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| Preface to the Second Edition | |
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| Preface to the First Edition | |
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| Interpolation | |
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| Lagrange Polynomial Interpolation | |
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| Cubic Spline Interpolation | |
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| Exercises | |
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| Further Reading | |
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| Numerical Differentiation - Finite Differences | |
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| Construction of Difference Formulas Using Taylor Series | |
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| A General Technique for Construction of Finite Difference Schemes | |
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| An Alternative Measure for the Accuracy of Finite Differences | |
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| Pad� Approximations | |
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| Non-Uniform Grids | |
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| Exercises | |
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| Further Reading | |
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| Numerical Integration | |
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| Trapezoidal and Simpson's Rules | |
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| Error Analysis | |
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| Trapezoidal Rule with End-Correction | |
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| Romberg Integration and Richardson Extrapolation | |
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| Adaptive Quadrature | |
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| Gauss Quadrature | |
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| Exercises | |
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| Further Reading | |
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| Numerical Solution of Ordinary Differential Equations | |
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| Initial Value Problems | |
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| Numerical Stability | |
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| Stability Analysis for the Euler Method | |
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| Implicit or Backward Euler | |
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| Numerical Accuracy Revisited | |
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| Trapezoidal Method | |
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| Linearization for Implicit Methods | |
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| Runge-Kutta Methods | |
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| Multi-Step Methods | |
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| System of First-Order Ordinary Differential Equations | |
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| Boundary Value Problems | |
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| Shooting Method | |
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| Direct Methods | |
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| Exercises | |
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| Further Reading | |
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| Numerical Solution of Partial Differential Equations | |
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| Semi-Discretization | |
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| von Neumann Stability Analysis | |
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| Modified Wavenumber Analysis | |
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| Implicit Time Advancement | |
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| Accuracy via Modified Equation | |
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| Du Fort-Frankel Method: An Inconsistent Scheme | |
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| Multi-Dimensions | |
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| Implicit Methods in Higher Dimensions | |
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| Approximate Factorization | |
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| Stability of the Factored Scheme | |
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| Alternating Direction Implicit Methods | |
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| Mixed and Fractional Step Methods | |
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| Elliptic Partial Differential Equations | |
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| Iterative Solution Methods | |
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| The Point Jacobi Method | |
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| Gauss-Seidel Method | |
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| Successive Over Relaxation Scheme | |
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| Multigrid Acceleration | |
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| Exercises | |
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| Further Reading | |
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| Discrete Transform Methods | |
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| Fourier Series | |
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| Discrete Fourier Series | |
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| Fast Fourier Transform | |
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| Fourier Transform of a Real Function | |
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| Discrete Fourier Series in Higher Dimensions | |
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| Discrete Fourier Transform of a Product of Two Functions | |
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| Discrete Sine and Cosine Transforms | |
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| Applications of Discrete Fourier Series | |
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| Direct Solution of Finite Differenced Elliptic Equations | |
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| Differentiation of a Periodic Function Using Fourier Spectral Method | |
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| Numerical Solution of Linear, Constant Coefficient Differential Equations with Periodic Boundary Conditions | |
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| Matrix Operator for Fourier Spectral Numerical Differentiation | |
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| Discrete Chebyshev Transform and Applications | |
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| Numerical Differentiation Using Chebyshev Polynomials | |
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| Quadrature Using Chebyshev Polynomials | |
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| Matrix Form of Chebyshev Collocation Derivative | |
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| Method of Weighted Residuals | |
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| The Finite Element Method | |
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| Application of the Finite Element Method to a Boundary Value Problem | |
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| Comparison with Finite Difference Method | |
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| Comparison with a Pad� Scheme | |
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| A Time-Dependent Problem | |
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| Application to Complex Domains | |
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| Constructing the Basis Functions | |
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| Exercises | |
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| Further Reading | |
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| A Review of Linear Algebra | |
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| Vectors, Matrices and Elementary Operations | |
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| System of Linear Algebraic Equations | |
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| Effects of Round-off Error | |
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| Operations Counts | |
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| Eigenvalues and Eigenvectors | |
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| Index | |