latest numerical solutions to initial value problems and boundary value problems described by ODEs and PDEs. The author offers practical methods that can be adapted to solve wide ranges of problems and illustrates them in the increasingly popular open source computer language R, allowing integration with more statistically based methods. The book begins with standard techniques, followed by an overview of 'high resolution' flux limiters and WENO to solve problems with solutions exhibiting high gradient phenomena. Meshless methods using radial basis functions are then discussed in the context of scattered data interpolation and the solution of PDEs on irregular grids. Three detailed case studies demonstrate how numerical methods can be used to tackle very different complex problems. With its focus on practical solutions to real-world problems, this book will be useful to students and practitioners in all areas of science and engineering, especially those using R.

This is the first book which describes completely the nontraditional difference schemes which combine the ideas of Pad‚-type approximation and upwind differencing. These possess some favorable properties and can be used to solve various problems in fluid dynamics and related disciplines. They were proposed by the author in the seventies and are extensively used in Russia. However, they seem to be relatively unknown outside the country. In this book, the author presents the theory of the schemes, to provide some sophisticated algorithms for different computational fluid dynamics problems, to supply readers with useful information which would permit them to construct a rich variety of algorithms of this type and to illustrate the applications of these methods to the numerical simulation of various fluid dynamics phenomena, ranging from supersonic viscous flows to some atmosphere and ocean processes. This book is an essential guide for anyone keenly interested in this field.

These notes developed from a course on the numerical solution of conservation laws first taught at the University of Washington in the fall of 1988 and then at ETH during the following spring. The overall emphasis is on studying the mathematical tools that are essential in de veloping, analyzing, and successfully using numerical methods for nonlinear systems of conservation laws, particularly for problems involving shock waves. A reasonable un derstanding of the mathematical structure of these equations and their solutions is first required, and Part I of these notes deals with this theory. Part II deals more directly with numerical methods, again with the emphasis on general tools that are of broad use. I have stressed the underlying ideas used in various classes of methods rather than present ing the most sophisticated methods in great detail. My aim was to provide a sufficient background that students could then approach the current research literature with the necessary tools and understanding. vVithout the wonders of TeX and LaTeX, these notes would never have been put together. The professional-looking results perhaps obscure the fact that these are indeed lecture notes. Some sections have been reworked several times by now, but others are still preliminary. I can only hope that the errors are not too blatant. Moreover, the breadth and depth of coverage was limited by the length of these courses, and some parts are rather sketchy.

Computational aeroacoustics is rapidly emerging as an essential element in the study of aerodynamic sound. As with all emerging technologies, it is paramount that we assess the various opportuni ties and establish achievable goals for this new technology. Essential to this process is the identification and prioritization of fundamental aeroacoustics problems which are amenable to direct numerical siIn ulation. Questions, ranging from the role numerical methods play in the classical theoretical approaches to aeroacoustics, to the correct specification of well-posed numerical problems, need to be answered. These issues provided the impetus for the Workshop on Computa tional Aeroacoustics sponsored by ICASE and the Acoustics Division of NASA LaRC on April 6-9, 1992. The participants of the Work shop were leading aeroacousticians, computational fluid dynamicists and applied mathematicians. The Workshop started with the open ing remarks by M. Y. Hussaini and the welcome address by Kristin Hessenius who introduced the keynote speaker, Sir James Lighthill. The keynote address set the stage for the Workshop. It was both an authoritative and up-to-date discussion of the state-of-the-art in aeroacoustics. The presentations at the Workshop were divided into five sessions - i) Classical Theoretical Approaches (William Zorumski, Chairman), ii) Mathematical Aspects of Acoustics (Rodolfo Rosales, Chairman), iii) Validation Methodology (Allan Pierce, Chairman), iv) Direct Numerical Simulation (Michael Myers, Chairman), and v) Unsteady Compressible Flow Computa tional Methods (Douglas Dwoyer, Chairman).

This textbook explores both the theoretical foundation of the Finite Volume Method (FVM) and its applications in Computational Fluid Dynamics (CFD). Readers will discover a thorough explanation of the FVM numerics and algorithms used for the simulation of incompressible and compressible fluid flows, along with a detailed examination of the components needed for the development of a collocated unstructured pressure-based CFD solver. Two particular CFD codes are explored. The first is uFVM, a three-dimensional unstructured pressure-based finite volume academic CFD code, implemented within Matlab. The second is OpenFOAM®, an open source framework used in the development of a range of CFD programs for the simulation of industrial scale flow problems. With over 220 figures, numerous examples and more than one hundred exercise on FVM numerics, programming, and applications, this textbook is suitable for use in an introductory course on the FVM, in an advanced course on numerics, and as a reference for CFD programmers and researchers.

A comparative study of five upwind schemes was performed to evaluate their ability accurately model the convective fluxes of the Euler equations for problems containing complex shock structure. The schemes investigated used a variety of Reimann solvers and obtained higher order accuracy using either a MUSCL or non-MUSCL approach. The MUSCL-type schemes included the flux vector split formulations of Steiger-Warming and van Leer and the flux difference split approach of Roe. The Non-MUSCL schemes included the Symmetric and Upwind TVD methods of Yee, and Harten and Yee. Two central difference schemes provide a basis for the evaluation of these upwind methods. The comparison was performed using identical meshes and convergence criteria. In a supersonic blunt body flow, all the upwind schemes displayed comparably resolved bow shocks, independent of free stream Mach number. However, a complex type IV shock on cowl lip example pointed out significant difference in the accuracy and convergence behavior of the schemes. A comparison of the flow structure shown by the various algorithms on identical grids indicated that the discrete solutions obtained with Upwind TVD and Roe flux difference splitting were the least diffusive of the upwind methods considered.

This book examines the well-posedness theory for nonlinear hyperbolic systems of conservation laws, recently completed by the author together with his collaborators. It covers the existence, uniqueness, and continuous dependence of classical entropy solutions. It also introduces the reader to the developing theory of nonclassical (undercompressive) entropy solutions. The systems of partial differential equations under consideration arise in many areas of continuum physics.

This book, first published in 2002, contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. These equations describe a wide range of wave propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a self-contained manner, along with much of the mathematical theory of hyperbolic problems. High-resolution versions of Godunov's method are developed, in which Riemann problems are solved to determine the local wave structure and limiters are then applied to eliminate numerical oscillations. These methods were originally designed to capture shock waves accurately, but are also useful tools for studying linear wave-propagation problems, particularly in heterogenous material. The methods studied are implemented in the CLAWPACK software package and source code for all the examples presented can be found on the web, along with animations of many of the simulations. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.