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“… a book which deserves a place on the desk of all who teach or seriously practice computational methods in fluid dynamics.”

Applied Mechanics Reviews
Vol. 52, No. 6, June 1999, p. B57

TABLE OF CONTENTS Copyright 1998 by Patrick J. Roache

Part I

Chapter 1 Introduction

1-A. The Realm of Computational Fluid Dynamics
1-B. Historical Outline of Computational Fluid Dynamics
1-C. Existence and Uniqueness of Solutions
1-D. Preliminary Remarks on Consistency, Convergence, and Stability of Solutions
1-E. Key Concepts of CFD

Chapter 2 Incompressible Flow Equations in Rectangular Coordinates

2-A. Pressure-Velocity Equations
2-B. Stream Function and Vorticity Transport Equations for Planar Flows
2-C. Conservation Form
2-D. Normalizing Systems
2-E. One-dimensional Model Transport Equations

Chapter 3 Basic Computational Methods for Incompressible Flow

3-A. Methods for Solving the Transport Equation
3-A-1. Some Basic Forms of Discretized Equations
3-A-2. Control Volume Approach; Finite Volume Methods; MMOC
3-A-3. The Conservative Property
3-A-4. A Description of Instability
3-A-5. Stability Analyses
3-A-5-a. Discrete Perturbation Stability Analysis
3-A-5-b. von Neumann Stability Analysis
3-A-5-c. Harts Heuristic Stability Analysis and the Modified Equation Approach
3-A-5-d. Summary and Evaluation of Stability Criteria
3-A-5-e. The von Neumann Analysis in Multidimensional Problems
3-A-6. One Step Explicit Methods: The Midpoint Leapfrog Method:Filtering
3-A-7. The DuFort-Frankel Leapfrog Method
3-A-8. The First Upwind Differencing Method; Artificial Viscosity Errors
3-A-9. The Transportive Property
3-A-10. Transportive and Conservative Differencing
3-A-11. The Second Upwind Differencing Method
3-A-12. Adams-Bashforth and Crocco Methods
3-A-13. Leith/Lax-Wendroff Method; Phase Errors, Aliasing Errors and Time-Splitting
3-A-14. Implicit Methods
3-A-15. Multi-Step Explicit Methods: Runge-Kutta Methods
3-A-16. ADI Methods
3-A-17. ADE Methods
3-A-18. The Hopscotch Method
3-A-19. The Fourth-Order Methods of Roberts And Weiss and of Crowley
3-A-20. Fromm’s Method of Zero Average Phase Error;Leonard’s QUICK Method
3-A-21. Arakawa’s Method
3-A-22. Remarks on Steady Flow Methods; Local Time Stepping
3-A-23. Remarks on Evaluating Methods; Behavioral Errors and Mimetic Properties; Compact Differencing
3-B. Methods for Solving the Stream Function Equations
3-B-1. Direct Methods
3-B-2. Richardson’s Method and Liebman’s Method
3-B-3. Southwell’s Residual Relaxation Method.
3-B-4. Successive Over-Relaxation (SOR) Method
3-B-5. Tactics and Strategy
3-B-6. ADI and Line SOR Methods
3-B-7. Other Iterative Methods, Coloring Schemes; Multigrid
3-B-8. EVP Method; Elliptic Marching Methods
3-B-9. Fourier-Series Methods
3.B.10. Higher Order Approximations
3-B-11. Remarks on Evaluating Methods
3-C. Boundary Conditions for the Vorticity and Stream Function Equations.
3-C-1. Remarks on the Dominant Importance of Computational Boundary Conditions
3-C-2. Walls in the First Mesh System
3-C-3. Walls in Alternate Mesh Systems
3-C-4. Symmetry Boundaries
3-C-5. Upper Boundary
3-C-6. Upstream Boundary
3-C-7. Outflow Boundary
3-C-8. “Wiggles” or “Ripples”
3-C-9. The Downstream Paradox
3-C-10. Computational versus Analytic versus “Fuzzy” Boundary Conditions
3-C-11. Conditions at “Infinity”
3-C-12. The Sharp Corners
3-C-12-A Boundary Conditions at the Sharp Convex Corner
3-C-12-B Convergence and Accuracy at the Sharp Convex Corner
3-C-13. Concluding Remarks on Vorticity-Stream Function Boundary Conditions and Coupling
3-D. Convergence Criteria and Initial Conditions
3-E. Pressure Solution
3-E-1. Numerical Cubature
3-E-2. Poisson Equation for Pressure
3-E-3. Boundary Conditions of the Second Kind on Pressure
3-E-4. Iterative Solution Methods with “Condensation”
3-E-5. Pressure Level
3-F. Temperature Solutions and Concentration Solutions
3-F-1. Basic Equations
3-F-2. Retention of Dissipation
3-F-3. Finite-Difference Representation of Dissipation
3-F-4. Boundary Conditions for Temperature and Concentration
3-F-5. Source Terms and Stiff Equations
3-G. Methods for Solving the Pressure-Velocity Equations
3-G-1. General Considerations
3-G-2. Basic Equations
3-G-3. Boundary Conditions in Primitive Variables
3-G-4. The MAC Method; Staggered Grids and Finite Volume Methods
3-G-5. Other Methods Using Primitive Variables
3-G-6. Relative Merits of the (
y,z) and (u, v, P) Systems
3-H. Three-Dimensional Flows
3-I. Other Discretization Methodologies

