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Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics Revised edition [Pehme köide]

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  • Formaat: Paperback, 239 pages, kõrgus x laius x paksus: 229x152x13 mm, kaal: 340 g, Illustrations
  • Sari: Classics in Applied Mathematics 67
  • Ilmumisaeg: 30-Jun-2011
  • Kirjastus: Society for Industrial & Applied Mathematics,U.S.
  • ISBN-10: 1611970741
  • ISBN-13: 9781611970746
Teised raamatud teemal:
Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics Revised editionSuurem pilt
  • Formaat: Paperback, 239 pages, kõrgus x laius x paksus: 229x152x13 mm, kaal: 340 g, Illustrations
  • Sari: Classics in Applied Mathematics 67
  • Ilmumisaeg: 30-Jun-2011
  • Kirjastus: Society for Industrial & Applied Mathematics,U.S.
  • ISBN-10: 1611970741
  • ISBN-13: 9781611970746
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781611970746.html
Märksõnad:
This classic text presents the best practices of developing multigrid solvers for large-scale computational problems in science and engineering. By representing a problem at multiple scales and employing suitable interscale interactions, multigrid avoids slowdown due to stiffness and reduces the computational cost of classical algorithms by orders of magnitude. Starting from simple examples, this book guides the reader through practical stages for developing reliable multigrid solvers, methodically supported by accurate performance predictors. The revised edition presents discretization and fast solution of linear and nonlinear partial differential systems; treatment of boundary conditions, global constraints and singularities; grid adaptation, high-order approximations, and system design optimization; applications to fluid dynamics, from simple models to advanced systems; new quantitative performance predictors, a MATLAB(R) sample code, and more. Readers will also gain access to the Multigrid Guide 2.0 Web site, where updates and new developments will be continually posted, including a chapter on Algebraic Multigrid.
List of Figures
xiii
List of Tables
xv
Preface to the Classics Edition xvii
Preface xxi
0 Introduction
1(6)
0.1 Where and why multigrid can help
1(2)
0.2 About this guide (the 1984 edition)
3(4)
1 Elementary Acquaintance With Multigrid
7(14)
1.1 Properties of slowly converging errors
7(2)
1.2 Error smoothing and its analysis: Example
9(3)
1.3 Coarse grid correction
12(1)
1.4 Multigrid cycle
12(2)
1.5 Model program and output
14(3)
1.6 Full Multigrid (FMG) algorithm
17(1)
1.7 General warnings. Boundary conditions. Nonlinearity
18(3)
I Stages in Developing Fast Solvers
21(62)
2 Stable Discretization
25(6)
2.1 Interior stability measures: h-ellipticity
26(3)
2.2 Boundaries, discontinuities
29(2)
3 Interior Relaxation and Smoothing Factors
31(16)
3.1 Local analysis of smoothing
31(2)
3.2 Work, robustness and other considerations
33(1)
3.3 Block relaxation rule. Semi smoothing
34(3)
3.4 Distributive, weighted, collective and box GS. Principal linearization
37(2)
3.5 Simultaneous displacement (Jacobi) schemes
39(1)
3.6 Relaxation ordering. Vector and parallel processing
40(2)
3.7 Principle of relaxing general PDE systems
42(2)
3.8 ILU smoothers
44(3)
4 Interior Two-Level Cycles
47(12)
4.1 Two-level cycling analysis. Switching criteria
49(2)
4.2 Choice of coarse grid
51(2)
4.2.1 Semi coarsening
52(1)
4.2.2 Modified and multiple coarse-grid functions
53(1)
4.3 Orders of interpolations and residual transfers
53(2)
4.4 Variable operators. Full weightings
55(1)
4.5 Coarse-grid operator. Variational and Galerkin coarsening
56(1)
4.6 Strongly discontinuous, strongly asymmetric operators
57(2)
5 Boundary Conditions and Two-Level Cycling
59(10)
5.1 Simplifications and debugging
60(1)
5.2 Interpolation near boundaries and singularities
61(1)
5.3 Relaxation on and near boundaries
62(1)
5.4 Residual transfers near boundaries
62(1)
5.5 Transfer of boundary residuals
63(1)
5.6 Treatment and use of global constraints
64(2)
5.7 Structural singularities. Reentrant corners. Local relaxation
66(3)
6 Many-Level Cycles
69(6)
6.1 Multigrid cycles. Initial and terminal relaxation
69(1)
6.2 Switching criteria. Types of cycles
70(1)
6.3 Coarsest grids. Inhomogeneous and indefinite operators
71(4)
7 Full Multi-Grid (FMG) Algorithms
75(8)
7.1 Order of the FMG interpolation
76(1)
7.2 Optimal switching to a new grid
77(1)
7.3 Total computational work. Termination criteria
78(1)
7.4 Two-level FMG mode analysis
79(2)
7.5 Half-space FMG mode analysis. First differential approximations
81(2)
II Advanced Techniques and Insights
