Finite Difference and Finite Volume Methods

Author: Sandip Mazumder

Publisher: Academic Press

ISBN: 0128035048

Category: Technology & Engineering

Page: 484

View: 3372

Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods focuses on two popular deterministic methods for solving partial differential equations (PDEs), namely finite difference and finite volume methods. The solution of PDEs can be very challenging, depending on the type of equation, the number of independent variables, the boundary, and initial conditions, and other factors. These two methods have been traditionally used to solve problems involving fluid flow. For practical reasons, the finite element method, used more often for solving problems in solid mechanics, and covered extensively in various other texts, has been excluded. The book is intended for beginning graduate students and early career professionals, although advanced undergraduate students may find it equally useful. The material is meant to serve as a prerequisite for students who might go on to take additional courses in computational mechanics, computational fluid dynamics, or computational electromagnetics. The notations, language, and technical jargon used in the book can be easily understood by scientists and engineers who may not have had graduate-level applied mathematics or computer science courses. Presents one of the few available resources that comprehensively describes and demonstrates the finite volume method for unstructured mesh used frequently by practicing code developers in industry Includes step-by-step algorithms and code snippets in each chapter that enables the reader to make the transition from equations on the page to working codes Includes 51 worked out examples that comprehensively demonstrate important mathematical steps, algorithms, and coding practices required to numerically solve PDEs, as well as how to interpret the results from both physical and mathematic perspectives
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Author: Stig Larsson,Vidar Thomee

Publisher: Springer-Verlag

ISBN: 3540274227

Category: Mathematics

Page: 272

View: 4354

Das Buch ist für Studenten der angewandten Mathematik und der Ingenieurwissenschaften auf Vordiplomniveau geeignet. Der Schwerpunkt liegt auf der Verbindung der Theorie linearer partieller Differentialgleichungen mit der Theorie finiter Differenzenverfahren und der Theorie der Methoden finiter Elemente. Für jede Klasse partieller Differentialgleichungen, d.h. elliptische, parabolische und hyperbolische, enthält der Text jeweils ein Kapitel zur mathematischen Theorie der Differentialgleichung gefolgt von einem Kapitel zu finiten Differenzenverfahren sowie einem zu Methoden der finiten Elemente. Den Kapiteln zu elliptischen Gleichungen geht ein Kapitel zum Zweipunkt-Randwertproblem für gewöhnliche Differentialgleichungen voran. Ebenso ist den Kapiteln zu zeitabhängigen Problemen ein Kapitel zum Anfangswertproblem für gewöhnliche Differentialgleichungen vorangestellt. Zudem gibt es ein Kapitel zum elliptischen Eigenwertproblem und zur Entwicklung nach Eigenfunktionen. Die Darstellung setzt keine tiefer gehenden Kenntnisse in Analysis und Funktionalanalysis voraus. Das erforderliche Grundwissen über lineare Funktionalanalysis und Sobolev-Räume wird im Anhang im Überblick besprochen.
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Eine Einführung in analytische und numerische Methoden

Author: Wolfgang Arendt,Karsten Urban

Publisher: Springer-Verlag

ISBN: 382742237X

Category: Mathematics

Page: 353

View: 8404

Dieses Lehrbuch gibt eine Einführung in die partiellen Differenzialgleichungen. Wir beginnen mit einigen ganz konkreten Beispielen aus den Natur- Ingenieur und Wirtschaftswissenschaften. Danach werden elementare Lösungsmethoden dargestellt, z.B. für die Black-Scholes-Gleichung aus der Finanzmathematik. Schließlich wird die analytische Untersuchung großer Klassen von partiellen Differenzialgleichungen dargestellt, wobei Hilbert-Raum-Methoden im Mittelpunkt stehen. Numerische Verfahren werden eingeführt und mit konkreten Beispielen behandelt. Zu jedem Kapitel finden sich Übungsaufgaben, mit deren Hilfe der Stoff eingeübt und vertieft werden kann. Dieses Buch richtet sich an Studierende im Bachelor oder im ersten Master-Jahr sowohl in der (Wirtschafts-)Mathematik als auch in den Studiengängen Informatik, Physik und Ingenieurwissenschaften.
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Finite Difference Methods

