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Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics)

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This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

The book is organized into two main sections and a set of appendices. Part I addresses steady-state boundary value problems, starting with two-point boundary value problems in one dimension, followed by coverage of elliptic problems in two and three dimensions. It concludes with a chapter on iterative methods for large sparse linear systems that emphasizes systems arising from difference approximations. Part II addresses time-dependent problems, starting with the initial value problem for ODEs, moving on to initial boundary value problems for parabolic and hyperbolic PDEs, and concluding with a chapter on mixed equations combining features of ODEs, parabolic equations, and hyperbolic equations. The appendices cover concepts pertinent to Parts I and II. Exercises and student projects, developed in conjunction with this book, are available on the book s webpage along with numerous MATLAB m-files.

Readers will gain an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods as well as the key concepts of stability theory, their relation to one another, and their practical implications. The author provides a foundation from which students can approach more advanced topics and further explore the theory and/or use of finite difference methods according to their interests and needs.

Audience: This book is designed as an introductory graduate-level textbook on finite difference methods and their analysis. It is also appropriate for researchers who desire an introduction to the use of these methods.

Contents: Preface; Part I: Boundary Value Problems and Iterative Methods. Chapter 1: Finite Difference Approximations; Chapter 2: Steady States and Boundary Value Problems; Chapter 3: Elliptic Equations; Chapter 4: Iterative Methods for Sparse Linear Systems; Part II: Initial Value Problems. Chapter 5: The Initial Value Problem for Ordinary Differential Equations; Chapter 6: Zero-Stability and Convergence for Initial Value Problems; Chapter 7: Absolute Stability for Ordinary Differential Equations; Chapter 8: Stiff Ordinary Differential Equations; Chapter 9: Diffusion Equations and Parabolic Problems; Chapter 10: Advection Equations and Hyperbolic Systems; Chapter 11: Mixed Equations; Appendix A: Measuring Errors; Appendix B: Polynomial Interpolation and Orthogonal Polynomials; Appendix C: Eigenvalues and Inner-Product Norms; Appendix D: Matrix Powers and Exponentials; Appendix E: Partial Differential Equations; Bibliography; Index.

Author(s): Randall Leveque
Series: Classics in Applied Mathematics
Publisher: SIAM, Society for Industrial and Applied Mathematics
Year: 2007

