Numerical Methods For Partial Differential Equations

✍ Scribed by William F. Ames

• Publisher: Academic Press
• Year: 1977
• Tongue: English
• Leaves: 380
• Series: Computer Science and Applied Mathmatics
• Edition: 2
• Category: Library

✦ Synopsis

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.Key Features 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

✦ Table of Contents

Cover
Title Page
Copyright Page
Dedication
Contents
Preface to second edition
Preface to first edition
Chapter 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
References
Chapter 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
(a) Diffusion in circular cylindrical coordinates
(b) Equations with reducible error
(c) Diffusion with spherical symmetry
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
(a) Application of the backward difference
(b) Crank–Nicolson forms
(c) Predictor–corrector methods
2-15 Concluding remarks
References
Chapter 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
(a) Jacobi method
(b) Gauss–Seidel method
(c) Successive over-relaxation (SOR)
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
(a) Gradient method
(b) Richardson's method
(c) Semi-iterative methods
3-11 Block iterative methods
3-12 Alternating direction methods
3-13 Summary of ADI results
3-14 Some nonlinear examples
(a) Mildly nonlinear elliptic equations

(c) Laminar flow of non-Newtonian fluids
References
Chapter 4. Hyperbolic equations
4-0 Introduction
4-1 The quasilinear system
4-2 Introductory examples
(a) Direct calculation of the primitive variables
(b) Stress wave propagation
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 Overstabiiity
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
(a) Method of Courant et al. [28]
(b) Lelevier's scheme
(c) Conservation form and Lax–Wendroff schemes
4-15 Lagrangian difference equations
4-16 Hopscotch methods for conservation laws
4-17 Explicit-implicit schemes for conservation laws
References
Chapter 5. Special topics
5-0 Introduction
5-1 Singularities
(a) Subtracting out the singularity
(b) Mesh refinement
(c) The method of Motz and Woods
(d) Removal of singularity
5-2 Shocks
(a) Pseudoviscosity
(b) Lax–Wendroff method
5-3 Eigenvalue problems
(a) The membrane eigenvalue problemt [Eqn (5-45)]
(b) Some methods of computation
(c) A nonlinear eigenvalue problem (Motz [48])
5-4 Parabolic equations in several space variables
(a) Generalizations of the elementary methods
(b) Alternating direction methods
5-5 Additional comments on elliptic equations
(a) Nonlinear over-relaxation
(b) Some alternative procedures
(c) Some questions associated with numerical weather prediction
5-6 Hyperbolic equations in higher dimensions
(a) Characteristics in three independent variables
(b) Finite difference methods
(c) Singularities in solutions of nonlinear hyperbolic equations
5-7 Mixed systems
5-8 Higher-order equations in elasticity and vibrations
(a) Elasticity
(b) Vibrations of a thin beam
5-9 Fluid mechanics: the Navier–Stokes equations
(a) Stream function—vorticity method
(b) Primitive variable methods
(c) Vector potential methods
(d) General comments and literature
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
References
Chapter 6. Weighted residuals and finite elements
6-0 introduction
6-1 Weighted residual methods (WRM)
(a) Subdomain (Biezeno and Koch [15, 1923])
(b) Collocation (Frazer et al. [16])
(c) Least squares (Gauss–Legendre; cf. Hall [18], Sorenson [19])
(d) Bubnov–Galerkin method (Bubnov [21], Galerkin [22])
(e) Moments (Yamada [23])
(f) General WRM
(g) Stationary functional method (Rayleigh [29], Ritz [30])
6-2 Orthogonal collocation
6-3 Bubnov–Galerkin (B-G) method
6-4 Remarks on completeness, convergence, and error bounds
6-5 Nagumo's lemma and application
6-6 Introduction to finite elements
References
Author Index
Subject Index
Computer Science and Applied Mathematics: A Series of Monographs and Textbooks