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9780521453097

High-Order Methods for Incompressible Fluid Flow

by
  • ISBN13:

    9780521453097

  • ISBN10:

    0521453097

  • Format: Hardcover
  • Copyright: 2002-08-19
  • Publisher: Cambridge University Press

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Summary

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 sttention given to enforcement of imcompressibility. 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.

Author Biography

M. O. Deville is a professor of numerical fluid mechanics at the Ecole Polytechnique Federale de Lausanne in Switzerland P. F. Fischer is a research scientist in the Mathematics and Computer Science Division of Argonne National Laboratory Since 1967, E. H. Mund has published extensively in the areas of nuclear reactor physics and computation, radiation transport, and applied mathematics. Dr. Mund is also part-time professor of nuclear engineering at the Universite Catholique de Louvain

Table of Contents

List of Figures
xvii
Preface xxv
Fluid Mechanics and Computation: An Introduction
1(32)
Viscous Fluid Flows
1(2)
Mass Conservation
3(2)
Momentum Equations
5(1)
Linear Momentum
5(1)
Angular Momentum
6(1)
Energy Conservation
6(1)
Thermodynamics and Constitutive Equations
7(1)
Fluid Flow Equations and Boundary Conditions
8(3)
Isothermal Incompressible Flow
8(1)
Thermal Convection: The Boussinesq Approximation
9(1)
Boundary and Initial Conditions
10(1)
Dimensional Analysis and Reduced Equations
11(4)
Vorticity Equation
15(1)
Simplified Models
16(1)
Turbulence and Challenges
17(5)
Numerical Simulation
22(11)
Hardware Issues
22(2)
Software Issues
24(2)
Algorithms
26(2)
Advantages of High-Order Methods
28(5)
Approximation Methods for Elliptic Problems
33(65)
Variational Form of Boundary-Value Problems
34(12)
Variational Functionals
34(5)
Boundary Conditions
39(1)
Sobolev Spaces and the Lax-Milgram Theorem
40(6)
An Approximation Framework
46(7)
Galerkin Approximations
47(4)
Collocation Approximation
51(2)
Finite-Element Methods
53(9)
The h-Version of Finite Elements
54(6)
The p-Version of Finite Elements
60(2)
Spectral-Element Methods
62(5)
Orthogonal Collocation
67(4)
Orthogonal Collocation in a Monodomain
67(2)
Orthogonal Collocation in a Multidomain
69(2)
Error Estimation
71(2)
Solution Techniques
73(19)
The Conditioning of a Matrix
74(7)
Basic Iterative Methods
81(2)
Preconditioning Schemes of High-Order Methods
83(3)
Iterative Methods Based on Projection
86(6)
A Numerical Example
92(6)
Parabolic and Hyperbolic Problems
98(64)
Introduction
98(1)
Time Discretization Schemes
99(20)
Linear Multistep Methods
100(10)
Predictor--Corrector Methods
110(3)
Runge--Kutta Methods
113(6)
Splitting Methods
119(5)
The Operator-Integration-Factor Splitting Method
121(2)
OIFS Example: The BDF3/RK4 Scheme
123(1)
The Parabolic Case: Unsteady Diffusion
124(5)
Spatial Discretization
126(1)
Time Advancement
127(2)
The Hyperbolic Case: Linear Convection
129(8)
Spatial Discretization
130(1)
Eigenvalues of the Discrete Problem and CFL Number
131(4)
Example of Temporal and Spatial Accuracy
135(2)
Inflow--Outflow Boundary Conditions
137(1)
Steady Advection--Diffusion Problems
137(8)
Spectral Elements and Bubble Stabilization
138(3)
Collocation and Staggered Grids
141(4)
Unsteady Advection--Diffusion Problems
145(6)
Spatial Discretization
146(3)
Temporal Discretization
149(1)
Outflow Conditions and Filter-Based Stabilization
149(2)
The Burgers Equation
151(4)
Space and Time Discretization
151(2)
Numerical Results
153(2)
The OIFS Method and Subcycling
155(3)
Taylor--Galerkin Time Integration
158(4)
Nonlinear Pure Advection
159(2)
Taylor--Galerkin and OIFS Methods
161(1)
Multidimensional Problems
162(72)
Introduction
162(1)
Tensor Products
162(7)
Elliptic Problems
169(9)
Weak Formulation and Sobolev Spaces
170(2)
A Constant-Coefficient Case
172(5)
The Variable-Coefficient Case
177(1)
Deformed Geometries
178(10)
Generation of Geometric Deformation
183(3)
Surface Integrals and Robin Boundary Conditions
186(2)
Spectral-Element Discretization
188(14)
Continuity and Direct Stiffness Summation
191(3)
Spectral--Element Operators
194(2)
Inhomogeneous Dirichlet Problems
196(1)
Iterative Solution Techniques
197(1)
Two-Dimensional Examples
198(4)
Collocation Discretizations
202(18)
The Diffusion Case
202(14)
The Advection--Diffusion Case
216(4)
Parabolic Problems
220(6)
Time-Dependent Projection
222(2)
Other Diffusion Systems
224(2)
Hyperbolic Problems
226(4)
Unsteady Advection--Diffusion Problems
230(2)
Further Reading
232(2)
Steady Stokes and Navier--Stokes Equations
234(57)
Steady Velocity--Pressure Formulation