Language: English
Pages: 356

Cover
Title Page
Copyright Page
Dedication
Table of Contents
Preface
I Boundary Value Problems and Iterative Methods
1 Finite Difference Approximations
1.1 Truncation errors
1.2 Deriving finite difference approximations
1.3 Second order derivatives
1.4 Higher order derivatives
1.5 A general approach to deriving the coefficients
2 Steady States and Boundary Value Problems
2.1 The heat equation
2.2 Boundary conditions
2.3 The steady-state problem
2.4 A simple finite difference method
2.5 Local truncation error
2.6 Global error
2.7 Stability
2.8 Consistency
2.9 Convergence
2.10 Stability in the 2-norm
2.11 Green’s functions and max-norm stability
2.12 Neumann boundary conditions
2.13 Existence and uniqueness
2.14 Ordering the unknowns and equations
2.15 A general linear second order equation
2.16 Nonlinear equations
2.16.1 Discretization of the nonlinear boundary value problem
2.16.2 Nonuniqueness
2.16.3 Accuracy on nonlinear equations
2.17 Singular perturbations and boundary layers
2.17.1 Interior layers
2.18 Nonuniform grids
2.18.1 Adaptive mesh selection
2.19 Continuation methods
2.20 Higher order methods
2.20.1 Fourth order differencing
2.20.2 Extrapolation methods
2.20.3 Deferred corrections
2.21 Spectral methods
3 Elliptic Equations
3.1 Steady-state heat conduction
3.2 The 5-point stencil for the Laplacian
3.3 Ordering the unknowns and equations
3.4 Accuracy and stability
3.5 The 9-point Laplacian
3.6 Other elliptic equations
3.7 Solving the linear system
3.7.1 Sparse storage in MATLAB
4 Iterative Methods for Sparse Linear Systems
4.1 Jacobi and Gauss–Seidel
4.2 Analysis of matrix splitting methods
4.2.1 Rate of convergence
4.2.2 Successive overrelaxation
4.3 Descent methods and conjugate gradients
4.3.1 The method of steepest descent
4.3.2 The A-conjugate search direction
4.3.3 The conjugate-gradient algorithm
4.3.4 Convergence of conjugate gradient
4.3.5 Preconditioners
4.3.6 Incomplete Cholesky and ILU preconditioners
4.4 The Arnoldi process and GMRES algorithm
4.4.1 Krylov methods based on three term recurrences
4.4.2 Other applications of Arnoldi
4.5 Newton–Krylov methods for nonlinear problems
4.6 Multigrid methods
4.6.1 Slow convergence of Jacobi
4.6.2 The multigrid approach
II Initial Value Problems
5 The Initial Value Problem for Ordinary Differential Equations
5.1 Linear ordinary differential equations
5.1.1 Duhamel’s principle
5.2 Lipschitz continuity
5.2.1 Existence and uniqueness of solutions
5.2.2 Systems of equations
5.2.3 Significance of the Lipschitz constant
5.2.4 Limitations
5.3 Some basic numerical methods
5.4 Truncation errors
5.5 One-step errors
5.6 Taylor series methods
5.7 Runge–Kutta methods
5.7.1 Embedded methods and error estimation
5.8 One-step versus multistep methods
5.9 Linear multistep methods
5.9.1 Local truncation error
5.9.2 Characteristic polynomials
5.9.3 Starting values
5.9.4 Predictor-corrector methods
6 Zero-Stability and Convergence for Initial Value Problems
6.1 Convergence
6.2 The test problem
6.3 One-step methods
6.3.1 Euler’s method on linear problems
6.3.2 Relation to stability for boundary value problems
6.3.3 Euler’s method on nonlinear problems
6.3.4 General one-step methods
6.4 Zero-stability of linear multistep methods
6.4.1 Solving linear difference equations
7 Absolute Stability for Ordinary Differential Equations
7.1 Unstable computations with a zero-stable method
7.2 Absolute stability
7.3 Stability regions for linear multistep methods
7.4 Systems of ordinary differential equations
7.4.1 Chemical kinetics
7.4.2 Linear systems
7.4.3 Nonlinear systems
7.5 Practical choice of step size
7.6 Plotting stability regions
7.6.1 The boundary locus method for linear multistep methods
7.6.2 Plotting stability regions of one-step methods
7.7 Relative stability regions and order stars
8 Stiff Ordinary Differential Equations
8.1 Numerical difficulties
8.2 Characterizations of stiffness
8.3 Numerical methods for stiff problems
8.3.1 A-stability and A(˛)-stability
8.3.2 L-stability
8.4 BDF methods
8.5 The TR-BDF2 method
8.6 Runge–Kutta–Chebyshev explicit methods
9 Diffusion Equations and Parabolic Problems
9.1 Local truncation errors and order of accuracy
9.2 Method of lines discretizations
9.3 Stability theory
9.4 Stiffness of the heat equation
9.5 Convergence
9.5.1 PDE versus ODE stability theory
9.6 Von Neumann analysis
9.7 Multidimensional problems
9.8 The locally one-dimensional method
9.8.1 Boundary conditions
9.8.2 The alternating direction implicit method
9.9 Other discretizations
10 Advection Equations and Hyperbolic Systems
10.1 Advection
10.2 Method of lines discretization
10.2.1 Forward Euler time discretization
10.2.2 Leapfrog
10.2.3 Lax–Friedrichs
10.3 The Lax–Wendroff method
10.3.1 Stability analysis
10.4 Upwind methods
10.4.1 Stability analysis
10.4.2 The Beam–Warming method
10.5 Von Neumann analysis
10.6 Characteristic tracing and interpolation
10.7 The Courant–Friedrichs–Lewy condition
10.8 Some numerical results
10.9 Modified equations
10.10 Hyperbolic systems
10.10.1 Characteristic variables
10.11 Numerical methods for hyperbolic systems
10.12 Initial boundary value problems
10.12.1 Analysis of upwind on the initial boundary value problem
10.12.2 Outflow boundary conditions
10.13 Other discretizations
11 Mixed Equations
11.1 Some examples
11.2 Fully coupled method of lines
11.3 Fully coupled Taylor series methods
11.4 Fractional step methods
11.5 Implicit-explicit methods
11.6 Exponential time differencing methods
11.6.1 Implementing exponential time differencing methods
III Appendices
A Measuring Errors
A.1 Errors in a scalar value
A.1.1 Absolute error
A.1.2 Relative error
A.2 “Big-oh” and “little-oh” notation
A.3 Errors in vectors
A.3.1 Norm equivalence
A.3.2 Matrix norms
A.4 Errors in functions
A.5 Errors in grid functions
A.5.1 Norm equivalence
A.6 Estimating errors in numerical solutions
A.6.1 Estimates from the true solution
A.6.2 Estimates from a fine-grid solution
A.6.3 Estimates from coarser solutions
B Polynomial Interpolation and Orthogonal Polynomials
B.1 The general interpolation problem
B.2 Polynomial interpolation
B.2.1 Monomial basis
B.2.2 Lagrange basis
B.2.3 Newton form
B.2.4 Error in polynomial interpolation
B.3 Orthogonal polynomials
B.3.1 Legendre polynomials
B.3.2 Chebyshev polynomials
C Eigenvalues and Inner-Product Norms
C.1 Similarity transformations
C.2 Diagonalizable matrices
C.3 The Jordan canonical form
C.4 Symmetric and Hermitian matrices
C.5 Skew-symmetric and skew-Hermitian matrices
C.6 Normal matrices
C.7 Toeplitz and circulant matrices
C.8 The Gershgorin theorem
C.9 Inner-product norms
C.10 Other inner-product norms
D Matrix Powers and Exponentials
D.1 The resolvent
D.2 Powers of matrices
D.2.1 Solving linear difference equations
D.2.2 Resolvent estimates
D.3 Matrix exponentials
D.3.1 Solving linear differential equations
D.4 Nonnormal matrices
D.4.1 Matrix powers
D.4.2 Matrix exponentials
D.5 Pseudospectra
D.5.1 Nonnormality of a Jordan block
D.6 Stable families of matrices and the Kreiss matrix theorem
D.7 Variable coefficient problems
E Partial Differential Equations
E.1 Classification of differential equations
E.1.1 Second order equations
E.1.2 Elliptic equations
E.1.3 Parabolic equations
E.1.4 Hyperbolic equations
E.2 Derivation of partial differential equations from conservation principles
E.2.1 Advection
E.2.2 Diffusion
E.2.3 Source terms
E.2.4 Reaction-diffusion equations
E.3 Fourier analysis of linear partial differential equations
E.3.1 Fourier transforms
E.3.2 The advection equation
E.3.3 The heat equation
E.3.4 The backward heat equation
E.3.5 More general parabolic equations
E.3.6 Dispersive waves
E.3.7 Even- versus odd-order derivatives
E.3.8 The Schrödinger equation
E.3.9 The dispersion relation
E.3.10 Wave packets
Bibliography
Index