Author: Gordon D. Smith

Publisher: Oxford University Press

ISBN: 9780198596509

Category: Mathematics

Page: 337

View: 5737

Substantially revised, this authoritative study covers the standard finite difference methods of parabolic, hyperbolic, and elliptic equations, and includes the concomitant theoretical work on consistency, stability, and convergence. The new edition includes revised and greatly expanded sections on stability based on the Lax-Richtmeyer definition, the application of Pade approximants to systems of ordinary differential equations for parabolic and hyperbolic equations, and a considerably improved presentation of iterative methods. A fast-paced introduction to numerical methods, this will be a useful volume for students of mathematics and engineering, and for postgraduates and professionals who need a clear, concise grounding in this discipline.
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Author: Stig Larsson,Vidar Thomee

Publisher: Springer Science & Business Media

ISBN: 3540887059

Category: Mathematics

Page: 262

View: 2651

The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.
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Author: Peter Knabner,Lutz Angerman

Publisher: Springer Science & Business Media

ISBN: 0387217622

Category: Mathematics

Page: 426

View: 8473

This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.
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An Introduction

Author: Vitoriano Ruas

Publisher: John Wiley & Sons

ISBN: 1119111358

Category: Technology & Engineering

Page: 300

View: 8252

Numerical Methods for Partial Differential Equations: An Introduction Vitoriano Ruas, Sorbonne Universités, UPMC - Université Paris 6, France A comprehensive overview of techniques for the computational solution of PDE's Numerical Methods for Partial Differential Equations: An Introduction covers the three most popular methods for solving partial differential equations: the finite difference method, the finite element method and the finite volume method. The book combines clear descriptions of the three methods, their reliability, and practical implementation aspects. Justifications for why numerical methods for the main classes of PDE's work or not, or how well they work, are supplied and exemplified. Aimed primarily at students of Engineering, Mathematics, Computer Science, Physics and Chemistry among others this book offers a substantial insight into the principles numerical methods in this class of problems are based upon. The book can also be used as a reference for research work on numerical methods for PDE’s. Key features: • A balanced emphasis is given to both practical considerations and a rigorous mathematical treatment. • The reliability analyses for the three methods are carried out in a unified framework and in a structured and visible manner, for the basic types of PDE's. • Special attention is given to low order methods, as practitioner's overwhelming default options for everyday use. • New techniques are employed to derive known results, thereby simplifying their proof. • Supplementary material is available from a companion website.
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Author: Christian Großmann,Hans-Görg Roos

Publisher: Springer-Verlag

ISBN: 9783519220893

Category: Mathematics

Page: 572

View: 4063

Mathematiker, Naturwissenschaftler und Ingenieure erhalten mit diesem Lehrbuch eine Einführung in die numerische Behandlung partieller Differentialgleichungen. Diskutiert werden die grundlegenden Verfahren - Finite Differenzen, Finite Volumen und Finite Elemente - für die wesentlichen Typen partieller Differentialgleichungen: elliptische, parabolische und hyperbolische Gleichungen. Einbezogen werden auch moderne Methoden zur Lösung der diskreten Probleme. Hinweise auf aktuelle Software sowie zahlreiche Beispiele und Übungsaufgaben runden diese Einführung ab.
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Author: Martin J. Gander,Felix Kwok

