Highly Oscillatory Problems

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.

Preface	p. vii
Oscillations over long times in numerical Hamiltonian systems	p. 1
Introduction	p. 1
Backward error analysis	p. 3
Modulated Fourier expansion	p. 7
Nonlinear wave equation	p. 16
Linear multistep methods	p. 19
Highly oscillatory quadrature	p. 25
Introduction	p. 25
Asymptotics	p. 28
Filon method	p. 33
Levin collocation method	p. 39
Steepest descent methods	p. 43
Other methods	p. 46
Rapid function approximation by modified Fourier series	p. 51
Introduction	p. 51
Univariate approximation	p. 53
Polyharmonic approximation	p. 60
Multivariate approximation	p. 62
Convergence acceleration	p. 66
Concluding remarks	p. 69
Approximation of high frequency wave propagation problems	p. 72
Introduction	p. 72
Geometrical optics	p. 74
Geometrical theory of diffraction	p. 82
Wavelet-based numerical homogenization	p. 98
Introduction	p. 98
Projection generated homogenization	p. 103
Wavelet-based numerical homogenization	p. 105
Numerical examples	p. 115
Plane wave methods for approximating the time harmonic wave equation	p. 127
Introduction	p. 127
Derivation of the UWVF	p. 131
Discretization of the UWVF	p. 135
Analysis of the UWVF	p. 138
Numerical implementation and results for the UWVF	p. 141
Conclusion	p. 149
Boundary integral methods in high frequency scattering	p. 154
Introduction	p. 154
Hybrid approximation spaces	p. 160
Stability and conditioning	p. 176
Implementation	p. 187
Novel analytical and numerical methods for elliptic boundary value problems	p. 194
Introduction	p. 194
The derivations of integral representations	p. 200
The elimination of the unknown boundary values using algebraic manipulations	p. 205
Formulation of Riemann-Hilbert problems	p. 223
A new numerical method	p. 226
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