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| Preface to the second edition | |
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| Preface to the first edition | |
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| Flowchart of contents | |
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| Ordinary differential equations | |
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| Euler's method and beyond | |
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| Ordinary differential equations and the Lipschitz condition | |
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| Euler's method | |
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| The trapezoidal rule | |
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| The theta method | |
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| Comments and bibliography | |
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| Exercises | |
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| Multistep methods | |
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| The Adams method | |
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| Order and convergence of multistep methods | |
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| Backward differentiation formulae | |
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| Comments and bibliography | |
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| Exercises | |
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| Runge-Kutta methods | |
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| Gaussian quadrature | |
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| Explicit Runge-Kutta schemes | |
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| Implicit Runge-Kutta schemes | |
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| Collocation and IRK methods | |
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| Comments and bibliography | |
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| Exercises | |
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| Stiff equations | |
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| What are stiff ODEs? | |
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| The linear stability domain and A-stability | |
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| A-stability of Runge-Kutta methods | |
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| A-stability of multistep methods | |
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| Comments and bibliography | |
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| Exercises | |
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| Geometric numerical integration | |
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| Between quality and quantity | |
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| Monotone equations and algebraic stability | |
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| From quadratic invariants to orthogonal flows | |
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| Hamiltonian systems | |
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| Comments and bibliography | |
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| Exercises | |
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| Error control | |
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| Numerical software vs. numerical mathematics | |
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| The Milne device | |
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| Embedded Runge-Kutta methods | |
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| Comments and bibliography | |
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| Exercises | |
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| Nonlinear algebraic systems | |
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| Functional iteration | |
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| The Newton-Raphson algorithm and its modification | |
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| Starting and stopping the iteration | |
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| Comments and bibliography | |
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| Exercises | |
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| The Poisson equation | |
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| Finite difference schemes | |
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| Finite differences | |
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| The five-point formula for ∇<sup>2</sup>u = f | |
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| Higher-order methods for ∇<sup>2</sup>u = f | |
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| Comments and bibliography | |
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| Exercises | |
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| The finite element method | |
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| Two-point boundary value problems | |
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| A synopsis of FEM theory | |
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| The Poisson equation | |
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| Comments and bibliography | |
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| Exercises | |
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| Spectral methods | |
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| Sparse matrices vs. small matrices | |
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| The algebra of Fourier expansions | |
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| The fast Fourier transform | |
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| Second-order elliptic PDEs | |
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| Chebyshev methods | |
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| Comments and bibliography | |
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| Exercises | |
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| Gaussian elimination for sparse linear equations | |
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| Banded systems | |
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| Graphs of matrices and perfect Cholesky factorization | |
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| Comments and bibliography | |
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| Exercises | |
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| Classical iterative methods for sparse linear equations | |
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| Linear one-step stationary schemes | |
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| Classical iterative methods | |
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| Convergence of successive over-relaxation | |
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| The Poisson equation | |
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| Comments and bibliography | |
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| Exercises | |
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| Multigrid techniques | |
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| In lieu of a justification | |
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| The basic multigrid technique | |
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| The full multigrid technique | |
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| Poisson by multigrid | |
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| Comments and bibliography | |
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| Exercises | |
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| Conjugate gradients | |
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| Steepest, but slow, descent | |
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| The method of conjugate gradients | |
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| Krylov subspaces and preconditioners | |
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| Poisson by conjugate gradients | |
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| Comments and bibliography | |
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| Exercises | |
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| Fast Poisson solvers | |
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| TST matrices and the Hockney method | |
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| Fast Poisson solver in a disc | |
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| Comments and bibliography | |
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| Exercises | |
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| Partial differential equations of evolution | |
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| The diffusion equation | |
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| A simple numerical method | |
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| Order, stability and convergence | |
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| Numerical schemes for the diffusion equation | |
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| Stability analysis I: Eigenvalue techniques | |
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| Stability analysis II: Fourier techniques | |
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| Splitting | |
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| Comments and bibliography | |
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| Exercises | |
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| Hyperbolic equations | |
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| Why the advection equation? | |
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| Finite differences for the advection equation | |
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| The energy method | |
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| The wave equation | |
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| The Burgers equation | |
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| Comments and bibliography | |
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| Exercises | |
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| Appendix Bluffer's guide to useful mathematics | |
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| Linear algebra | |
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| Vector spaces | |
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| Matrices | |
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| Inner products and norms | |
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| Linear systems | |
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| Eigenvalues and eigenvectors | |
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| Bibliography | |
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| Analysis | |
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| Introduction to functional analysis | |
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| Approximation theory | |
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| Ordinary differential equations | |
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| Bibliography | |
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| Index | |