An introduction to fast fourier transform methods for partial differential equations, with applications

목차 (Table of Contents)

• CONTENTS
• Chapter 1. Basic Preliminaries
• 1.1 Introduction = 1
• 1.2 Poisson's equation in a rectangle = 2
• 1.3 Discussion = 6
• 1.4 Summation of complex finite Fourier series = 8
• 1.5 The solution of tridiagonal systems = 14
• 1.6 A numerical example = 18
• Chapter 2. Algorithms
• 2.1 Introduction = 23
• 2.2 The case n ＝ 4 ＝ $$2^2$$ = 23
• 2.3 The case n ＝ $$2^k$$ = 27
• 2.4 Discussion = 30
• 2.5 Some useful definitions and properties = 34
• 2.6 The Fourier transform of two real data sequences using a complex DFT routine = 36
• 2.7 The Fourier transform of 2n (complex) points from two separate n-point transforms (the 'doubling' algorithm) = 38
• 2.8 The Fourier transform of 2n real data points = 40
• 2.9 Calculation of Fourier series for real data = 41
• 2.10 Calculation of cosine series for real data = 42
• 2.11 Calculation of sine series for real data = 45
• Chapter 3. FFT Solution of Partial Differential Equations
• 3.1 Introduction = 49
• 3.2 Discrete representations ; general considerations ; boundary conditions = 51
• 3.3 Two-dimensional second-order problems = 55
• 3.3.1 General form soluble by FFT methods = 55
• 3.3.2 Elliptic equations in cartesian coordinates = 58
• 3.3.3 Parabolic equations = 66
• 3.3.4 Hyperbolic equations = 70
• 3.3.5 Other coordinate systems and grid configurations = 73
• 3.4 Higher order equations = 74
• 3.5 Three-dimensional problems = 75
• 3.5.1 General considerations = 75
• 3.5.2 Poisson's equation in cartesian coordinates = 76
• 3.6 General separable problems = 83
• Chapter 4. Cyclic Reduction
• 4.1 Introduction = 87
• 4.2 The reduction procedure = 88
• 4.3 Matrix factorisation = 94
• 4.4 Stability = 98
• 4.5 Bunemann algorithms = 101
• 4.6 More general cyclic reduction algorithms = 105
• 4.7 The FACR(A) method: optimised reduction = 106
• Chapter 5. Irregular Regions
• 5.1 Introduction = 111
• 5.2 Linear combinations of solutions: unit source method = 113
• 5.3 Discussion = 120
• 5.4 Matrix formulation = 122
• 5.5 Applications using splitting rather than imbedding = 129
• 5.6 Discussion = 132
• Chapter 6. Two methods for more general problems
• 6.1 Introduction = 135
• 6.2 Non-separable elliptic problems = 136
• 6.3 Use of the Laplace transform for linear time-dependent problems = 143
• Appendix 1
• A1.1 Introduction = 153
• A1.2 The Dirichlet problem = 154
• A1.3 The Neumann problem = 155
• A1.4 The periodic problem = 157
• A1.5 The symmetric Dirichlet problem (Dirichlet-Neumann conditions) = 159
• Appendix 2 FFT Fortran computer subroutine = 163
• References = 165
• Index = 176