目录
- Table of Contents
Preface iii
1.The basic equations of fiuid dynamics 1
1.1 Introduction 1
1.2 Vector analysis 5
1.3 The total derivative and the transport theorem 9
1.4 Conservation of mass 12
1.5 Conservation of momentum 13
1.6 Conservation of energy 19
1.7 Thermodynamic aspects 22
1.8 Bernoulli's theorem 26
1.9 Kelvin's circulation theorem and potential flow 28
1.10 The Euler equations 32
1.11 The convection-diffusion equation 33
1.12 Conditions for incompressible flow 34
1.13 Turbulence 37
1.14 Stratified flow and free convection 43
1.15 Moving frame of reference 47
1.16 The shallow-water equations 48
2.Partial differential equations: analytic aspects 53
2.1 Introduction 53
2.2 Classification of partial differential equations 54
2.3 Boundary conditions 61
2.4 Maximum principles 66
2.5 Boundary layer theory 70
3.Finite volume and fìnite difference discretization on
nonuniform grids 81
3.1 Introduction 81
3.2 An elliptic equation 82
3.3 A one-dimensional example 84
3.4 Vertex-centered discretization 88
3.5 Cell-centered discretization , 94
3.6 Upwind discretization 96
3.7 Nonuniform grids in one dimension 99
4.The stationary convection-diffusion equation 111
4.1 Introduction 111
4.2 Finite volume discretization of the stationary convectiondiffusion equation in one dimension 113
4.3 Numerical experiments on locally refined one-dimensional grid 120
4.4 Schemes of positive type 122
4.5 Upwind discretization 126
4.6 Defect correction 129
4.7 Péclet-independent accuracy in two dimensions 133
4.8 More accurate discretization of the convection term 148
5.The nonstationary convection-diffusion equation 163
5.1 Introduction 163
5.2 Example of instability 164
5.3 Stability definitions 166
5 4 The discrete maximum principle 170
5.5 Fourier stability analysis 171
5.6 Principles of von Neumann stability analysis 174
5.7 Useful properties of the symbol 178
5.8 Derivation of von Neumann stability conditions 184
5.9 Numerical experiments 208
5.10 Strong stability 217
6.The incompressible Navier-Stokes equations 227
6.1 Introduction 227
6.2 Equations of motion and boundary conditions 227
6.3 Spatial discretization on colocated grid 232
6.4 Spatial discretization on staggered grid 240
6.5 On the choice of boundary conditions 244
6.6 Temporal discretization on staggered grid 249
6.7 Temporal discretization on colocated grid 261
7.Iterative methods 263
7.1 Introduction 263
7.2 Stationary iterative methods 264
7.3 Krylov subspace methods 270
7.4 Multigrid methods 285
7.5 Fast Poisson solvers 292
7.6 Iterative methods for the incompressible Navier-Stokes equations 293
8.The shallow-water equations 305
8.1 Introduction 305
8.2 The one-dimensional case 305
8.3 The two-dimensional case 323
9.Scalar conservation laws 339
9.1 Introduction 339
9.2 Godunov's order barrier theorem 339
9.3 Linear schemes 346
9.4 Scalar conservation laws 361
10.The Euler equations in one space dimension 397
10.1 Introduction 397
10.2 Analytic aspects 397
10.3 The approximate Riemann solver of Roe 414
10.4 The Osher scheme 425
10.5 Flux splitting schemes 436
10.6 Numerical stability 442
10.7 The Jameson-Schmidt-Turkel scheme 447
10.8 Higher order schemes 456
11.Discretization in general domains 467
11.1 Introduction 467
11.2 Three types of grid 467
11.3 Boundary-fitted grids 470
11.4 Basic geometric properties of grid cells 474
11.5 Introduction to tensor analysis 484
11.5.1 Invariance 485
11.5.2 The geometric quantities 490
11.5.3 Tensor calculus 498
11.5.4 The equations of motion in general coordinates 501
12.Numerical solution of the Euler equations in general domains 503
12.1 Introduction 503
12.2 Analytic aspects 503
12.3 Cell-centered finite volume discretization on boundary-fitted grids 511
12.4 Numerical boundary conditions 518
12.5 Temporal discretization 525
13.Numerical solution of the Navier-Stokes equations in general domains 531
13.1 Introduction 531
13.2 Analytic aspects 531
13.3 Colocated scheme for the compressible Navier-Stokes equations 533
13.4 Colocated scheme for the incompressible Navier-Stokes equations 535
13.5 Staggered scheme for the incompressible Navier-Stokes equations 538
13.6 An application 557
13.7 Verification and validation 559
14.Unified methods for computing incompressible and compressible fiow 567
14.1 The need for unified methods 567
14.2 Difficulties with the zero Mach number limit 568
14.3 Preconditioning 571
14.4 Mach-uniform dimensionless Euler equations 578
14.5 A staggered scheme for fully compressible flow 583
14.6 Unified schemes for incompressible and compressible flow 589
References 603
Index 633
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