1. The basic equations if fluid 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 finite 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 convection-diffusion 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 Peclet-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
maximim 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
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 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
coordinates ............................................. 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 flow .................................... 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