A universal solution to one-dimensional oscillatory integrals

How to calculate the highly oscillatory integrals is the bottleneck that restraints the research of light wave and electromagnetic wave’s propagation and scattering. Levin method is a classical quadrature method for this type of integrals. Unfortunately it is susceptible to the system of linear equations’ ill-conditioned behavior. We bring forward a universal quadrature method in this paper, which adopts Chebyshev differential matrix to solve the ordinary differential equation (ODE). This method can not only obtain the indefinite integral’ function values directly, but also make the system of linear equations well-conditioned for general oscillatory integrals. Furthermore, even if the system of linear equations in our method is ill-conditioned, TSVD method can be adopted to solve them properly and eventually obtain accurate integral results, thus making a breakthrough in Levin method’s susceptivity to the system of linear equations’ ill-conditioned behavior.

[1]  V. I. Tatarskii.  Theory of single scattering by random distributed scatterers , 2003 .

[2]  Arieh Iserles,et al.  Quadrature methods for multivariate highly oscillatory integrals using derivatives , 2006, Math. Comput..

[3]  Rudolph E. Langer,et al.  On Numerical Approximation , 1959 .

[4]  P. Grandclément.  Introduction to spectral methods , 2006, gr-qc/0609020.

[5]  W. Chew.  Waves and Fields in Inhomogeneous Media , 1990 .

[6]  J. Kong.  Electromagnetic Wave Theory , 1986 .

[7]  Elsayed M. E. Elbarbary,et al.  Higher order pseudospectral differentiation matrices , 2005 .

[8]  R. Baltensperger.  Improving the accuracy of the matrix differentiation method for arbitrary collocation points , 2000 .

[9]  N. S. Bakhvalov,et al.  Evaluation of the integrals of oscillating functions by interpolation at nodes of gaussian quadratures , 1968 .

[10]  W. Cheney,et al.  Numerical analysis: mathematics of scientific computing (2nd ed) , 1991 .

[11]  A. Iserles,et al.  Efficient quadrature of highly oscillatory integrals using derivatives , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  L. Filon.  III.—On a Quadrature Formula for Trigonometric Integrals. , 1930 .

[13]  G. A. Evans,et al.  An expansion method for irregular oscillatory integrals , 1997, Int. J. Comput. Math..

[14]  David Levin,et al.  Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations , 1982 .

[15]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[16]  Sheehan Olver,et al.  Moment-free numerical integration of highly oscillatory functions , 2006 .

[17]  Bruno Costa,et al.  On the computation of high order pseudospectral derivatives , 2000 .

[18]  Timothy Nigel Phillips,et al.  Mass- and momentum-conserving spectral methods for Stokes flow , 1994 .

[19]  J. R. Webster,et al.  A comparison of some methods for the evaluation of highly oscillatory integrals , 1999 .

[20]  J. R. Webster,et al.  A high order, progressive method for the evaluation of irregular oscillatory integrals , 1997 .

[21]  G. Evans,et al.  An alternative method for irregular oscillatory integrals over a finite range , 1994 .

[22]  Jean-Paul Berrut,et al.  The errors in calculating the pseudospectral differentiation matrices for C̆ebys̆ev-Gauss-Lobatto points , 1999 .

[23]  Akira Ishimaru,et al.  Wave propagation and scattering in random media , 1997 .