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What is included with this book?
Preface to the Second Edition | p. ix |
Preface to the First Edition | p. xi |
Interpolation | p. 1 |
Lagrange Polynomial Interpolation | p. 1 |
Cubic Spline Interpolation | p. 4 |
Exercises | p. 8 |
Further Reading | p. 12 |
Numerical Differentiation - Finite Differences | p. 13 |
Construction of Difference Formulas Using Taylor Series | p. 13 |
A General Technique for Construction of Finite Difference Schemes | p. 15 |
An Alternative Measure for the Accuracy of Finite Differences | p. 17 |
Padé Approximations | p. 20 |
Non-Uniform Grids | p. 23 |
Exercises | p. 25 |
Further Reading | p. 29 |
Numerical Integration | p. 30 |
Trapezoidal and Simpson's Rules | p. 30 |
Error Analysis | p. 31 |
Trapezoidal Rule with End-Correction | p. 34 |
Romberg Integration and Richardson Extrapolation | p. 35 |
Adaptive Quadrature | p. 37 |
Gauss Quadrature | p. 40 |
Exercises | p. 44 |
Further Reading | p. 47 |
Numerical Solution of Ordinary Differential Equations | p. 48 |
Initial Value Problems | p. 48 |
Numerical Stability | p. 50 |
Stability Analysis for the Euler Method | p. 52 |
Implicit or Backward Euler | p. 55 |
Numerical Accuracy Revisited | p. 56 |
Trapezoidal Method | p. 58 |
Linearization for Implicit Methods | p. 62 |
Runge-Kutta Methods | p. 64 |
Multi-Step Methods | p. 70 |
System of First-Order Ordinary Differential Equations | p. 74 |
Boundary Value Problems | p. 78 |
Shooting Method | p. 79 |
Direct Methods | p. 82 |
Exercises | p. 84 |
Further Reading | p. 100 |
Numerical Solution of Partial Differential Equations | p. 101 |
Semi-Discretization | p. 102 |
von Neumann Stability Analysis | p. 109 |
Modified Wavenumber Analysis | p. 111 |
Implicit Time Advancement | p. 116 |
Accuracy via Modified Equation | p. ll9 |
Du Fort-Frankel Method: An Inconsistent Scheme | p. 121 |
Multi-Dimensions | p. 124 |
Implicit Methods in Higher Dimensions | p. 126 |
Approximate Factorization | p. 128 |
Stability of the Factored Scheme | p. 133 |
Alternating Direction Implicit Methods | p. 134 |
Mixed and Fractional Step Methods | p. 136 |
Elliptic Partial Differential Equations | p. 137 |
Iterative Solution Methods | p. 140 |
The Point Jacobi Method | p. l41 |
Gauss-Seidel Method | p. 143 |
Successive Over Relaxation Scheme | p. 144 |
Multigrid Acceleration | p. 147 |
Exercises | p. 154 |
Further Reading | p. 166 |
Discrete Transform Methods | p. 167 |
Fourier Series | p. 167 |
Discrete Fourier Series | p. 168 |
Fast Fourier Transform | p. 169 |
Fourier Transform of a Real Function | p. 170 |
Discrete Fourier Series in Higher Dimensions | p. 172 |
Discrete Fourier Transform of a Product of Two Functions | p. 173 |
Discrete Sine and Cosine Transforms | p. 175 |
Applications of Discrete Fourier Series | p. 176 |
Direct Solution of Finite Differenced Elliptic Equations | p. 176 |
Differentiation of a Periodic Function Using Fourier Spectral Method | p. 180 |
Numerical Solution of Linear, Constant Coefficient Differential Equations with Periodic Boundary Conditions | p. 182 |
Matrix Operator for Fourier Spectral Numerical Differentiation | p. 185 |
Discrete Chebyshev Transform and Applications | p. 188 |
Numerical Differentiation Using Chebyshev Polynomials | p. 192 |
Quadrature Using Chebyshev Polynomials | p. 195 |
Matrix Form of Chebyshev Collocation Derivative | p. 196 |
Method of Weighted Residuals | p. 200 |
The Finite Element Method | p. 201 |
Application of the Finite Element Method to a Boundary Value Problem | p. 202 |
Comparison with Finite Difference Method | p. 207 |
Comparison with a Padé Scheme | p. 209 |
A Time-Dependent Problem | p. 210 |
Application to Complex Domains | p. 213 |
Constructing the Basis Functions | p. 215 |
Exercises | p. 221 |
Further Reading | p. 226 |
A Review of Linear Algebra | p. 227 |
Vectors, Matrices and Elementary Operations | p. 227 |
System of Linear Algebraic Equations | p. 230 |
Effects of Round-off Error | p. 230 |
Operations Counts | p. 231 |
Eigenvalues and Eigenvectors | p. 232 |
Index | p. 235 |
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