Preface |
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vii | |
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1 | (34) |
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1 | (2) |
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3 | (2) |
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1.3 Algorithm, Numerical Method, Implementation and Simulation |
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5 | (4) |
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9 | (6) |
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15 | (3) |
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1.6 A Hierarchy of Computation |
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18 | (8) |
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1.7 Ramification, Turbulence and the Complexity of Fluid Flows |
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26 | (5) |
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1.8 The Development of Computer Hardware |
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31 | (2) |
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1.9 Some Remarks on Software |
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33 | (2) |
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35 | (34) |
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2.1 Numerical Methods as Mappings |
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35 | (2) |
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2.2 Fixed Points: Stability, Instability and Superstability |
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37 | (3) |
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2.3 Stability of Exact Solution, Implicit and Explicit Approximations |
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40 | (3) |
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43 | (7) |
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2.5 Random Number Generation |
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50 | (2) |
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2.6 Newton's Method in the Complex Plane |
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52 | (4) |
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2.7 Mappings and Fluid Flows |
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56 | (13) |
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3 Ordinary Differential Equations: Initial Value Problem |
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69 | (49) |
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3.1 Some Conventional Wisdom |
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69 | (2) |
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3.2 Explicit Euler and Implicit Euler Schemes |
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71 | (6) |
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77 | (7) |
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3.4 Adams--Bashforth--Moulton Methods |
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84 | (6) |
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3.5 Other Methods and Considerations |
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90 | (2) |
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3.6 Bashforth's Problem: Sessile Drop on a Flat Plate |
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92 | (3) |
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3.7 Flow Due to a Collapsing Bubble |
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95 | (8) |
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3.8 Motion of a Solid in Ideal Fluid |
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103 | (3) |
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3.9 The Point-Vortex Equations |
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106 | (5) |
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3.10 Vortex Sheet Roll-up |
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111 | (7) |
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118 | (48) |
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4.1 Forward, Backward and Central Difference |
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119 | (8) |
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4.2 Matrix Derivative Operators |
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127 | (11) |
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138 | (3) |
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4.4 Non-uniform Discretization |
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141 | (3) |
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4.5 Numerical Interpolation |
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144 | (5) |
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4.6 Numerical Integration |
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149 | (17) |
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5 Boundary Value and Eigenvalue ODEs |
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166 | (44) |
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5.1 Linear Boundary Value Problems |
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167 | (8) |
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5.2 Nonlinear Boundary Value Problems |
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175 | (9) |
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5.3 Boundary Value Problems in Viscous Flow |
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184 | (4) |
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188 | (4) |
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5.5 Hydrodynamic Instability |
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192 | (18) |
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6 Methods Based on Functional Expansions |
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210 | (34) |
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210 | (1) |
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6.2 Fourier Approximation |
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211 | (7) |
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6.3 Polynomial Approximation |
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218 | (8) |
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6.4 Galerkin, Tau, Collocation and Pseudo-spectral Methods |
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226 | (14) |
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240 | (4) |
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7 Partial Differential Equations |
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244 | (49) |
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7.1 Definitions and Preliminaries |
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244 | (6) |
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7.2 The Advection Equation |
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250 | (9) |
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7.3 The Diffusion Equation |
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259 | (3) |
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7.4 The Advection--Diffusion Equation |
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262 | (5) |
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267 | (3) |
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7.6 More on Stability: Non-periodic Boundary Conditions |
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270 | (9) |
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279 | (4) |
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7.8 Implicit Time-differencing |
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283 | (4) |
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7.9 Direct Solution with Matrix Representation |
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287 | (6) |
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8 Multi-dimensional Partial Differential Equations |
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293 | (82) |
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293 | (16) |
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8.2 Navier--Stokes Equations |
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309 | (19) |
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8.3 Navier--Stokes Equations in Spectral Form |
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328 | (6) |
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8.4 Finite Volume Formulation |
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334 | (3) |
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8.5 CFD for Complex Geometries |
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337 | (19) |
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8.6 Sharp Interface Cartesian Grid Method |
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356 | (4) |
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8.7 Immersed Boundary Method |
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360 | (15) |
References |
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375 | (15) |
Index |
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390 | |