Spectral computation of highly oscillatory integral equations in laser theory

Abstract

We are concerned in this paper with the numerical computation of the spectra of highly oscillatory integrals that arise in laser simulations. Discretised using the modified Fourier basis, the spectral problem for the integral equation is converted into two independent infinite systems of linear equations whose unknowns are the coefficients of the modified Fourier functions, namely the cosine and shifted sine functions, respectively. Each (m,n) entry of the resulting coefficient matrices can be represented exactly by expressions involving the error function with an argument that involves the oscillatory parameter ω and the numbers m and n. Moreover, considering the behaviour of the error function for a large argument, the asymptotics for each entry are analysed for large ω or for large m and n and this enables efficient truncation of the infinite systems. Numerical experiments are provided to illustrate the effectiveness of this method.