Fundamentals of Computational Fluid Dynamics

ISBN-10: 3540416072
ISBN-13: 9783540416074
Edition: 2001
List price: $89.95 Buy it from $43.58
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Description: This book is intended as a textbook for a first course in computational fluid dynamics and will be of interest to researchers and practitioners as well. It emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods  More...

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Book details

List price: $89.95
Copyright year: 2001
Publisher: Springer
Publication date: 5/12/2003
Binding: Hardcover
Pages: 250
Size: 6.50" wide x 9.50" long x 0.75" tall
Weight: 1.144
Language: English

This book is intended as a textbook for a first course in computational fluid dynamics and will be of interest to researchers and practitioners as well. It emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. The linear convection and diffusion equations are used to illustrate concepts throughout. The chosen approach, in which the partial differential equations are reduced to ordinary differential equations, and finally to difference equations, gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics.

Introduction
Motivation
Problem Specification and Geometry Preparation
Selection of Governing Equations and Boundary Conditions
Selection of Gridding Strategy and Numerical Method
Assessment and Interpretation of Results
Overview
Notation
Conservation Laws and the Model Equations
Conservation Laws
The Navier-Stokes and Euler Equations
The Linear Convection Equation
Differential Form
Solution in Wave Space
The Diffusion Equation
Differential Form
SolutioninWaveSpace
Linear Hyperbolic Systems
Exercises
Finite-Difference Approximations
Meshes and Finite-Difference Notation
Space DerivativeApproximations
Finite-Difference Operators
Point Difference Operators
Matrix Difference Operators
Periodic Matrices
CirculantMatrices
Constructing Differencing Schemes of Any Order
TaylorTables
Generalization of Difference Formulas
Lagrange and Hermite Interpolation Polynomials
Practical Application of Pad� Formulas
OtherHigher-OrderSchemes
FourierErrorAnalysis
ApplicationtoaSpatialOperator
Difference Operators at Boundaries
TheLinearConvectionEquation
The Diffusion Equation
Exercises
The Semi-Discrete Approach
Reduction of PDE's to ODE's
The Model ODE's
TheGenericMatrixForm
ExactSolutionsofLinearODE's
EigensystemsofSemi-discreteLinearForms
Single ODE's of First and Second Order
CoupledFirst-OrderODE's
General Solution of Coupled ODE's with Complete Eigensystems
RealSpaceandEigenspace
Definition
EigenvalueSpectrumsforModelODE's
Eigenvectors of the Model Equations
Solutions of the Model ODE's
TheRepresentative Equation
Exercises
Finite-Volume Methods
Basic Concepts
ModelEquations in Integral Form
TheLinearConvectionEquation
The Diffusion Equation
One-DimensionalExamples
A Second-Order Approximation to the Convection Equation
A Fourth-Order Approximation to the Convection Equation
A Second-Order Approximation to the Diffusion Equation
ATwo-Dimensional Example
Exercises
Time-Marching Methods for ODE'S
Notation
Converting Time-Marching Methods to O�E's
Solution of Linear O�E's with Constant Coefficients
First- and Second-Order Difference Equations
Special Cases of Coupled First-Order Equations
Solution of the Representative O�E's
The Operational Form and its Solution
Examples of Solutions to Time-Marching O�E's
The �-� Relation
Establishing the Relation
The Principal �-Root
Spurious �-Roots
One-Root Time-Marching Methods
Accuracy Measures of Time-Marching Methods
Local and Global Error Measures
Local Accuracy of the Transient Solution (er<sub>�</sub>�, er<sub>&omgea;)
Local Accuracy of the Particular Solution (er<sub>�</sub>)
Time Accuracy for Nonlinear Applications
Global Accuracy
Linear Multistep Methods
The General Formulation
Examples
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
Local Linearization for Coupled Sets of Nonlinear Equations
Exercises
Stability of Linear Systems
Dependence on the Eigensystem
Inherent Stability of ODE's
The Criterion
Complete Eigensystems
Defective Eigensystems
Numerical Stability of O�E's
The Criterion
Complete Eigensystems
Defective Eigensystems
Time-Space Stability and Convergence of O�E's
Numerical Stability Concepts in the Complex �-Plane
�-Root Traces Relative to the Unit Circle
Stability for Small �t
Numerical Stability Concepts in the Complex �h Plane
Stability for Large h
Unconditional Stability, A-Stable Methods
Stability Contours in the Complex �h Plane
Fourier Stability Analysis
The Basic Procedure
Some Examples
Relation to Circulant Matrices
Consistency
Exercises
Choosing a Time-Marching Method
Stiffness Definition for ODE's
Relation to �-Eigenvalues
Driving and Parasitic Eigenvalues
Stiffness Classifications
Relation of Stiffness to Space Mesh Size
Practical Considerations for Comparing Methods
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
Steady Problems
Exercises
Relaxation Methods
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
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
Exercises
Multigrid
Motivation
Eigenvector and Eigenvalue Identification with Space Frequencies
Properties of the Iterative Method
The Basic Process
A Two-Grid Process
Exercises
Numerical Dissipation
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
Exercises
Split and Factored Forms
The Concept
Factoring Physical Representations -Time Splitting
Factoring Space MatrixOperators in 2D
Mesh Indexing Convention
Data-Bases and Space Vectors
Data-Base Permutations
Space Splitting and Factoring
Second-Order Factored Implicit Methods
Importance of Factored Forms in Two and Three Dimensions
The Delta Form
Exercises
Analysis of Split and Factored Forms
The Representative Equation for Circulant Operators
Example Analysis of Circulant Systems
Stability Comparisons of Time-Split Methods
Analysisofa Second-Order Time-Split Method
The Representative Equation for Space-Split Operators
Example Analysis of the 2D Model Equation
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 3D Model Equation
Exercises
Appendices
Useful Relations from Linear Algebra
Notation
Definitions
Algebra
Eigensystems
Vector and Matrix Norms
Some Properties of Tridiagonal Matrices
Standard Eigensystem for Simple Tridiagonal Matrices
Generalized Eigensystem for Simple Tridiagonal Matrices
The Inverse of a Simple Tridiagonal Matrix
Eigensystems of Circulant Matrices
Standard Tridiagonal Matrices
General Circulant Systems
Special Cases Found from Symmetries
Special Cases Involving Boundary Conditions
The Homogeneous Property of the Euler Equations
Index

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