Chapter 4 Compressible Flow Equations
in Rectangular Coordinates                                                     backtotop2

4-A. Fundamental Difficulties
4-B. Customary Equations
4-C. Conservation Form
4-D. Supplemental Relations
4-E. Normalized Conservation Equations
4-F. Short-Form Equations
4-G. Existence of Shocks - Physical and Mathematical
4-H. Non-Uniqueness of Nonlinear Solutions

Chapter 5 Basic Computational Methods for Compressible Flow

5-A. Preliminary Considerations
5-A-1. Shock-Free Methods and Shock-Patching Methods
5-A-2. Stability Considerations
5-A-3. Early Attempts at Implicit Methods
5-B. Methods for the Numerical Treatment of Shocks
5-C. Shock Smearing by Artificial Dissipation
5-D. Methods Using Explicit Artificial Viscosities
5-D-1. von Neumann-Richtmyer Method
5-D-2. Landshoff’s Method and Longley’s Method
5-D-3. Rusanov’s Method
5-D-4. Errors Arising from Artificial Viscosities
5-E. Methods Using Intrinsic Artificial Damping
5-E-1. Upwind Differencing
5-E-2. The Domain of Influence and Truncation Error
5-E-3. PIC and FLIC
5-E-4. Lax’s Method
5-E-5. Lax-Wendroff Method
5-E-6. Two-Step Lax-Wendroff and Second-Order Upwind Methods
5-E-7. The Method of Abarbanel and Zwas
5-E-8. Other Methods; Riemann Solvers; FCT
5-F. Viscous Terms in the Compressible Flow Equations
5-F-1. Spatial Difference Forms
5-F-2. General Considerations
5-F-3. Methods for the Viscous Terms
5-G. Boundary Conditions for Compressible Flow
5-G-1. Slip Walls
5-G-1-a. Slip Walls in the First Mesh System
5-G-1-b. Slip Walls in the Second Mesh System: FVM
5-G-2. No-Slip Walls
5-G-2-a. No-Slip Walls in the First Mesh System
5-G-2-b. No-Slip Walls in the Second Mesh System
5-G-2-c. Staggered Mesh Evaluation of Density
5-G-3. Sharp Corners
5-G-4. Symmetry Surfaces
5-G-5. Upstream Boundary
5-G-6. Downstream Boundary
5-G-7. Upper Boundary: The Simple Wave Condition
5-H. Convergence Criteria and Initial Conditions
5-I. Remarks on Subsonic and Supersonic Solutions
5-J. Higher Order Systems for Compressible Flows
5-K. Implicit Methods for Compressible Flow
5-L. Incompressible Solutions with Implicit Compressible Flow Codes
5-M. Nonlinear Flux Limiters And Related Methods

