Finite difference schemes and partial differential equations /

Finite difference schemes and partial differential equations /

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Bibliographic Details
Author / Creator:Strikwerda, John C., 1947-
Imprint:Philadelphia : Society for Industrial and Applied Mathematics, ©2004.
Description:1 online resource (xii, 435 pages) : illustrations
Language:English
Subject:
Differential equations, Partial -- Numerical solutions.
Finite differences.
Differential equations, Partial -- Numerical solutions.
Finite differences.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/12577219
Hidden Bibliographic Details
ISBN:0898715679
9780898715675
Table of Contents:
  • Preface to the Second Edition
  • Preface to the First Edition
  • 1. Hyperbolic Partial Differential Equations
  • 1.1. Overview of Hyperbolic Partial Differential Equations
  • 1.2. Boundary Conditions
  • 1.3. Introduction to Finite Difference Schemes
  • 1.4. Convergence and Consistency
  • 1.5. Stability
  • 1.6. The Courant-Friedrichs-Lewy Condition
  • 2. Analysis of Finite Difference Schemes
  • 2.1. Fourier Analysis
  • 2.2. Von Neumann Analysis
  • 2.3. Comments on Instability and Stability
  • 3. Order of Accuracy of Finite Difference Schemes
  • 3.1. Order of Accuracy
  • 3.2. Stability of the Lax-Wendroff and Crank-Nicolson Schemes
  • 3.3. Difference Notation and the Difference Calculus
  • 3.4. Boundary Conditions for Finite Difference Schemes
  • 3.5. Solving Tridiagonal Systems
  • 4. Stability for Multistep Schemes
  • 4.1. Stability for the Leapfrog Scheme
  • 4.2. Stability for General Multistep Schemes
  • 4.3. The Theory of Schur and von Neumann Polynomials
  • 4.4. The Algorithm for Schur and von Neumann Polynomials
  • 5. Dissipation and Dispersion
  • 5.1. Dissipation
  • 5.2. Dispersion
  • 5.3. Group Velocity and the Propagation of Wave Packets
  • 6. Parabolic Partial Differential Equations
  • 6.1. Overview of Parabolic Partial Differential Equations
  • 6.2. Parabolic Systems and Boundary Conditions
  • 6.3. Finite Difference Schemes for Parabolic Equations
  • 6.4. The Convection-Diffusion Equation
  • 6.5. Variable Coefficients
  • 7. Systems of Partial Differential Equations in Higher Dimensions
  • 7.1. Stability of Finite Difference Schemes for Systems of Equations
  • 7.2. Finite Difference Schemes in Two and Three Dimensions
  • 7.3. The Alternating Direction Implicit Method
  • 8. Second-Order Equations
  • 8.1. Second-Order Time-Dependent Equations
  • 8.2. Finite Difference Schemes for Second-Order Equations
  • 8.3. Boundary Conditions for Second-Order Equations
  • 8.4. Second-Order Equations in Two and Three Dimensions
  • 9. Analysis of Well-Posed and Stable Problems
  • 9.1. The Theory of Well-Posed Initial Value Problems
  • 9.2. Well-Posed Systems of Equations
  • 9.3. Estimates for Inhomogeneous Problems
  • 9.4. The Kreiss Matrix Theorem
  • 10. Convergence Estimates for Initial Value Problems
  • 10.1. Convergence Estimates for Smooth Initial Functions
  • 10.2. Related Topics
  • 10.3. Convergence Estimates for Nonsmooth Initial Functions
  • 10.4. Convergence Estimates for Parabolic Differential Equations
  • 10.5. The Lax-Richtmyer Equivalence Theorem
  • 10.6. Analysis of Multistep Schemes
  • 10.7. Convergence Estimates for Second-Order Differential Equations
  • 11. Well-Posed and Stable Initial-Boundary Value Problems
  • 11.1. Preliminaries
  • 11.2. Analysis of Boundary Conditions for the Leapfrog Scheme
  • 11.3. The General Analysis of Boundary Conditions
  • 11.4. Initial-Boundary Value Problems for Partial Differential Equations
  • 11.5. The Matrix Method for Analyzing Stability
  • 12. Elliptic Partial Differential Equations and Difference Schemes
  • 12.1. Overview of Elliptic Partial Differential Equations
  • 12.2. Regularity Estimates for Elliptic Equations
  • 12.3. Maximum Principles
  • 12.4. Boundary Conditions for Elliptic Equations
  • 12.5. Finite Difference Schemes for Poisson's Equation
  • 12.6. Polar Coordinates
  • 12.7. Coordinate Changes and Finite Differences
  • 13. Linear Iterative Methods
  • 13.1. Solving Finite Difference Schemes for Laplace's Equation in a Rectangle
  • 13.2. Eigenvalues of the Discrete Laplacian
  • 13.3. Analysis of the Jacobi and Gauss-Seidel Methods
  • 13.4. Convergence Analysis of Point SOR
  • 13.5. Consistently Ordered Matrices
  • 13.6. Linear Iterative Methods for Symmetric, Positive Definite Matrices
  • 13.7. The Neumann Boundary Value Problem
  • 14. The Method of Steepest Descent and the Conjugate Gradient Method
  • 14.1. The Method of Steepest Descent
  • 14.2. The Conjugate Gradient Method
  • 14.3. Implementing the Conjugate Gradient Method
  • 14.4. A Convergence Estimate for the Conjugate Gradient Method
  • 14.5. The Preconditioned Conjugate Gradient Method
  • A. Matrix and Vector Analysis
  • A.1. Vector and Matrix Norms
  • A.2. Analytic Functions of Matrices
  • B. A Survey of Real Analysis
  • B.1. Topological Concepts
  • B.2. Measure Theory
  • B.3. Measurable Functions
  • B.4. Lebesgue Integration
  • B.5. Function Spaces
  • C. A Survey of Results from Complex Analysis
  • C.1. Basic Definitions
  • C.2. Complex Integration
  • C.3. A Phragmen-Lindelof Theorem
  • C.4. A Result for Parabolic Systems
  • References
  • Index

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