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9780521134439

Highly Oscillatory Problems

by
  • ISBN13:

    9780521134439

  • ISBN10:

    0521134439

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2009-07-31
  • Publisher: Cambridge University Press

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Summary

The first book to approach high oscillation as a subject of its own, Highly Oscillatory Problems begins a new dialogue and lays the groundwork for future research. It ensues from the six-month programme held at the Newton Institute of Mathematical Sciences, which was the first time that different specialists in highly oscillatory research, from diverse areas of mathematics and applications, had been brought together for a single intellectual agenda. This ground-breaking volume consists of eight review papers by leading experts in subject areas of active research, with an emphasis on computation: numerical Hamiltonian problems, highly oscillatory quadrature, rapid approximation of functions, high frequency wave propagation, numerical homogenization, discretization of the wave equation, high frequency scattering and the solution of elliptic boundary value problems.

Table of Contents

Prefacep. vii
Oscillations over long times in numerical Hamiltonian systemsp. 1
Introductionp. 1
Backward error analysisp. 3
Modulated Fourier expansionp. 7
Nonlinear wave equationp. 16
Linear multistep methodsp. 19
Highly oscillatory quadraturep. 25
Introductionp. 25
Asymptoticsp. 28
Filon methodp. 33
Levin collocation methodp. 39
Steepest descent methodsp. 43
Other methodsp. 46
Rapid function approximation by modified Fourier seriesp. 51
Introductionp. 51
Univariate approximationp. 53
Polyharmonic approximationp. 60
Multivariate approximationp. 62
Convergence accelerationp. 66
Concluding remarksp. 69
Approximation of high frequency wave propagation problemsp. 72
Introductionp. 72
Geometrical opticsp. 74
Geometrical theory of diffractionp. 82
Wavelet-based numerical homogenizationp. 98
Introductionp. 98
Projection generated homogenizationp. 103
Wavelet-based numerical homogenizationp. 105
Numerical examplesp. 115
Plane wave methods for approximating the time harmonic wave equationp. 127
Introductionp. 127
Derivation of the UWVFp. 131
Discretization of the UWVFp. 135
Analysis of the UWVFp. 138
Numerical implementation and results for the UWVFp. 141
Conclusionp. 149
Boundary integral methods in high frequency scatteringp. 154
Introductionp. 154
Hybrid approximation spacesp. 160
Stability and conditioningp. 176
Implementationp. 187
Novel analytical and numerical methods for elliptic boundary value problemsp. 194
Introductionp. 194
The derivations of integral representationsp. 200
The elimination of the unknown boundary values using algebraic manipulationsp. 205
Formulation of Riemann-Hilbert problemsp. 223
A new numerical methodp. 226
Table of Contents provided by Ingram. All Rights Reserved.

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