Publisher: SIAM

ISBN: 161197531X

Category: Science

Page: 153

View: 8099

This book provides an elementary yet comprehensive introduction to the numerical solution of partial differential equations (PDEs). Used to model important phenomena, such as the heating of apartments and the behavior of electromagnetic waves, these equations have applications in engineering and the life sciences, and most can only be solved approximately using computers. Numerical Analysis of Partial Differential Equations Using Maple and MATLAB provides detailed descriptions of the four major classes of discretization methods for PDEs (finite difference method, finite volume method, spectral method, and finite element method) and runnable MATLAB® code for each of the discretization methods and exercises. It also gives self-contained convergence proofs for each method using the tools and techniques required for the general convergence analysis but adapted to the simplest setting to keep the presentation clear and complete. This book is intended for advanced undergraduate and early graduate students in numerical analysis and scientific computing and researchers in related fields. It is appropriate for a course on numerical methods for partial differential equations.
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Author: J.W. Thomas

Publisher: Springer Science & Business Media

ISBN: 1489972781

Category: Mathematics

Page: 437

View: 7205

What makes this book stand out from the competition is that it is more computational. Once done with both volumes, readers will have the tools to attack a wider variety of problems than those worked out in the competitors' books. The author stresses the use of technology throughout the text, allowing students to utilize it as much as possible.
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Author: P. L. Roe,J. J. Chattot

Publisher: World Scientific

ISBN: 9810248105

Category: Mathematics

Page: 382

View: 9252

This book consists of 20 review articles dedicated to Prof. Philip Roe on the occasion of his 60th birthday and in appreciation of his original contributions to computational fluid dynamics. The articles, written by leading researchers in the field, cover many topics, including theory and applications, algorithm developments and modern computational techniques for industry.
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Author: William F. Ames

Publisher: Academic Press

ISBN: 0080571301

Category: Mathematics

Page: 451

View: 7975

This volume is designed as an introduction to the concepts of modern numerical analysis as they apply to partial differential equations. The book contains many practical problems and their solutions, but at the same time, strives to expose the pitfalls--such as overstability, consistency requirements, and the danger of extrapolation to nonlinear problems methods used on linear problems. Numerical Methods for Partial Differential Equations, Third Edition reflects the great accomplishments that have taken place in scientific computation in the fifteen years since the Second Edition was published. This new edition is a drastic revision of the previous one, with new material on boundary elements, spectral methods, the methods of lines, and invariant methods. At the same time, the new edition retains the self-contained nature of the older version, and shares the clarity of its exposition and the integrity of its presentation. Material on finite elements and finite differences have been merged, and now constitute equal partners Additional material has been added on boundary elements, spectral methods, the method of lines, and invariant methods References have been updated, and reflect the additional material Self-contained nature of the Second Edition has been maintained Very suitable for PDE courses
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Author: Uri M. Ascher

Publisher: SIAM

ISBN: 0898718910

Category: Evolution equations

Page: 395

View: 8053

Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. Audience: suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method.
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Problem Solving Using Mathematica

Author: Victor Grigor'e Ganzha,Evgenii Vasilev Vorozhtsov

Publisher: CRC Press

ISBN: 1351427504

Category: Mathematics

Page: 347

View: 4424

Partial differential equations (PDEs) play an important role in the natural sciences and technology, because they describe the way systems (natural and other) behave. The inherent suitability of PDEs to characterizing the nature, motion, and evolution of systems, has led to their wide-ranging use in numerical models that are developed in order to analyze systems that are not otherwise easily studied. Numerical Solutions for Partial Differential Equations contains all the details necessary for the reader to understand the principles and applications of advanced numerical methods for solving PDEs. In addition, it shows how the modern computer system algebra Mathematica® can be used for the analytic investigation of such numerical properties as stability, approximation, and dispersion.
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With Applications to Geophysics