83(108)
8 Full Approximation Scheme (FAS) and Applications
87(12)
8.1 From CS to FAS
87(2)
8.2 FAS: dual point of view
89(1)
8.3 Nonlinear problems
89(4)
8.3.1 Eigenvalue problems
91(1)
8.3.2 Continuation (embedding) techniques
91(2)
8.4 Estimating truncation errors. τ-extrapolation
93(1)
8.5 FAS interpolations and transfers
94(1)
8.6 Application to integral equations
95(1)
8.7 Small storage algorithms
96(3)
9 Local Refinements and Grid Adaptation
99(10)
9.1 Non-uniformity organized by uniform grids
99(2)
9.2 Anisotropic refinements
101(1)
9.3 Local coordinate transformations
102(2)
9.4 Sets of rotated cartesian grids
104(1)
9.5 Self-adaptive techniques
104(3)
9.6 Exchange rate algorithms. λ-FMG
107(2)
10 Higher-Order Techniques
109(4)
10.1 Fine-grid defect corrections. Pseudo spectral methods
109(2)
10.2 Double discretization: High-order residual transfers
111(1)
10.3 Relaxation with only subprincipal terms
112(1)
11 Coarsening Guided By Discretization
113(4)
12 True Role of Relaxation
117(4)
13 Dealgebraization of Multigrid
121(4)
13.1 Reverse trend: Algebraic multigrid
123(2)
14 Practical Role of Rigorous Analysis and Quantitative Predictions
125(8)
14.1 Rigorous qualitative analysis
125(3)
14.2 Quantitative predictors
128(1)
14.3 Direct numerical performance predictors
129(4)
14.3.1 Compatible relaxation
129(3)
14.3.2 Other idealized cycles
132(1)
15 Chains of Problems. Frozen τ
133(2)
16 Time Dependent Problems
135(4)
III Applications to Fluid Dynamics
139(4)
17 Cauchy-Riemann Equations
143(10)
17.1 The differential problem
143(1)
17.2 Discrete Cauchy-Riemann equations
144(3)
17.3 DGS relaxation and its smoothing rate
147(1)
17.4 Multigrid procedures
148(2)
17.5 Numerical results
150(1)
17.6 Remark on non-staggered grids
151(2)
18 Steady-State Stokes Equations
153(12)
18.1 The differential problem
153(1)
18.2 Finite-difference equations
154(3)
18.3 Distributive relaxation
157(2)
18.4 Multi-grid procedures
159(1)
18.5 Numerical results
160(1)
18.6 Non-staggered grids
161(4)
19 Steady-State Incompressible Navier-Stokes Equations
165(6)
19.1 The differential problem
105(55)
19.2 Staggered finite-difference approximations
160(7)
19.3 Distributive relaxation
167(2)
19.4 Multigrid procedures and numerical results
169(1)
19.5 Results for non-staggered grids
169(2)
20 Compressible Navier-Stokes and Euler Equations
171(18)
20.1 The differential equations
171(6)
20.1.1 Conservation laws and simplification
171(2)
20.1.2 The viscous principal part
173(1)
20.1.3 Elliptic singular perturbation
174(1)
20.1.4 Inviscid (Euler) and subprincipal operators
175(1)
20.1.5 Incompressible and small Mach limits
176(1)
20.2 Stable staggered discretization
177(3)
20.2.1 Discretization of the subprincipal part
177(1)
20.2.2 The full quasi-linear discretization
178(1)
20.2.3 Simplified boundary conditions
179(1)
20.3 Distributive relaxation for the simplified system
180(7)
20.3.1 General approach to relaxation design
180(2)
20.3.2 Possible relaxation scheme for inviscid flow
182(1)
20.3.3 Distributed collective Gauss-Seidel
183(1)
20.3.4 Relaxation ordering and smoothing rates
184(1)
20.3.5 Summary: relaxation of the full system
185(2)
20.4 Multigrid procedures
187(2)
21 Remarks On Solvers For Transonic Potential Equations
189(2)
21.1 Multigrid improvements
189(1)
21.2 Artificial viscosity in quasi-linear formulations
190(1)
A TestCycle: Matlab Code
191(14)
A.1 addflops.m
191(1)
A.2 Bilinearlnterpolation.m
191(1)
A.3 Cycle.m
192(3)
A.4 errornorm.m
195(1)
A.5 flops.m
195(1)
A.6 FwLinearRestrictor.m
195(1)
A.7 GaussSeidelSmoother.m
196(1)
A.8 Level.m
197(3)
A.9 MultilevelBuilder.m
200(1)
A.10 Operator.m
201(1)
A.11 Options.m
202(1)
A.12 TestCycle.m
203(2)
Bibliography 205(12)
Index 217
Achi Brandt is Professor Emeritus at the Weizmann Institute of Science, Professor in Residence at the University of California, Los Angeles and the Chief Scientist of VideoSurf, Inc. In 2005 he won the SIAM/ACM Prize in Computational Science and Engineering 'for pioneering modern multilevel methods, from multigrid solvers for partial differential equations to multiscale techniques for statistical physics, and for influencing almost every aspect of contemporary computational science and engineering'. Oren Livne is a Senior Software Engineer at the Office of the Associate Vice President for Health Sciences Information Technology at the University of Utah. He received his Ph.D. in applied mathematics from the Weizmann Institute of Science. His doctoral work, supervised by Achi Brandt, focused on multigrid methods for electronic structure computations in quantum chemistry.