Fundamentals of Computational Fluid Dynamics

Harvard Lomax and Thomas H. Pulliam
NASA Ames Research Center

David W. Zingg
University of Toronto Institute for Aerospace Studies

This book is intended for use as a textbook in a first or second year introductory course in CFD at the graduate level. It is currently being used in that form at both Stanford and U. of Toronto.

TABLE OF CONTENTS [postscript 88Kb] ....[pdf format 745Kb]

• Chapter 1: INTRODUCTION [postscript 82Kb] ....[pdf format 640Kb]
  • Motivation
  • Background
  • Notation
• Chapter 2: THE MODEL EQUATIONS [postscript 216Kb] ....[pdf format 1.2Mb]
  • Conservation Laws
  • The 1-D Euler and Navier-Stokes Equations
  • The Linear Convection Equation
    • Differential Form
    • Solution in Wave Space
  • The Diffusion Equation
    • Differential Form
    • Solution in Wave Space
  • Hyperbolic Systems
• Chapter 3: FINITE-DIFFERENCE APPROXIMATIONS [postscript 989Kb] ....[pdf format 3.5Mb]
  • Meshes and Finite-Difference Notation
  • Space Derivative Approximations
  • Finite-Difference Operators
    • Point Difference Operators
    • Matrix Difference Operators
    • Periodic Matrices
    • Circulant Matrices
  • Constructing Differencing Schemes of Any Order
    • Taylor Tables
    • Generalization of Difference Formulas
    • Lagrange and Hermite Interpolation Polynomials
    • Practical Application of Padé Formulas
    • Other Higher-Order Schemes
  • Fourier Error Analysis
    • Application to a Spatial Operator
  • Difference Operators at Boundaries
    • The Linear Convection Equation
    • The Diffusion Equation
• Chapter 4: THE SEMI-DISCRETE APPROACH [postscript 199Kb] ....[pdf format 2.0Mb]
  • Reduction of PDE's to ODE's --- Semi-Discrete Methods
    • The Model ODE's
      • Model ODE for diffusion
      • Model ODE for biconvection
    • The Generic Matrix Form
  • Eigensystems of Semi-Discrete Linear Forms
    • Complete Systems
    • Defective Systems
  • Exact Solutions of Linear Autonomous ODE's
    • Classification of ODE's
    • Single ODE's of First- and Second-Order
      • First-Order Equations
      • Second-Order Equations
    • Coupled First-Order ODE's
      • A Complete System
      • A Derogatory System
      • A Defective System
    • General Solution of Coupled ODE's with Complete Eigensystems
  • Real Space and Eigenspace
    • Definition
    • Eigenvalue Spectrums for Model ODE's
    • Eigenvectors of the Model Equations
      • The Diffusion Model
      • The Biconvection Model
  • The Isolation Theorem and the Representative Equation
• Chapter 5: FINITE-VOLUME METHODS [postscript 158Kb] ....[pdf format 1.5Mb]
  • Basic Concepts
  • Model Equations in Integral Form
    • The Linear Convection Equation
    • The Diffusion Equation
  • One-Dimensional Examples
    • A Second-Order Approximation to the Convection Equation
    • A Fourth-Order Approximation to the Convection Equation
    • A Second-Order Approximation to the Diffusion Equation
  • A Two-Dimensional Example
• Chapter 6: TIME-MARCHING METHODS FOR ODE'S [postscript 293Kb] ....[pdf format 4.0Mb]
  • Converting Time-Marching Methods to O(Delta)E 's
  • Solution of Linear O(Delta)E's With Constant Coefficients
    • First- and Second-Order Difference Equations
      • First-Order Equations
      • Second-Order Equations
    • Special Cases of Coupled First-Order Equations
      • A Complete System
      • A Defective System
  • Solution of the Representative O(Delta)E's
    • The Operational Form and its Solution
    • Examples of Solutions to Time-Marching O(Delta)E's
  • The (lambda)-(sigma) Relation
    • Establishing the Relation
    • The Principal (sigma)-Root
    • Spurious (sigma)-Roots
    • One-Root Time-Marching Methods
  • Accuracy Measures of Time-Marching Methods
    • Local and Global Error Measures
    • Local Accuracy of the Transient Solution
      • Transient error
      • Amplitude and Phase Error
    • Local Accuracy of the Particular Solution
    • Time Accuracy For Nonlinear Applications
    • Global Accuracy
      • Global error in the transient
      • Global error in amplitude and phase
      • Global error in the particular solution
  • Linear Multistep Methods
    • The General Formulation
    • Simple Examples
      • Explicit Methods
      • Implicit Methods
    • Two-Step Linear Multistep Methods
  • Predictor-Corrector Methods
  • Runge-Kutta Methods
  • Implementation of Implicit Methods
    • Application to Systems of Equations
    • Application to Nonlinear Equations
    • Local Linearization for Scalar Equations
      • General Development
      • Implementation of the Trapezoidal Method
      • Implementation of the Implicit Euler Method
      • Newton's Method
    • Local Linearization for Coupled Sets of Nonlinear Equations
• Chapter 7: STABILITY OF LINEAR SYSTEMS [postscript 849Kb] ....[pdf format 3.9Mb]
  • Dependence on the Eigensystem
  • Inherent Stability of ODE's
    • The Criterion
    • Complete Eigensystems
    • Defective Eigensystems
  • Numerical Stability of O(Delta)E 's
    • The Criterion
    • Complete Eigensystems
    • Defective Eigensystems
  • Time-Space Stability and Convergence of O(Delta)E's
  • Numerical Stability Concepts in the Complex (sigma)-Plane
    • (sigma)-Root Traces Relative to the Unit Circle
