Contents......Page 5
Preface......Page 7
1.1 The Nature of CFD......Page 11
1.2 Overview of the Book......Page 13
1.3 Algorithm, Numerical Method, Implementation and Simulation......Page 15
1.4 Models and Methods......Page 19
1.5 Round-off Error......Page 25
1.6 A Hierarchy of Computation......Page 28
1.7 Ramification, Turbulence and the Complexity of Fluid Flows......Page 36
1.8 The Development of Computer Hardware......Page 41
1.9 Some Remarks on Software......Page 43
2.1 Numerical Methods as Mappings......Page 45
2.2 Fixed Points: Stability, Instability and Superstability......Page 47
2.3 Stability of Exact Solution, Implicit and Explicit Approximations......Page 50
2.4 More on Mappings......Page 53
2.5 Random Number Generation......Page 60
2.6 Newton’s Method in the Complex Plane......Page 62
2.7 Mappings and Fluid Flows......Page 66
3.1 Some Conventional Wisdom......Page 79
3.2 Explicit Euler and Implicit Euler Schemes......Page 81
3.3 Runge–Kutta Methods......Page 87
3.4 Adams–Bashforth–Moulton Methods......Page 94
3.5 Other Methods and Considerations......Page 100
3.6 Bashforth’s Problem: Sessile Drop on a Flat Plate......Page 102
3.7 Flow Due to a Collapsing Bubble......Page 105
3.8 Motion of a Solid in Ideal Fluid......Page 113
3.9 The Point-Vortex Equations......Page 116
3.10 Vortex Sheet Roll-up......Page 121
4 Spatial Discretization......Page 128
4.1 Forward, Backward and Central Difference......Page 129
4.2 Matrix Derivative Operators......Page 137
4.3 Compact Differences......Page 148
4.4 Non-uniform Discretization......Page 151
4.5 Numerical Interpolation......Page 154
4.6 Numerical Integration......Page 159
5 Boundary Value and Eigenvalue ODEs......Page 176
5.1 Linear Boundary Value Problems......Page 177
5.2 Nonlinear Boundary Value Problems......Page 185
5.3 Boundary Value Problems in Viscous Flow......Page 194
5.4 Eigenvalue Problems......Page 198
5.5 Hydrodynamic Instability......Page 202
6.1 Introduction......Page 220
6.2 Fourier Approximation......Page 221
6.3 Polynomial Approximation......Page 228
6.4 Galerkin, Tau, Collocation and Pseudo-spectral Methods......Page 236
6.5 Some Examples......Page 250
7.1 Definitions and Preliminaries......Page 254
7.2 The Advection Equation......Page 260
7.3 The Diffusion Equation......Page 269
7.4 The Advection–Diffusion Equation......Page 272
7.5 Godunov’s Theorem......Page 277
7.6 More on Stability: Non-periodic Boundary Conditions......Page 280
7.7 Burgers’ Equation......Page 289
7.8 Implicit Time-differencing......Page 293
7.9 Direct Solution with Matrix Representation......Page 297
8.1 Multi-dimensions......Page 303
8.2 Navier–Stokes Equations......Page 319
8.3 Navier–Stokes Equations in Spectral Form......Page 338
8.4 Finite Volume Formulation......Page 344
8.5 CFD for Complex Geometries......Page 347
8.6 Sharp Interface Cartesian Grid Method......Page 366
8.7 Immersed Boundary Method......Page 370
References......Page 385
Index......Page 400