Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations

2nd Edition - August 28, 1977

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  • Author: William F. Ames
  • eBook ISBN: 9781483262420

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Numerical Methods for Partial Differential Equations, Second Edition deals with the use of numerical methods to solve partial differential equations. In addition to numerical fluid mechanics, hopscotch and other explicit-implicit methods are also considered, along with Monte Carlo techniques, lines, fast Fourier transform, and fractional steps methods. Comprised of six chapters, this volume begins with an introduction to numerical calculation, paying particular attention to the classification of equations and physical problems, asymptotics, discrete methods, and dimensionless forms. Subsequent chapters focus on parabolic and hyperbolic equations, elliptic equations, and special topics ranging from singularities and shocks to Navier-Stokes equations and Monte Carlo methods. The final chapter discuss the general concepts of weighted residuals, with emphasis on orthogonal collocation and the Bubnov-Galerkin method. The latter procedure is used to introduce finite elements. This book should be a valuable resource for students and practitioners in the fields of computer science and applied mathematics.

Table of Contents

  • Preface to Second Edition

    Preface to First Edition

    1 Fundamentals

    1-0 Introduction

    1-1 Classification of Physical Problems

    1-2 Classification of Equations

    1-3 Asymptotics

    1-4 Discrete Methods

    1-5 Finite Differences and Computational Molecules

    1-6 Finite Difference Operators

    1-7 Errors

    1-8 Stability and Convergence

    1-9 Irregular Boundaries

    1-10 Choice of Discrete Network

    1-11 Dimensionless Forms


    2 Parabolic Equations

    2-0 Introduction

    2-1 Simple Explicit Methods

    2-2 Fourier Stability Method

    2-3 Implicit Methods

    2-4 An Unconditionally Unstable Difference Equation

    2-5 Matrix Stability Analysis

    2-6 Extension of Matrix Stability Analysis

    2-7 Consistency, Stability, and Convergence

    2-8 Pure Initial Value Problems

    2-9 Variable Coefficients

    2-10 Examples of Equations with Variable Coefficients

    2-11 General Concepts of Error Reduction

    2-12 Explicit Methods for Nonlinear Problems

    2-13 An Application of the Explicit Method

    2-14 Implicit Methods for Nonlinear Problems

    2-15 Concluding Remarks


    3 Elliptic Equations

    3-0 Introduction

    3-1 Simple Finite Difference Schemes

    3-2 Iterative Methods

    3-3 Linear Elliptic Equations

    3-4 Some Point Iterative Methods

    3-5 Convergence of Point Iterative Methods

    3-6 Rates of Convergence

    3-7 Accelerations—Successive Over-Relaxation (SOR)

    3-8 Extensions of SOR

    3-9 Qualitative Examples of Over-Relaxation

    3-10 Other Point Iterative Methods

    3-11 Block Iterative Methods

    3-12 Alternating Direction Methods

    3-13 Summary of ADI Results

    3-14 Some Nonlinear Examples


    4 Hyperbolic Equations

    4-0 Introduction

    4-1 The Quasilinear System

    4-2 Introductory Examples

    4-3 Method of Characteristics

    4-4 Constant States and Simple Waves

    4-5 Typical Application of Characteristics

    4-6 Explicit Finite Difference Methods

    4-7 Overstability

    4-8 Implicit Methods for Second-Order Equations

    4-9 Nonlinear Examples

    4-10 Simultaneous First-Order Equations—Explicit Methods

    4-11 An Implicit Method for First-Order Equations

    4-12 Hybrid Methods for First-Order Equations

    4-13 Gas Dynamics in One-Space Variable

    4-14 Eulerian Difference Equations

    4-15 Lagrangian Difference Equations

    4-16 Hopscotch Methods for Conservation Laws

    4-17 Explicit-Implicit Schemes for Conservation Laws


    5 Special Topics

    5-0 Introduction

    5-1 Singularities

    5-2 Shocks

    5-3 Eigenvalue Problems

    5-4 Parabolic Equations in Several Space Variables

    5-5 Additional Comments on Elliptic Equations

    5-6 Hyperbolic Equations in Higher Dimensions

    5-7 Mixed Systems

    5-8 Higher-Order Equations in Elasticity and Vibrations

    5-9 Fluid Mechanics: The Navier-Stokes Equations

    5-10 Introduction to Monte Carlo Methods

    5-11 Method of Lines

    5-12 Fast Fourier Transform and Applications

    5-13 Method of Fractional Steps


    6 Weighted Residuals and Finite Elements

    6-0 Introduction

    6-1 Weighted Residual Methods (WRM)

    6-2 Orthogonal Collocation

    6-3 Bubnov-Galerkin (B-G) Method

    6-4 Remarks on Completeness, Convergence, and Errors Bounds

    6-5 Nagumo's Lemma and Application

    6-6 Introduction to Finite Elements


    Author Index

    Subject Index

Product details

  • No. of pages: 380
  • Language: English
  • Copyright: © Academic Press 1977
  • Published: August 28, 1977
  • Imprint: Academic Press
  • eBook ISBN: 9781483262420

About the Author

William F. Ames

About the Editors

Werner Rheinboldt

Alan Jeffrey

Affiliations and Expertise

University at Newcastle Upon Tyne, UK

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