Think Before You Compute: A Prelude to Computational Fluid Dynamics

Contents
Preface
PART I A FIRST PROBLEM
	1 The driven cavity
		1.1 The problem
		1.2 Know your physics
		1.3 Know your PDEs
		1.4 Special physics of the corner
		1.5 Nondimensionalisation
		1.6 Steady vs transient calculations
		1.7 Pressure!
	2 Streamfunction-vorticity formulation
		2.1 Formulation
		2.2 Finite differences (simple)
		2.3 Poisson problem
		2.4 Test the code
		2.5 Code quality
		2.6 Simple graphs
		2.7 Vorticity evolution
		2.8 Time-step instability
		2.9 Accuracy consistency
		2.10 Results
	3 Primitive variable formulation
		3.1 Formulation
		3.2 Pressure equation
		3.3 Algorithm 1 with pressure equation
		3.4 Incompressibility as a constraint: split time-step
		3.5 Algorithm 2 by projection – spurious pressure modes
		3.6 Algorithm 3 with a staggered grid
		3.7 Results from algorithm 3
PART II GENERALITIES
	4 Finite differences
		4.1 Higher orders
			4.1.1 Central differencing
			4.1.2 One-sided differencing
			4.1.3 Nonequispaced points
		4.2 Compact fourth-order Poisson solver
			4.2.1 One-dimensional version
			4.2.2 Two dimensions
		4.3 Upwinding
		4.4 Other grids
		4.5 Conservative schemes
	5 Finite elements
		5.1 The two ideas
		5.2 Representations in one dimension
			5.2.1 Constant elements
			5.2.2 Linear elements
			5.2.3 Quadratic elements
			5.2.4 Cubic elements
			5.2.5 Basis functions
		5.3 Representations in two dimensions
			5.3.1 Constant elements
			5.3.2 Linear elements
			5.3.3 Quadratic elements
			5.3.4 Cubic elements
			5.3.5 Basis functions
			5.3.6 Rectangles
		5.4 Variational statement of the Poisson problem
		5.5 Details in one dimension
		5.6 Details in two dimensions
		5.7 Galerkin formulation
		5.8 Diffusion equation
			5.8.1 Weak formulation
			5.8.2 In one dimension
			5.8.3 In two dimensions
		5.9 Navier–Stokes equation
			5.9.1 Weak formulation
			5.9.2 Time integration
			5.9.3 Pressure problem – locking
			5.9.4 Pressure problem – spurious modes
	6 Spectral methods
		6.1 The two ideas
		6.2 Global vs local
		6.3 Choice of spectral basis functions
		6.4 Chebyshev polynomials
		6.5 Rates of convergence
		6.6 Gibbs phenomenon
		6.7 Discrete Fourier Transform
		6.8 Aliasing
		6.9 Fast Fourier Transform (FFT)
		6.10 Differential matrix
		6.11 Navier–Stokes
		6.12 Bridging the gap
	7 Time integration
		7.1 Stability
		7.2 Forward Euler
		7.3 Backward Euler
		7.4 Midpoint Euler
		7.5 Crank–Nicolson
		7.6 Leapfrog
		7.7 Runge–Kutta
		7.8 Multistep methods
		7.9 Symplectic integrators
		7.10 Navier–Stokes
	8 Linear algebra
		8.1 Lapack
		8.2 Gaussian elimination
			8.2.1 Pivoting
			8.2.2 LU decomposition
			8.2.3 Errors
		8.3 QR decomposition
			8.3.1 QR by Gram–Schmidt
			8.3.2 QR by Givens rotations
			8.3.3 QR by Householder reflections
		8.4 Sparse matrices
		8.5 Conjugate gradients
		8.6 Eigenproblems
		8.7 Power iteration
		8.8 Jacobi
		8.9 Main method
PART III SPECIAL TOPICS
9 Software packages and FreeFem++
	9.1 Poisson problem
	9.2 Driven cavity
10 Hyperbolic equations
	10.1 Simplest, but unstable
	10.2 Lax–Friedricks, too stable
	10.3 Upwinding
	10.4 Crank–Nicolson
	10.5 Lax–Wendroff
	10.6 Angled Derivative
	10.7 Propagation of discontinuities
	10.8 Flux limiters
	10.9 Nonlinear advection
	10.10 Godunov method
11 Representation of surfaces
	11.1 Curves in two dimensions
		11.1.1 Splines
	11.2 Surfaces in three dimensions
		11.2.1 Redistributing points
		11.2.2 Curvature
	11.3 Volume of Fluid (VoF) method
	11.4 Diffuse interface method
	11.5 Level sets
		11.5.1 Fast Marching Method
12 Boundary integral method
	12.1 Integral equation for Laplace equation
		12.1.1 Greens functions
		12.1.2 Eigensolutions
		12.1.3 Singular integrals
	12.2 Discretisation
		12.2.1 Evaluation of the matrix elements
		12.2.2 Avoiding the eigensolution
		12.2.3 Tests
		12.2.4 Costs
	12.3 Free-surface potential flows
	12.4 Stokes flows
13 Fast Poisson solvers
	13.1 Multigrid method
		13.1.1 A V-cycle
		13.1.2 Accuracy and costs
	13.2 Fast Fourier Transforms
	13.3 Domain decomposition
		13.3.1 Costs
14 Fast Multipole Method
	14.1 Trees, roots and leaves
	14.2 Barnes–Hut algorithm
	14.3 Fast Multipole algorithm
		14.3.1 Upward pass
		14.3.2 Downward pass
		14.3.3 Errors
		14.3.4 Costs
15 Nonlinear considerations
	15.1 Finding steady states
		15.1.1 Finding the Jacobian
		15.1.2 Example of the limit cycle of the Van der Pol oscillator
	15.2 Parameter continuation
	15.3 Searching for singularities of physical problems
		15.3.1 Use of computer algebra
16 Particle methods
	16.1 Molecular dynamics
	16.2 Lattice Gas
	16.3 Lattice Boltzmann
	16.4 Dissipative particle dynamics
	16.5 Stokesian dynamics
		16.5.1 Hydrodynamic interactions
		16.5.2 Brownian motion
	16.6 Force Coupling Method
	16.7 Granular media simulations
	16.8 Smooth Particle Hydrodynamics
17 Wavelets
	17.1 Continuous Wavelet Transform
	17.2 Discrete Wavelet Transform
	17.3 Fast Wavelet Transform
	17.4 Daubechies wavelets
Index