Preface
Preface to the First Edition
Contents
1 Numerical Approximation of Model Partial Differential Equations
1.1 Discrete Integration Methods for Ordinary Differential Equations
1.1.1 Construction of Numerical Integration Schemes
1.1.2 General Form of Numerical Schemes
1.1.3 Application to the Absorption Equation
1.1.4 Stability of a Numerical Scheme
1.2 Model Partial Differential Equations
1.2.1 The Convection Equation
1.2.2 The Wave Equation
1.2.3 The Heat Equation
1.3 Solutions and Programs
2 Nonlinear Differential Equations: Application to Chemical Kinetics
2.1 Physical Problem and Mathematical Modeling
2.2 Stability of the System
2.3 Model for the Maintained Reaction
2.3.1 Existence of a Critical Point and Stability
2.3.2 Numerical Solution
2.4 Model of Reaction with a Delay Term
2.5 Solutions and Programs
3 Fourier Approximation
3.1 Introduction
3.2 Fourier Series
3.3 Trigonometric Interpolation
3.4 L2-Approximation
3.5 The Dirichlet kernel
3.6 The Fast Fourier Transform
3.7 Further Reading
3.8 Solutions and Programs
Chapter References
4 Polynomial Approximation
4.1 Introduction
4.2 Interpolation
4.2.1 Lagrange Interpolation
4.2.2 Hermite Interpolation
4.3 Best Polynomial Approximation
4.3.1 Best Uniform Approximation
4.3.2 Best Hilbertian Approximation
4.4 Piecewise Polynomial Approximation
4.5 Further Reading
4.6 Solutions and Programs
Chapter References
5 Solving an Advection–Diffusion Equation by a Finite Element Method
5.1 Variational Formulation of the Problem
5.2 A P1 Finite Element Method
5.3 A P2 Finite Element Method
5.4 A Stabilization Method
5.4.1 Computation of the Solution at the Endpoints of the Intervals
5.4.2 Analysis of the Stabilized Method
5.5 The Case of a Variable Source Term
5.6 Solutions and Programs
Chapter References
6 Solving a Differential Equation by a Legendre Spectral Method
6.1 Some Properties of the Legendre Polynomials
6.2 Gauss–Legendre Quadrature
6.3 Legendre series expansions
6.4 A Spectral Discretization
6.5 Extensions to other orthogonal polynomials
6.6 Solutions and Programs
Chapter References
7 High-order finite difference methods
7.1 Taylor series expansion and finite difference grid
7.2 Explicit finite difference schemes
7.3 Implicit finite difference schemes
7.4 Compact finite difference schemes
7.5 Boundary conditions
7.5.1 Periodic Boundary Conditions
7.5.2 Non-Periodic Boundary Conditions
7.6 Summary of FD schemes
7.7 Test of high-order FD schemes
7.8 Solving boundary-value problems with high-order FD schemes
7.9 Solutions and programs
Chapter References
8 Signal Processing: Multiresolution Analysis
8.1 Introduction
8.2 Approximation of a Function: Theoretical Aspect
8.2.1 Piecewise Constant Functions
8.2.2 Decomposition of the Space VJ
8.2.3 Decomposition and Reconstruction Algorithms
8.2.4 Importance of Multiresolution Analysis
8.3 Multiresolution Analysis: Practical Aspect
8.4 Multiresolution Analysis: Implementation
8.5 Introduction to Wavelet Theory
8.5.1 Scaling Functions and Wavelets
8.5.2 The Schauder Wavelet
8.5.3 Implementation of the Schauder Wavelet
8.5.4 The Daubechies Wavelet
8.5.5 Implementation of the Daubechies Wavelet D4
8.6 Generalization: Image Processing
8.6.1 Image Processing: Implementation
8.7 Solutions and Programs
Chapter References
9 Elasticity: Elastic Deformation of a Thin Plate
9.1 Introduction
9.2 Modeling Elastic Deformations (Linear Problem)
9.3 Modeling Electrostatic Forces (Nonlinear Problem)
9.4 Numerical Discretization of the Problem
9.5 Programming Tips
