Numerical Approximation Methods

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This second edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; a

Prenter introduces the numerical solution of boundary value problems by variational methods to serious students of engineering and mathematics who are familiar with basic applied linear algebra and sl

The subject of computational plasticity encapsulates the numerical methods used for the finite element simulation of the behaviour of a wide range of engineering materials considered to be plastic &#

Unifying the most important methodology in this field, Multi-Resolution Methods for Modeling and Control of Dynamical Systems explores existing approximation methods as well as develops new ones for t

A number of methods currently exist for the analysis and design of slopes. This book provides a critical review of these and offers several more appropriate approaches for overcoming numerical converg

The behaviour of systems occurring in real life is often modelled by partial differential equations. This book investigates how a user or observer can influence the behaviour of such systems mathematically and computationally. A thorough mathematical analysis of controllability problems is combined with a detailed investigation of methods used to solve them numerically, these methods being validated by the results of numerical experiments. In Part I of the book the authors discuss the mathematics and numerics relating to the controllability of systems modelled by linear and non-linear diffusion equations; Part II is dedicated to the controllability of vibrating systems, typical ones being those modelled by linear wave equations; finally, Part III covers flow control for systems governed by the Navier-Stokes equations modelling incompressible viscous flow. The book is accessible to graduate students in applied and computational mathematics, engineering and physics; it will also be of us

Observations of ocean circulation have increased as a result of international field programmes and of remote sensing systems on artificial earth satellites. Oceanographers are increasingly turning to inverse methods for combining these observations with numerical models of ocean circulation. Professor Bennett's work explores the potential for inverse theory, emphasizing possibilities rather than expedient or rudimentary applications. In addition to interpolating the data and adding realism to the model solutions, the methods can yield estimates for unobserved flow variables, forcing fields, and model parameters. Inverse formulations can resolve ill-posed modelling problems, lead to design criteria for oceanic observing systems, and enable the testing of models as scientific hypothesis. Exercises of varying difficulty rehearse technical skills and supplement the central theoretical development. Thus this book will be invaluable for environmental scientists and engineers, advanced underg

Elden (numerical analysis, Linkoping U.) uses as examples handwritten digits, text summarization, pagerank computations made famous by a certain very popular search engine (the name of which is now us

Increases in computer power and technology have rapidly advanced the applications of numerical modelling in environmental and earth sciences. The progress of numerical modelling in atmospheric, oceanic and geophysical sciences was the topic of an international conference held by the National Autonomous University of Mexico. The review articles and research papers in this volume constitute a wide-ranging account of modelling environmental and earth processes through a variety of numerical simulations. Sections are devoted to such topics as general circulation models and global change, methods of geophysical data assimilation, and mathematical and computational methods with geophysical applications. This book forms an excellent introduction and overview for graduate students as well as a critical update for researchers.

This book provides a thorough introduction to the specialized techniques needed to carry out numerical simulations of QCD: a description of lattice discretizations of fermions and gauge fields, method

This comprehensive text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis laid on the magnetohydrodynamics of liquid metals, and on

This set consists of the third edition of this highly acclaimed undergraduate textbook and its solutions manual containing complete worked solutions to half of the problems. Suitable for teaching all the mathematics for an undergraduate course in any of the physical sciences, the text provides lucid descriptions of all the topics, many worked examples, and over 800 exercises. New stand-alone chapters give a systematic account of the 'special functions' of physical science, cover an extended range of practical applications of complex variables, and give an introduction to quantum operators. Further tabulations, of relevance in statistics and numerical integration, have been added. In this edition, the remaining exercises have no hints, answers or worked solutions and can be used for unaided homework; full solutions are available to instructors on a password-protected web site, www.cambridge.org/9780521679718.

This estimation reference text thoroughly describes matrix factorization methods successfully employed by numerical analysts, familiarizing readers with the techniques that lead to efficient, economic

This book covers numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. It presents

This 2006 textbook provides a concise introduction to the key concepts and tools of statistical mechanics. It also covers advanced topics such as non-relativistic quantum field theory and numerical methods. After introducing classical analytical techniques, such as cluster expansion and Landau theory, the authors present important numerical methods with applications to magnetic systems, Lennard-Jones fluids and biophysics. Quantum statistical mechanics is discussed in detail and applied to Bose-Einstein condensation and topics in astrophysics and cosmology. In order to describe emergent phenomena in interacting quantum systems, canonical non-relativistic quantum field theory is introduced and then reformulated in terms of Feynman integrals. Combining the authors' many years' experience of teaching courses in this area, this textbook is ideal for advanced undergraduate and graduate students in physics, chemistry and mathematics.

