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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 | |