Author: Dale R. Durran

Publisher: Springer Science & Business Media

ISBN: 9781441964120

Category: Mathematics

Page: 516

View: 4122

This scholarly text provides an introduction to the numerical methods used to model partial differential equations, with focus on atmospheric and oceanic flows. The book covers both the essentials of building a numerical model and the more sophisticated techniques that are now available. Finite difference methods, spectral methods, finite element method, flux-corrected methods and TVC schemes are all discussed. Throughout, the author keeps to a middle ground between the theorem-proof formalism of a mathematical text and the highly empirical approach found in some engineering publications. The book establishes a concrete link between theory and practice using an extensive range of test problems to illustrate the theoretically derived properties of various methods. From the reviews: "...the books unquestionable advantage is the clarity and simplicity in presenting virtually all basic ideas and methods of numerical analysis currently actively used in geophysical fluid dynamics." Physics of Atmosphere and Ocean
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Author: Joe D. Hoffman,Steven Frankel

Publisher: CRC Press

ISBN: 9780824704438

Category: Mathematics

Page: 840

View: 7482

Emphasizing the finite difference approach for solving differential equations, the second edition of Numerical Methods for Engineers and Scientists presents a methodology for systematically constructing individual computer programs. Providing easy access to accurate solutions to complex scientific and engineering problems, each chapter begins with objectives, a discussion of a representative application, and an outline of special features, summing up with a list of tasks students should be able to complete after reading the chapter- perfect for use as a study guide or for review. The AIAA Journal calls the book "...a good, solid instructional text on the basic tools of numerical analysis."
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Eine anwendungsorientierte Einführung

Author: Hans-Joachim Bungartz,Stefan Zimmer,Martin Buchholz,Dirk Pflüger

Publisher: Springer-Verlag

ISBN: 3642376568

Category: Computers

Page: 400

View: 2759

Dieses Buch gibt eine Einführung in die mathematische und informatische Modellierung sowie in die Simulation als universelle Methodik. Und so geht es um Klassen von Modellen, um deren Herleitung und um die Vielfalt an Beschreibungsarten, die eingesetzt werden können – diskret oder kontinuierlich, deterministisch oder stochastisch. Aber immer geht es auch darum, wie aus unterschiedlichen abstrakten Modellen ganz konkrete Simulationsergebnisse gewonnen werden können. Nach einem kompakten Repetitorium zum benötigten mathematischen Apparat wird das Konzept „Über das Modell zur Simulation" anhand von 14 Szenarien aus den Bereichen „Spielen – entscheiden – planen", „Verkehr auf Highways und Datenhighways", „Dynamische Systeme" sowie „Physik im Rechner" umgesetzt. Ob Spieltheorie oder Finanzmathematik, Verkehr oder Regelung, ob Populationsdynamik oder Chaos, Molekulardynamik, Kontinuumsmechanik oder Computergraphik – der Leser erhält auf anschauliche und doch systematische Weise Einblicke in die Welt der Modelle und Simulationen.
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Analytical and Numerical Methods, Second Edition

Author: Mark S. Gockenbach

Publisher: SIAM

ISBN: 0898719356

Category: Mathematics

Page: 654

View: 3998

A fresh, forward-looking undergraduate textbook that treats the finite element method and classical Fourier series method with equal emphasis.
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Author: Arieh Iserles

Publisher: Cambridge University Press

ISBN: 9780521556552

Category: Mathematics

Page: 378

View: 6599

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 point of departure is mathematical but the exposition strives to maintain a balance between theoretical, algorithmic and applied aspects of the subject. In detail, topics covered include numerical solution of ordinary differential equations by multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; a variety of algorithms to solve large, sparse algebraic systems; methods for parabolic and hyperbolic differential equations and techniques of their analysis. The book is accompanied by an appendix that presents brief back-up in a number of mathematical topics. Dr Iserles concentrates on fundamentals: deriving methods from first principles, analysing them with a variety of mathematical techniques and occasionally discussing questions of implementation and applications. By doing so, he is able to lead the reader to theoretical understanding of the subject without neglecting its practical aspects. The outcome is a textbook that is mathematically honest and rigorous and provides its target audience with a wide range of skills in both ordinary and partial differential equations.
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