A first course in the numerical analysis of differential equations / Arieh Iserles.

Includes bibliographical references and index.

Preface to the second editionp. ix Preface to the first editionp. xiii Flowchart of contentsp. xix Ordinary differential equationsp. 1 Euler's method and beyondp. 3 Ordinary differential equations and the Lipschitz conditionp. 3 Euler's methodp. 4 The trapezoidal rulep. 8 The theta methodp. 13 Comments and bibliographyp. 15 Exercisesp. 16 Multistep methodsp. 19 The Adams methodp. 19 Order and convergence of multistep methodsp. 21 Backward differentiation formulaep. 26 Comments and bibliographyp. 28 Exercisesp. 31 Runge-Kutta methodsp. 33 Gaussian quadraturep. 33 Explicit Runge-Kutta schemesp. 38 Implicit Runge-Kutta schemesp. 41 Collocation and IRK methodsp. 43 Comments and bibliographyp. 48 Exercisesp. 50 Stiff equationsp. 53 What are stiff ODEs?p. 53 The linear stability domain and A-stabilityp. 56 A-stability of Runge-Kutta methodsp. 59 A-stability of multistep methodsp. 63 Comments and bibliographyp. 68 Exercisesp. 70 Geometric numerical integrationp. 73 Between quality and quantityp. 73 Monotone equations and algebraic stabilityp. 77 From quadratic invariants to orthogonal flowsp. 83 Hamiltonian systemsp. 87 Comments and bibliographyp. 95 Exercisesp. 99 Error controlp. 105 Numerical software vs. numerical mathematicsp. 105 The Milne devicep. 107 Embedded Runge-Kutta methodsp. 113 Comments and bibliographyp. 119 Exercisesp. 121 Nonlinear algebraic systemsp. 123 Functional iterationp. 123 The Newton-Raphson algorithm and its modificationp. 127 Starting and stopping the iterationp. 130 Comments and bibliographyp. 132 Exercisesp. 133 The Poisson equationp. 137 Finite difference schemesp. 139 Finite differencesp. 139 The five-point formula for ∇2u = fp. 147 Higher-order methods for ∇2u = fp. 158 Comments and bibliographyp. 163 Exercisesp. 166 The finite element methodp. 171 Two-point boundary value problemsp. 171 A synopsis of FEM theoryp. 184 The Poisson equationp. 192 Comments and bibliographyp. 200 Exercisesp. 201 Spectral methodsp. 205 Sparse matrices vs. small matricesp. 205 The algebra of Fourier expansionsp. 211 The fast Fourier transformp. 214 Second-order elliptic PDEsp. 219 Chebyshev methodsp. 222 Comments and bibliographyp. 225 Exercisesp. 230 Gaussian elimination for sparse linear equationsp. 233 Banded systemsp. 233 Graphs of matrices and perfect Cholesky factorizationp. 238 Comments and bibliographyp. 243 Exercisesp. 246 Classical iterative methods for sparse linear equationsp. 251 Linear one-step stationary schemesp. 251 Classical iterative methodsp. 259 Convergence of successive over-relaxationp. 270 The Poisson equationp. 281 Comments and bibliographyp. 286 Exercisesp. 288 Multigrid techniquesp. 291 In lieu of a justificationp. 291 The basic multigrid techniquep. 298 The full multigrid techniquep. 302 Poisson by multigridp. 303 Comments and bibliographyp. 307 Exercisesp. 308 Conjugate gradientsp. 309 Steepest, but slow, descentp. 309 The method of conjugate gradientsp. 312 Krylov subspaces and preconditionersp. 317 Poisson by conjugate gradientsp. 323 Comments and bibliographyp. 325 Exercisesp. 327 Fast Poisson solversp. 331 TST matrices and the Hockney methodp. 331 Fast Poisson solver in a discp. 336 Comments and bibliographyp. 342 Exercisesp. 344 Partial differential equations of evolutionp. 347 The diffusion equationp. 349 A simple numerical methodp. 349 Order, stability and convergencep. 355 Numerical schemes for the diffusion equationp. 362 Stability analysis I: Eigenvalue techniquesp. 368 Stability analysis II: Fourier techniquesp. 372 Splittingp. 378 Comments and bibliographyp. 381 Exercisesp. 383 Hyperbolic equationsp. 387 Why the advection equation?p. 387 Finite differences for the advection equationp. 394 The energy methodp. 403 The wave equationp. 407 The Burgers equationp. 413 Comments and bibliographyp. 418 Exercisesp. 422 Appendix Bluffer's guide to useful mathematicsp. 427 Linear algebrap. 428 Vector spacesp. 428 Matricesp. 429 Inner products and normsp. 432 Linear systemsp. 434 Eigenvalues and eigenvectorsp. 437 Bibliographyp. 439 Analysisp. 439 Introduction to functional analysisp. 439 Approximation theoryp. 442 Ordinary differential equationsp. 445 Bibliographyp. 446 Indexp. 447

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.

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