Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Authors
PART Ⅰ Fundamentals
Chapter 1 Introduction
1.1 General Remarks
1.2 Comparison of Experimental, Theoretical, and Computational Approaches
1.3 Historical Perspective
Chapter 2 Partial Differential Equations
2.1 Introduction
2.1.1 Partial Differential Equations
2.2 Physical Classification
2.2.1 Equilibrium Problems
2.2.2 Eigenvalue Problems
2.2.3 Marching Problems
2.3 Mathematical Classification
2.3.1 Hyperbolic PDEs
2.3.2 Parabolic PDEs
2.3.3 Elliptic PDEs
2.4 Well-Posed Problem
2.5 Systems of Partial Differential Equations
2.6 Other PDEs of Interest
Problems
Chapter 3 Basics of Discretization Methods
3.1 Introduction
3.2 Finite Differences
3.3 Difference Representation of Partial Differential Equations
3.3.1 Truncation Error
3.3.2 Round-Off and Discretization Errors
3.3.3 Consistency
3.3.4 Stability
3.3.5 Convergence for Marching Problems
3.3.6 Comment on Equilibrium Problems
3.3.7 Conservation Form and Conservative Property
3.4 Further Examples of Methods for Obtaining Finite-Difference Equations
3.4.1 Use of Taylor Series
3.4.2 Use of Polynomial Fitting
3.4.3 Integral Method
3.5 Finite-Volume Method
3.6 Introduction to the Use of Irregular Meshes
3.6.1 Irregular Mesh due to Shape of a Boundary
3.6.2 Irregular Mesh Not Caused by Shape of a Boundary
3.6.3 Concluding Remarks
3.7 Stability Considerations
3.7.1 Fourier or von Neumann Analysis
3.7.2 Stability Analysis for Systems of Equations
Problems
Chapter 4 Application of Numerical Methods to Selected Model Equations
4.1 Wave Equation
4.1.1 Euler Explicit Methods
4.1.2 Upstream (First-Order Upwind or Windward) DifferencingMethod
4.1.3 Lax Method
4.1.4 Euler Implicit Method
4.1.5 Leap Frog Method
4.1.6 Lax-Wendroff Method
4.1.7 Two-Step Lax-Wendroff Method
4.1.8 MacCormack Method
4.1.9 Second-Order Upwind Method
4.1.10 Time-Centered Implicit Method (Trapezoidal Differencing Method)
4.1.11 Rusanov (Burstein-Mirin) Method
4.1.12 Warming-Kutler-Lomax Method
4.1.13 Runge-Kutta Methods
4.1.14 Additional Comments
4.2 Heat Equation
4.2.1 Simple Explicit Method
4.2.2 Richardson's Method
4.2.3 Simple Implicit (Laasonen) Method
4.2.4 Crank-Nicolson Method
4.2.5 Combined Method A
4.2.6 Combined Method B
4.2.7 DuFort-Frankel Method
4.2.8 Keller Box and Modified Box Methods
4.2.9 Methods for the Two-Dimensional Heat Equation
4.2.10 ADI Methods
4.2.11 Splitting or Fractional-Step Methods
4.2.12 ADE Methods
4.2.13 Hopscotch Method
4.2.14 Additional Comments
4.3 Laplace's Equation
4.3.1 Finite-Difference Representations for Laplace's Equation
4.3.1.1 Five-Point Formula
……
PART Ⅱ Application of Numerical Methods to the Equations of Fluid Mechanics and Heat Transfer
Appendix A: Subroutine for Solving a Tridiagonal System of Equations
Appendix B: Subroutines for Solving Block Tridiagonal Systems of Equations
Appendix C: Modified Strongly Implicit Procedure
Nomenclature
References
Index
