Preface to the Second Edition |
|
ix | |
Preface to the First Edition |
|
xi | |
|
|
1 | (12) |
|
1.1 Lagrange Polynomial Interpolation |
|
|
1 | (3) |
|
1.2 Cubic Spline Interpolation |
|
|
4 | (4) |
|
|
8 | (4) |
|
|
12 | (1) |
|
2 Numerical Differentiation - Finite Differences |
|
|
13 | (17) |
|
2.1 Construction of Difference Formulas Using Taylor Series |
|
|
13 | (2) |
|
2.2 A General Technique for Construction of Finite Difference Schemes |
|
|
15 | (2) |
|
2.3 An Alternative Measure for the Accuracy of Finite Differences |
|
|
17 | (3) |
|
|
20 | (3) |
|
|
23 | (2) |
|
|
25 | (4) |
|
|
29 | (1) |
|
|
30 | (18) |
|
3.1 Trapezoidal and Simpson's Rules |
|
|
30 | (1) |
|
|
31 | (3) |
|
3.3 Trapezoidal Rule with End-Correction |
|
|
34 | (1) |
|
3.4 Romberg Integration and Richardson Extrapolation |
|
|
35 | (2) |
|
|
37 | (3) |
|
|
40 | (4) |
|
|
44 | (3) |
|
|
47 | (1) |
|
4 Numerical Solution of Ordinary Differential Equations |
|
|
48 | (53) |
|
4.1 Initial Value Problems |
|
|
48 | (2) |
|
|
50 | (2) |
|
4.3 Stability Analysis for the Euler Method |
|
|
52 | (3) |
|
4.4 Implicit or Backward Euler |
|
|
55 | (1) |
|
4.5 Numerical Accuracy Revisited |
|
|
56 | (2) |
|
|
58 | (4) |
|
4.7 Linearization for Implicit Methods |
|
|
62 | (2) |
|
|
64 | (6) |
|
|
70 | (4) |
|
4.10 System of First-Order Ordinary Differential Equations |
|
|
74 | (4) |
|
4.11 Boundary Value Problems |
|
|
78 | (6) |
|
|
79 | (3) |
|
|
82 | (2) |
|
|
84 | (16) |
|
|
100 | (1) |
|
5 Numerical Solution of Partial Differential Equations |
|
|
101 | (66) |
|
|
102 | (7) |
|
5.2 von Neumann Stability Analysis |
|
|
109 | (2) |
|
5.3 Modified Wavenumber Analysis |
|
|
111 | (5) |
|
5.4 Implicit Time Advancement |
|
|
116 | (3) |
|
5.5 Accuracy via Modified Equation |
|
|
119 | (2) |
|
5.6 Du Fort-Frankel Method: An Inconsistent Scheme |
|
|
121 | (3) |
|
|
124 | (2) |
|
5.8 Implicit Methods in Higher Dimensions |
|
|
126 | (2) |
|
5.9 Approximate Factorization |
|
|
128 | (9) |
|
5.9.1 Stability of the Factored Scheme |
|
|
133 | (1) |
|
5.9.2 Alternating Direction Implicit Methods |
|
|
134 | (2) |
|
5.9.3 Mixed and Fractional Step Methods |
|
|
136 | (1) |
|
5.10 Elliptic Partial Differential Equations |
|
|
137 | (17) |
|
5.10.1 Iterative Solution Methods |
|
|
140 | (1) |
|
5.10.2 The Point Jacobi Method |
|
|
141 | (2) |
|
5.10.3 Gauss-Seidel Method |
|
|
143 | (1) |
|
5.10.4 Successive Over Relaxation Scheme |
|
|
144 | (3) |
|
5.10.5 Multigrid Acceleration |
|
|
147 | (7) |
|
|
154 | (12) |
|
|
166 | (1) |
|
6 Discrete Transform Methods |
|
|
167 | (60) |
|
|
167 | (9) |
|
6.1.1 Discrete Fourier Series |
|
|
168 | (1) |
|
6.1.2 Fast Fourier Transform |
|
|
169 | (1) |
|
6.1.3 Fourier Transform of a Real Function |
|
|
170 | (2) |
|
6.1.4 Discrete Fourier Series in Higher Dimensions |
|
|
172 | (1) |
|
6.1.5 Discrete Fourier Transform of a Product of Two Functions |
|
|
173 | (2) |
|
6.1.6 Discrete Sine and Cosine Transforms |
|
|
175 | (1) |
|
6.2 Applications of Discrete Fourier Series |
|
|
176 | (9) |
|
6.2.1 Direct Solution of Finite Differenced Elliptic Equations |
|
|
176 | (4) |
|
6.2.2 Differentiation of a Periodic Function Using Fourier Spectral Method |
|
|
180 | (2) |
|
6.2.3 Numerical Solution of Linear, Constant Coefficient Differential Equations with Periodic Boundary Conditions |
|
|
182 | (3) |
|
6.3 Matrix Operator for Fourier Spectral Numerical Differentiation |
|
|
185 | (3) |
|
6.4 Discrete Chebyshev Transform and Applications |
|
|
188 | (12) |
|
6.4.1 Numerical Differentiation Using Chebyshev Polynomials |
|
|
192 | (3) |
|
6.4.2 Quadrature Using Chebyshev Polynomials |
|
|
195 | (1) |
|
6.4.3 Matrix Form of Chebyshev Collocation Derivative |
|
|
196 | (4) |
|
6.5 Method of Weighted Residuals |
|
|
200 | (1) |
|
6.6 The Finite Element Method |
|
|
201 | (12) |
|
6.6.1 Application of the Finite Element Method to a Boundary Value Problem |
|
|
202 | (5) |
|
6.6.2 Comparison with Finite Difference Method |
|
|
207 | (2) |
|
6.6.3 Comparison with a Pade Scheme |
|
|
209 | (1) |
|
6.6.4 A Time-Dependent Problem |
|
|
210 | (3) |
|
6.7 Application to Complex Domains |
|
|
213 | (8) |
|
6.7.1 Constructing the Basis Functions |
|
|
215 | (6) |
|
|
221 | (5) |
|
|
226 | (1) |
|
A A Review of Linear Algebra |
|
|
227 | (8) |
|
A.1 Vectors, Matrices and Elementary Operations |
|
|
227 | (3) |
|
A.2 System of Linear Algebraic Equations |
|
|
230 | (1) |
|
A.2.1 Effects of Round-off Error |
|
|
230 | (1) |
|
|
231 | (1) |
|
A.4 Eigenvalues and Eigenvectors |
|
|
232 | (3) |
Index |
|
235 | |