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Deville M.O., Fischer P.F., Mund E.H. High Order Methods for Incompressible Fluid Flow

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Deville M.O., Fischer P.F., Mund E.H. High Order Methods for Incompressible Fluid Flow
First published in printed format 2002. - Cambridge University Press, 2004. 529 p. – ISBN:0-521-45309-7 hardback, ISBN:0-511-03760-0 eBook
High-order numerical methods provide an efficient approach to simulating many physical problems. This book considers the range of mathematical, engineering, and computer science topics that form the foundation of high-order numerical methods for the simulation of incompressible fluid flows in complex domains. Introductory chapters present high-order spatial and temporal discretizations for one-dimensional problems. These are extended to multiple space dimensions with a detailed discussion of tensor-product forms, multi-domain methods, and preconditioners for iterative solution techniques. Numerous discretizations of the steady and unsteady Stokes and Navier-Stokes equations are presented, with particular attention given to enforcement of incompressibility. Advanced discretizations, implementation issues, and parallel and vector performance are considered in the closing sections. Numerous examples are provided throughout to illustrate the capabilities of high-order methods in actual applications. Computer scientists, engineers and applied mathematicians interested in developing software for solving flow problems will find this book a valuable reference.
Contents
List of Figures
Preface
Fluid Mechanics and Computation: An Introduction
Viscous Fluid Flows
Mass Conservation
Momentum Equations
Linear Momentum
Angular Momentum
Energy Conservation
Thermodynamics and Constitutive Equations
Fluid Flow Equations and Boundary Conditions
Isothermal Incompressible Flow
Thermal Convection: The Boussinesq Approximation
Boundary and Initial Conditions
Dimensional Analysis and Reduced Equations
Vorticity Equation
Simplified Models
Turbulence and Challenges
Numerical Simulation
Hardware Issues
Software Issues
Algorithms
Advantages of High-Order Methods
Approximation Methods for Elliptic Problems
Variational Form of Boundary-Value Problems
Variational Functionals
Boundary Conditions
Sobolev Spaces and the Lax-Milgram Theorem
An Approximation Framework
Galerkin Approximations
Collocation Approximation
Finite-Element Methods
The h-Version of Finite Elements
The p-Version of Finite Elements
Spectral-Element Methods
Orthogonal Collocation
Orthogonal Collocation in a Monodomain
Orthogonal Collocation in a Multidomain
Error Estimation
Solution Techniques
The Conditioning of a Matrix
Basic Iterative Methods
Preconditioning Schemes of High-Order Methods
Iterative Methods Based on Projection
A Numerical Example
Parabolic and Hyperbolic Problems
Introduction
Time Discretization Schemes
Linear Multistep Methods
Predictor–Corrector Methods
Runge–Kutta Methods
Splitting Methods
The Operator-Integration-Factor Splitting Method
OIFS Example: The BDF3/RK4 Scheme
The Parabolic Case: Unsteady Diffusion
Spatial Discretization
Time Advancement
The Hyperbolic Case: Linear Convection
Spatial Discretization
Eigenvalues of the Discrete Problem and CFL Number
Example of Temporal and Spatial Accuracy
Inflow–Outflow Boundary Conditions
Steady Advection–Diffusion Problems
Spectral Elements and Bubble Stabilization
Collocation and Staggered Grids
Unsteady Advection–Diffusion Problems
Spatial Discretization
Temporal Discretization
Outflow Conditions and Filter-Based Stabilization
The Burgers Equation
Space and Time Discretization
Numerical Results
The OIFS Method and Subcycling
Taylor–Galerkin Time Integration
Nonlinear Pure Advection
Taylor–Galerkin and OIFS Methods
Multidimensional Problems
Introduction
Tensor Products
Elliptic Problems
Weak Formulation and Sobolev Spaces
A Constant-Coefficient Case
The Variable-Coefficient Case
Deformed Geometries
Generation of Geometric Deformation
Surface Integrals and Robin Boundary Conditions
Spectral-Element Discretizations
Continuity and Direct Stiffness Summation
Spectral–Element Operators
Inhomogeneous Dirichlet Problems