Thermochronology, the study of the thermal history of rocks, enables us to quantify the nature and timing of tectonic processes. First published in 2006, Quantitative Thermochronology is a robust review of isotopic ages, and presents a range of numerical modeling techniques to allow the physical implications of isotopic age data to be explored. The authors provide analytical, semi-analytical and numerical solutions to the heat transfer equation in a range of tectonic settings and under varying boundary conditions. They then illustrate their modeling approach built around a large number of case studies. The benefits of different thermochronological techniques are also described. Computer programs on an accompanying website at www.cambridge.org/9781107407152 are introduced through the text and provide a means of solving the heat transport equation in the deforming Earth to predict the ages of rocks and compare them directly to geological and geochronological data. Several short tutorials

Perturbation series expansion methods are sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. This book gives a comprehensive guide to the use of series expansion methods for investigating phase transitions and critical phenomena, and lattice models of quantum magnetism, strongly correlated electron systems and elementary particles. Early chapters cover the classical treatment of critical phenomena through high-temperature expansions, and introduce graph theoretical and combinatorial algorithms. The book then discusses high-order linked-cluster perturbation expansions for quantum lattice models, finite temperature expansions, and lattice gauge models. Also included are numerous detailed examples and case studies, and an accompanying resources website, www.cambridge.org/9780521842426, contains programs for implementing these powerful numerical techniques. A valuable resource for graduate students and postdoctoral researchers work

Contents List of Figures; List of Tables; Preface; Chapter 1: Introduction; Chapter 2: Flow and Transport Equations; Chapter 3: Rock and Fluid Properties; Chapter 4: Numerical Methods; Chapter 5: Solu

This is the first comprehensive monograph that features state-of-the-art multigrid methods for enhancing the modeling versatility, numerical robustness, and computational efficiency of one of the most

The finite element method is one of the major tools used in the numerical solution of partial differential equations in science and engineering. This book offers a fundamental and practical introducti

Here is a book for anyone who would like to become better acquainted with the modern tools of numerical analysis for several significant computational problems arising in finance. The authors review s

Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for stu

The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing gives concrete examples of how to justify the validity of every single digit of a numerical answer. Methods range from carefu

This book is an introduction to numerical analysis and intends to strike a balance between analytical rigor and the treatment of particular methods for engineering problemsEmphasizes the earlier stage

This book describes a modern numerical technique, a lattice Boltzmann method, for shallow water flows with or without flow turbulence. This method requires only a simple microscopic equation to determ

This book describes a modern numerical technique, a lattice Boltzmann method, for shallow water flows with or without flow turbulence. This method requires only a simple microscopic equation to determ

This graduate textbook derives the theory of convex sets and functions, convex programming, nonlinear convex duality, and generalized convexity, then introduces numerical solution techniques for nonli

This book is an effective, concise text for students and researchers that combines the tools of dynamic programming with numerical techniques and simulation-based econometric methods. Doing so, it br

By giving a detailed account of bootstrap methods and their properties for dependent data, this book provides illustrative numerical examples throughout. The book fills a gap in the literature coverin

In light of increasing uses for direct numerical approximations of Maxwell's equations in science and engineering, this text provides mathematics graduate students and researchers with a theoretical f

In recent years there have been significant developments in the development of stable and accurate finite element procedures for the numerical approximation of a wide range of fluid mechanics problems

This book promotes the use of level set methods by scientists and engineers working on moving interface problems. It clearly presents the essential mathematical and numerical techniques. From the revi

A comprehensive introduction to the exploding field of data miningWe are surrounded by data, numerical and otherwise, which must be analyzed and processed to convert it into information that informs,

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 al

High-order numerical methods provide an efficient approach to simulating many physical problems. This book considers the range of mathematical, engineering, and computer science topics that form the foundation of high-order numerical methods for the simulation of incompressible fluid flows in complex domains. Introductory chapters present high-order spatial and temporal discretizations for one-dimensional problems. These are extended to multiple space dimensions with a detailed discussion of tensor-product forms, multi-domain methods, and preconditioners for iterative solution techniques. Numerous discretizations of the steady and unsteady Stokes and Navier-Stokes equations are presented, with particular attention given to enforcement of incompressibility. Advanced discretizations, implementation issues, and parallel and vector performance are considered in the closing sections. Numerous examples are provided throughout to illustrate the capabilities of high-order methods in actual app

Domain decomposition methods provide powerful and flexible tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applications in science a

The first of its kind in the field, this title examines the use of modern, shock-capturing finite volume numerical methods, in the solution of partial differential equations associated with free-surfa

Domain decomposition methods are designed to allow the effective numerical solution of partial differential equations on parallel computer architectures. They comprise a relatively new field of study

Partial differential equations arise in almost all areas of science, engineering, modeling, and forecasting. During the last two decades pseudospectral methods have emerged as successful alternatives to better known computational procedures, (e.g. finite difference and finite element methods of numerical solution), in several key application areas. These areas include computational fluid dynamics, wave motion, and weather forecasting. This book explains how, when and why this pseudospectral approach works. In order to make the subject accessible to students as well as researchers and engineers, the subject is presented using illustrations, examples, heuristic explanations, and algorithms rather than rigorous theoretical arguments. This book will be of interest to graduate students, scientists and engineers interested in applying pseudospectral methods to real problems.