234(2)
Stokes Equations
236(17)
The Weak Formulation
236(2)
The Spectral-Element Method
238(7)
Collocation Methods on Single and Staggered Grids
245(8)
Linear Systems, Algorithms, and Preconditioners
253(6)
Spectral-Element Methods and Uzawa Algorithm
253(4)
Collocation Methods
257(2)
Poisson Pressure Solver and Green's-Function Technique
259(5)
General Considerations
259(1)
The Green's-Function Method
260(3)
Implementation
263(1)
Divergence-Free Bases
264(5)
Stabilization of the PN--PN Approximation by Bubble Functions
269(3)
hp-Methods for Stokes Problems
272(1)
Steady Navier--Stokes Equations
273(5)
Weak Formulation
274(1)
Collocation Approximation of the Navier--Stokes Equations
275(2)
Solution Algorithms: Iterative and Newton Methods
277(1)
Applications
278(10)
Stokes Problems
278(5)
Navier--Stokes Problems
283(5)
Complements and Engineering Considerations
288(3)
Unsteady Stokes and Navier--Stokes Equations
291(42)
Unsteady Velocity--Pressure Formulation
291(2)
Unsteady Stokes Equations
293(7)
The Weak Formulation
293(2)
Uzawa Algorithm
295(1)
Splitting and Decoupling Algorithms
296(4)
Pressure Preconditioning
300(3)
Unsteady Navier--Stokes Equations
303(6)
Weak Formulation
303(1)
Advection Treatment
304(5)
Projection Methods
309(6)
Fractional-Step Method
310(3)
Pressure Correction Method
313(2)
Stabilizing Unsteady Flows
315(3)
Arbitrary Lagrangian--Eulerian Formulation and Free-Surface Flows
318(8)
ALE Formulation
319(1)
Free-Surface Conditions
320(2)
Variational Formulation of Free-Surface Flows
322(3)
Space and Time Discretization
325(1)
Unsteady Applications
326(3)
Extrusion from a Die
326(1)
Vortex-Sheet Roll-Up
327(2)
Unsteady Flow in Arteriovenous Grafts
329(1)
Further Reading and Engineering Considerations
329(4)
Domain Decomposition
333(46)
Introduction
333(1)
Preconditioning Methods
334(18)
Substructuring and the Steklov--Poincare Operator
334(4)
Overlapping Schwarz Procedures
338(7)
Schwarz Preconditioners for High-Order Methods
345(3)
Spectral-Element Multigrid
348(4)
The Mortar Element Method
352(16)
Elliptic Problems
357(1)
Implementation
358(5)
Steady Stokes Problems
363(2)
Applications
365(3)
Adaptivity and Singularity Treatment
368(10)
Coupling between Finite and Spectral Elements
369(1)
Singularity Treatment
370(1)
Triangular and Tetrahedral Elements
371(6)
Error Estimates and Adaptivity
377(1)
Further Reading
378(1)
Vector and Parallel Implementations
379(38)
Introduction
379(1)
Serial Architectures
380(4)
Pipelining
381(1)
Memory, Bandwidth, and Caches
382(2)
Tensor-Product Operator Evaluation
384(7)
Tensor-Product Evaluation
385(5)
Other Operations
390(1)
Parallel Programming
391(8)
Communication Characteristics
393(3)
Vector Reductions
396(3)
Parallel Multidomain Methods
399(9)
Data Distribution and Operator Evaluation
399(2)
Direct Stiffness Summation
401(5)
Domain Partitioning
406(1)
Coarse-Grid Solves
407(1)
Applications
408(8)
Hairpin Vortices
408(2)
Driven Cavity
410(2)
Backward-Facing Step
412(4)
Further Reading
416(1)
A Preliminary Mathematical Concepts 417(25)
Metric Spaces
417(4)
Definition
417(1)
Open Set, Closed Set, Neighborhood
418(1)
Cauchy Sequence, Limit Points, Dense Sets
419(1)
Mapping, Domain, Range, Continuity
419(1)
Convergence, Completeness, Completion Process
420(1)
Normed Spaces
421(2)
Definition
421(1)
Banach Spaces
422(1)
Linear Operators and Functionals in Normed Spaces
423(4)
Linear Operator, Domain, Range, Nullspace
423(1)
The Inverse Operator
423(1)
Bounded Operators, Compact Operators
424(1)
Bounded Linear Functionals, Dual Spaces
425(1)
The Frechet Derivative of an Operator
425(2)
Inner-Product Spaces
427(7)
Definition
427(1)
Hilbert Spaces
428(1)
Cauchy--Schwarz Inequality
429(1)
The Riesz Representation
430(1)
Orthogonality, Orthogonal Projection
431(1)
Separable Hilbert Spaces, Basis
432(1)
Gram--Schmidt Orthonormalization Process
433(1)
Distributions
434(8)
Definitions
434(4)
Basic Properties of Distributions
438(4)
B Orthogonal Polynomisls and Discrete Transforms 442(25)
Systems of Orthogonal Polynomials
442(6)
Eigensolutions of Sturm--Liouville Problems
444(1)
The Legendre Polynomials
445(2)
The Chebyshev Polynomials
447(1)
Gaussian-Type Quadratures
448(6)
Fundamental Theorems
448(2)
Gaussian Rules Based on Legendre Polynomials
450(1)
Gaussian Rules Based on Chebyshev Potynomials
451(1)
Discrete Inner Products and Norms
452(2)
Spectral Approximate and Interpolation
454(13)
Preliminaries
454(1)
Discrete Spectral Transforms
455(3)
Approximate Evaluation of Derivatives
458(6)
Estimates for Truncation and Interpolation Errors
464(3)
Bibliography 467(22)
Index 489

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