Iterative Solution Techniques
Two-Dimensional Examples
Collocation Discretizations
The Diffusion Case
The Advection–Diffusion Case
Parabolic Problems
Time-Dependent Projection
Other Diffusion Systems
Hyperbolic Problems
Unsteady Advection–Diffusion Problems
Further Reading
Steady Stokes and Navier–Stokes Equations
Steady Velocity–Pressure Formulation
Stokes Equations
The Weak Formulation
The Spectral-Element Method
CollocationMethods on Single and Staggered Grids
Linear Systems, Algorithms, and Preconditioners
Spectral-Element Methods and Uzawa Algorithm
Collocation Methods
Poisson Pressure Solver and Green’s-Function Technique
General Considerations
The Green’s-Function Method
Implementation
Divergence-Free Bases
Stabilization of the PN –PN Approximation by Bubble Functions
hp-Methods for Stokes Problems
Steady Navier–Stokes Equations
Weak Formulation
Collocation Approximation of the Navier–Stokes
Equations
Solution Algorithms: Iterative and NewtonMethods
Applications
Stokes Problems
Navier–Stokes Problems
Complements and Engineering Considerations
Unsteady Stokes and Navier–Stokes Equations
Unsteady Velocity–Pressure Formulation
Unsteady Stokes Equations
The Weak Formulation
Uzawa Algorithm
Splitting and Decoupling Algorithms
Pressure Preconditioning
Unsteady Navier–Stokes Equations
Weak Formulation
Advection Treatment
Projection Methods
Fractional-Step Method
Pressure Correction Method
Stabilizing Unsteady Flows
Arbitrary Lagrangian–Eulerian Formulation and Free-Surface Flows
ALE Formulation
Free-Surface Conditions
Variational Formulation of Free-Surface Flows
Space and Time Discretization
Unsteady Applications
Extrusion from a Die
Vortex-Sheet Roll-Up
Unsteady Flow in Arteriovenous Grafts
Further Reading and Engineering Considerations
Domain Decomposition
Introduction
Preconditioning Methods
Substructuring and the Steklov–Poincar´ e Operator
Overlapping SchwarzProcedures
SchwarzPreconditioners for High-Order Methods
Spectral-Element Multigrid
The Mortar Element Method
Elliptic Problems
Implementation
Steady Stokes Problems
Applications
Adaptivity and Singularity Treatment
Coupling between Finite and Spectral Elements
Singularity Treatment
Triangular and Tetrahedral Elements
Error Estimates and Adaptivity
Further Reading
Vector and Parallel Implementations
Introduction
Serial Architectures
Pipelining
Memory, Bandwidth, and Caches
Tensor-Product Operator Evaluation
Tensor-Product Evaluation
Other Operations
Parallel Programming
Communication Characteristics
Vector Reductions
Parallel Multidomain Methods
Data Distribution and Operator Evaluation
Direct Stiffness Summation
Domain Partitioning
Coarse-Grid Solves
Applications
Hairpin Vortices
Driven Cavity
Backward-Facing Step
Further Reading
A Preliminary Mathematical Concepts
A.Metric Spaces
A.Definition
A. Open Set, Closed Set, Neighborhood
A. Cauchy Sequence, Limit Points, Dense Sets
A.Mapping, Domain, Range, Continuity
A.Convergence, Completeness, Completion Process
A. Normed Spaces
A. Definition
A.Banach Spaces
A.Linear Operators and Functionals in Normed Spaces
A.Linear Operator, Domain, Range, Nullspace
A.The Inverse Operator
A.Bounded Operators, Compact Operators
A. Bounded Linear Functionals, Dual Spaces
A.The Fr´ echet Derivative of an Operator
A.Inner-Product Spaces
A.Definition
A. Hilbert Spaces
A.Cauchy–SchwarzInequality
A.The RieszRepresentation
A.Orthogonality, Orthogonal Projection
A.Separable Hilbert Spaces, Basis
A.Gram–Schmidt Orthonormalization Process
A.Distributions
A.Definitions
A.Basic Properties of Distributions
B Orthogonal Polynomials and Discrete Transforms
B. Systems of Orthogonal Polynomials
B. Eigensolutions of Sturm–Liouville Problems
B.The Legendre Polynomials
B.The Chebyshev Polynomials
B.Gaussian-Type Quadratures
B.Fundamental Theorems
B. Gaussian Rules Based on Legendre Polynomials
B. Gaussian Rules Based on Chebyshev Polynomials
B.Discrete Inner Products and Norms
B.Spectral Approximation and Interpolation
B. Preliminaries
B. Discrete Spectral Transforms
B. Approximate Evaluation of Derivatives
B. Estimates for Truncation and Interpolation Errors
Bibliography
Index
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