      • Locus of the exact trace
      • Examples of some methods
      • a. Explicit Euler Method
      • b. Leapfrog Method
      • c. Second-Order Adams-Bashforth Method
      • d. Trapezoidal Method
      • e. Gazdag Method
      • f,g. Second- and Fourth-Order Runge-Kutta Methods, RK2 and RK4
    • Stability for Small
      • Mild instability
      • Catastrophic instability
      • Milne and Adams type methods
  • Numerical Stability Concepts in the Complex (lambda)h Plane
    • Stability for Large h.
    • Unconditional Stability, A-Stable Methods
    • Stability Contours in the Complex (lambda)h Plane.
      • Contours for explicit methods
      • Contours for unconditionally stable implicit methods
      • Contours for conditionally stable implicit methods
  • Fourier or von Neumann Stability Analysis
    • The Basic Procedure
    • Some Examples
    • Relation to Circulant Matrices
  • Consistency
• Chapter 8: CHOICE OF TIME-MARCHING METHODS [postscript 189Kb] ....[pdf format 1.8Mb]
  • Stiffness Definition for ODE's
    • Relation to (lambda)-Eigenvalues
    • Driving and Parasitic Eigenvalues
    • Stiffness Classifications
  • Relation of Stiffness to Space Mesh Size
  • Practical Considerations for Comparing Methods
    • Events
    • Derivative Evaluations
  • Comparing the Efficiency of Explicit Methods
    • Imposed Constraints
    • An Example Involving Diffusion
    • An Example Involving Periodic Convection
  • Coping With Stiffness
    • Explicit Methods
    • Implicit Methods
    • A Perspective
• Chapter 9: RELAXATION METHODS [postscript 733Kb] ....[pdf format 2.9Mb]
  • Formulation of the Model Problem
    • Preconditioning the Basic Matrix
    • The Model Equations
  • Classical Relaxation
    • The Delta Form of an Iterative Scheme
    • The Converged Solution, the Residual, and the Error
    • The Classical Methods
      • Point Operator Schemes in One Dimension
      • The General Form
  • The ODE Approach to Classical Relaxation
    • The Ordinary Differential Equation Formulation
    • ODE Form of the Classical Methods
  • Eigensystems of the Classical Methods
    • The Point-Jacobi System
    • The Gauss-Seidel System
    • The SOR System
  • Nonstationary Processes
• Chapter 10: MULTIGRID STRATEGIES [postscript 379Kb] ....[pdf format 891Kb]
  • Motivation
    • Eigenvector and Eigenvalue Identification with Space Frequencies
    • Properties of the Iterative Process
• Chapter 11: NUMERICAL DISSIPATION [postscript 173Kb] ....[pdf format 1.6Mb]
  • One-Sided First-Derivative Space Differencing
  • The Modified Partial Differential Equation
  • The Lax-Wendroff Method
  • Upwind Schemes
    • Flux-Vector Splitting
    • Flux-Difference Splitting
  • Artificial Dissipation
  • The Upwind Connection To Artificial Dissipation
• Chapter 12: SPLIT AND FACTORED FORMS IN TIME-ACCURATE METHODS [postscript 184Kb] ....[pdf format 1.8Mb]
  • The Concept
  • Factoring Physical Representations --- Time Splitting
  • Factoring Space Matrix Operators in 2--D
    • Mesh Indexing Convention
    • Data Bases and Space Vectors
    • Data Base Permutations
    • Space Splitting and Factoring
  • Second-Order, Implicit, Split & Factored Methods
  • Importance of Factored Forms in 2 and 3 Dimensions
  • The Delta Form
  • Factored Forms Employing Flux Splitting
• Chapter 13: LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS [postscript 869Kb] ....[pdf format 2.1Mb]
  • Introduction
  • The Representative Equation for Circulant Operators
  • Example Analysis of Circulant Systems
    • Stability Comparisons of Time-Split Methods
      • 1. The Explicit-Implicit Method
      • 2. The Explicit-Explicit Method
    • Analysis of a Second-Order Time-Split Method
      • Accuracy
      • Stability
  • The Representative Equation for Space-Split 2-D Operators
  • Example Analysis of 2-D Model Equations
    • The Unfactored Implicit Euler Method
    • The Factored Nondelta Form of the Implicit Euler Method
    • The Factored Delta Form of the Implicit Euler Method
    • The Factored Delta Form of the Trapezoidal Method
  • Example Analysis of the 3-D Model Equation
• Appendix A: USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA [postscript 106Kb] ....[pdf format 636Kb]
  • Notation
  • Definitions
  • Algebra
  • Eigensystems
  • Vector and Matrix Norms
• Appendix B: SOME PROPERTIES OF TRIDIAGONAL MATRICES [postscript 131Kb] ....[pdf format 831Kb]
  • Standard Eigensystem for Simple Tridiagonals
  • Generalized Eigensystem for Simple Tridiagonals
  • The Inverse of a Simple Tridiagonal
  • Eigensystems of Circulant Matrices
    • Standard Tridiagonals
    • General Circulant Systems
  • Special Cases Found From Symmetries
  • Special Cases Involving Boundary Conditions
• Appendix C: LOCAL LINEARIZATION FOR THE EULER EQUATIONS [postscript 91Kb] ....[pdf format 360Kb]
  • Derivation of the Flux Jacobians
  • The Homogeneous Property of the Euler Equations
  
  A tar file of all the postscript files is available : Book.tar.Z
  Please feel free to download it.
  

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  Please Note: Theses are notes in progress and will by periodically updated. If you find errors, typos or have suggestions PLEASE contact the authors.
  Thomas H. Pulliam
  Mon Sept 16 11:16:40 PDT 1996