Chapter 6 Other Mesh Systems, Coordinate Systems,           backtotop2
 and Equation Systems

6-A. Special Mesh Systems
6-B. Coordinate Transformations
6-C. Other Orthogonal Coordinate Systems
6-D. Other Systems of Equations
6-D-1. Gross Simplifications to Navier-Stokes Equations
6-D-2. Minor Simplifications of Navier-Stokes Equations
6-D-3. Complifications to Navier-Stokes Equations
6-D-4. Alternate Mathematical Formulations
6-E. Future Developments

Chapter 7 Recommendations on Programming, Testing, and Information Processing

7-A. Computer Programming
7-B. Debugging and Testing
7-C. Information Processing
7-C-1. Numbers
7-C-2. Plots and Motion Pictures
7-C-3. Diagnostic Functionals
7-D. Closure

Introduction to Part II

Chapter 8 Finite Element vs. Finite Difference Methods          backtotop2

References for Chapter 8

Chapter 9 Operation Count for Direct Gaussian Elimination

9-A. Introduction
9-B. Round-Off Error
9-C. Speed and Storage Penalty
9-D. Operation Count as An Index Of Merit
9-E. Operation Count and Storage Penalty for GE In 2D
9-F. Operation Count and Storage Penalty for GE In 3D
9-G. Conclusions
References for Chapter 9

Chapter 10 Multigrid Solvers

10-A. Overview of Multigrid Methods
10-B. The Basic Multigrid Method
10-B-1. Motivation For The Cycling
10-B-2. Transferring the Solutions
10-B-2-a. Coarse to Fine Transfer
10-C. Other Multigrid Methods
10-D. Subgrid Coefficient Generation for Black Box Multigrid
10-D-1. Background and Motivation
10-D-2. Subgrid Generation
10-D-3. Upwinding In Subgrid Generation
10-D-4. Supergrid Generation
10-D-5. Performance
References for Chapter 10

Chapter 11 A Sixth-Order Accurate Direct Solver for Poisson and Helmholtz Equations in Polar Coordinates

11-A. Introduction
11-B. The Method
11-C. Numerical Verification
11-D. Discussion
Acknowledgments
References For Chapter 11

Chapter 12 The Legitimacy of the Poisson Pressure Equation  backtotop2

12-A. Summary
12-B. Introduction
12-C. Discussion of Derived Boundary Conditions
12-D. The Homegeneous Gradient Boundary Condition
12-E. Equivalence of The Poisson Pressure Approach and Continuity
12-F. Additional Comments
12-G. Acknowledgements and Publication Note
12-H. Additional Discussion
References for Chapter 12

Chapter 13 A Flux-Based Modified Method of Characteristics

13-A. Summary
13-B. Introduction
13-C. Non-Flux-Based MMOC Derivations of Difference Methods
13-D. Two Problems with Non-Flux-Based MMOC
13-E. Flux-Based MMOC Derivations Of Difference Methods
13-F. Erroneous Flux-Based MMOC Methodology For CFL > 1
13-G. Correct Flux-Based MMOC Methodology For CFL > 1
13-G-2. The Characteristic-Tracking Velocities
13-G-3. The Flux Contribution From the Core Cell
13-G-4. Flux Limiters
13-H. Recommended Method
13-I. Velocity Reversals
13-J. Multidimensions And Additional Terms
Acknowledgements
References for Chapter 13

Chapter 14 Solution Adaptive Grids and Time Steps              backtotop2

14-A. A Perfect Coordinate Transformation for the 1-D Advection-Diffusion Equation
14-B. Transformation of Governing Equations
14-C. Multidimensional Transformed Equations
14-D. Solution Adaptive Strategies
14-E. Time Accuracy Estimation And Adaptive Time Steps
14-F. Time Resolution of Source Terms
14-G. Domain Decomposition
References for Chapter 14