9.5.1 Modular Programming
9.5.2 Program Validation
9.6 Solving the Linear Problem
9.7 Solving the Nonlinear Problem
9.7.1 A Fixed-Point Algorithm
9.7.2 Numerical Solution
9.8 Solutions and Programs
9.8.1 Solution of Exercise 9.1
9.8.2 Solution of Exercise 9.2
9.8.3 Further Comments
Chapter References
10 Domain Decomposition Using a Schwarz Method
10.1 Principle and Application Field of Domain Decomposition
10.2 1D Finite Difference Solution
10.3 Schwarz Method in One Dimension
10.3.1 Discretization
10.4 Extension to the 2D Case
10.4.1 Finite Difference Solution
10.4.2 Domain Decomposition in the 2D Case
10.4.3 Implementation of Realistic Boundary Conditions
10.4.4 Possible Extensions
10.5 Solutions and Programs
Chapter References
11 Geometrical Design: Bézier Curves and Surfaces
11.1 Introduction
11.2 Bézier Curves
11.3 Basic Properties of Bézier Curves
11.3.1 Convex Hull of the Control Points
11.3.2 Multiple Control Points
11.3.3 Tangent Vector to a Bézier Curve
11.3.4 Junction of Bézier Curves
11.3.5 Generation of the Point P(t)
11.4 Generation of Bézier Curves
11.5 Splitting Bézier Curves
11.6 Intersection of Bézier Curves
11.6.1 Implementation
11.7 Bézier Surfaces
11.8 Basic properties of Bézier Surfaces
11.8.1 Convex Hull
11.8.2 Tangent Vector
11.8.3 Junction of Bézier Patches
11.8.4 Construction of the Point P(t)
11.9 Construction of Bézier Surfaces
11.10 Solutions and Programs
11.10.1 Solution of Exercise 11.1
11.10.2 Solution of Exercise 11.2
11.10.3 Solution of Exercise 11.3
Chapter References
12 Gas Dynamics: The Riemann Problem and Discontinuous Solutions: Application to the Shock Tube Problem
12.1 Physical Description of the Shock Tube Problem
12.2 Euler Equations of Gas Dynamics
12.2.1 Dimensionless Equations
12.2.2 Exact Solution
12.3 Numerical Solution
12.3.1 Lax–Wendroff and MacCormack Centered Schemes
12.3.2 Upwind Schemes (Roe's Approximate Solver)
12.4 Solutions and Programs
Chapter References
13 Optimization Applied to Model Fitting
13.1 Principle and examples of continuous optimization
13.2 Mathematical overview in the case J:mathbbRnrightarrowmathbbR
13.3 Numerical methods
13.3.1 Descent algorithms
13.3.2 Newton-like algorithms
13.4 Application to fitting of model parameters
13.4.1 General Least-Square problem
13.4.2 Epidemic modeling
13.5 Solutions and Programs
Chapter References
14 Thermal Engineering: Optimization of an Industrial Furnace
14.1 Introduction
14.2 Formulation of the Problem
14.3 Finite Element Discretization
14.4 Implementation
14.4.1 Matrix Computation
14.4.2 Right-Hand-Side Computation
14.4.3 The Linear System
14.5 Boundary Conditions
14.5.1 Modular Implementation
14.5.2 Numerical Solution of the Problem
14.6 Inverse Problem Formulation
14.7 Implementation of the Inverse Problem
14.8 Solutions and Programs
14.8.1 Solution of Exercise 14.1
14.8.2 Solution of Exercise 14.2
14.8.3 Further Comments
Chapter References
15 Fluid Dynamics: Solving the 2D Navier–Stokes Equations
15.1 Introduction
15.2 The Incompressible Navier–Stokes Equations
15.3 Numerical Algorithm
15.4 Computational Domain, Staggered Grids, and Boundary Conditions
15.5 Finite Difference Discretization
15.6 Flow Visualization
15.7 Initial Condition
15.8 Step-by-Step Implementation
15.8.1 Solving a Linear System with Tridiagonal, Periodic Matrix
15.8.2 Solving the Unsteady Heat Equation
15.8.3 Solving the Steady Heat Equation Using FFTs
15.8.4 Solving the 2D Navier–Stokes Equations
15.9 Solutions and Programs
Chapter References
Index
Index
Index of Programs
Lake Index