Chapter 15 Elliptic Grid Generation and Hybrid Grid Adaptation

15-A. Summary
15-B. Introduction
15-C. Adaptive Poisson Grid Generation
15-D. Anderson’s Adaptive Poisson Grid Generator
15-E. Details of the Hybrid Adaptive Poisson Grid Generator
15-F. Example Calculations
15-G. Comparisons, Costs and Extensions of the Hybrid Technique
15-H. Grid Smoothness
15-I. Conclusions
Acknowledgements
References for Chapter 15

Chapter 16 Semidirect High Order Accuracy Solutions in Non-Orthogonal Grids by Richardson Extrapolation

16-A. Introduction
16-B. Basic Numerical Method
16-C. Coordinate Transformation
16-D. Boundary Conditions at Outflow
16-E. Boundary Conditions at Inflow And Upper Boundary
16-F. Boundary Conditions on the No Slip Wall
16-G. Initial Conditions And Channel Shape
16-H. Iterative Convergence Criterion And Performance
16-I. Richardson Extrapolation
16-J. Discretization Error Convergence
16-K. Reynolds Number Scaling Behavior
16-L. Limit Analysis
16-M. Effect of Other Boundary Conditions
16-N. Conclusion
Acknowledgements
References for Chapter 16

Chapter 17 Nonlinear Flux Limiters                                      backtotop2
Applied to Groundwater Contaminant Transport

17-A. Summary
17-B. Introduction
17-C. Algorithm
17-D. Results
17-E. Conclusions
Acknowledgement
References for Chapter 17

Chapter 18 Verification of Codes and Calculations

18-A. Summary
18-B. Introduction
18-C. Semantics
18-D. Code Verification and Validation: Numerical Vs. Conceptual Modeling
18-E. Verification of Calculations
18-F. Code Confirmation and Certification
18-G. Grid Convergence vs. Iterative Convergence
18-H. Error Taxonomies
18-I. Truncation Error vs. Discretization Error
18-J. Calibration and Tuning
18-K. Customer Illusions vs. Customer Care
18-L. Other Distinctions
18-M. Code Verification Via Systematic Grid Convergence Testing
18-N. Grid Convergence Index
18-O. Sensitivity of Grid Convergence Testing
18-P. Esoteric Coding Errors
18-Q. Special Considerations for Turbulence Modeling
18-R. Extraction of Observed Order From Grid Convergence Tests
18-S. Cafe Curves
18-T. Surrogate Single-Grid Indexes
18-U. Conclusion
Acknowledgement
References for Chapter 18

Chapter 19 The Grid Convergence Index

19.1 Introduction
19.2 Background on Grid Convergence Reporting
19.3 Richardson Extrapolation
19.4 A Generalization of Richardson Extrapolation
19.5 Richardson’s Extrapolation for
p
19.6 Grid Convergence Index for the Fine Grid Solution
19.7 Grid Convergence Index for the Coarse Grid Solution
19.8 Example GCI Calculation
19.9 Should the Coefficient Be “1” Or “3” Or “1.25” ?
19.10. Additional Features of Grid Convergence Studies For Verification of Codes and Calculations
19.10.1 Non-Integer Grid Refinement
19.10. 2 Independent Coordinate Refinement and Mixed Order Methods
19.10.3 Non-Cartesian Grids, Boundary Fitted Grids, Unstructured Grids, Adaptive Grids
19.10.4 Shocks, Discontinuities, Singularities
19.10.5 Achieving the Asymptotic Range
19.10.6 Extraction of the Observed Order of Convergence from Grid Convergence Tests
19.10.7 Method of Characteristics and Spectral Methods
19.10.8 Non-Smooth Property Variation and the GCI
19.10.9 Non-Smooth Property Variation and Geostatistical Realizations
19.10.10 Stopping Criteria for Iterative Convergence
19.11. Conclusion
References for Chapter 19

Appendix A Tridiagonal Algorithm                                         backtotop2

Appendix B On Artificial Viscosity

Problems

References & Bibliography for Part I

Suggestions for a Course Using This Text

Subject Index

backtotop

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