A comparison of numerical and analytical methods for the reduced wave equation with multiple spatial scales

Abstract

We compare four different techniques for solving the ordinary differential equation uxx +/- u = [finite part integral]([var epsilon]x) on the unbounded interval, x [set membership, variant] [-[infinity], [infinity]], when [finite part integral]([var epsilon]x) decays rapidly as |x| --> [infinity]. This problem, although very simple, is representative of problems that arise in such diverse fields as numerical weather prediction, plasma physics, and weakly non-local solitary waves. When [var epsilon] We find that the perturbation series is asymptotic but almost always divergent. The effectiveness of the other methods depends on the sign of the coefficient in the differential equation. When the sign is negative, u(x) decays rapidly as |x| --> [infinity]. Pade approximants converge and the rational Chebyshev pseudospectral method is very accurate. One might suppose that the numerical method would be ineffective for small [var epsilon] because of the difficulty of simultaneously resolving two very disparate length scales. However, because that part of u(x) which varies on the "fast" O(1) scale is exponentially small in 1/[var epsilon], as few as twenty basis functions give six decimal place accuracy for a smooth [finite part integral](x) for all [var epsilon].When the sign of the differential equation is negative, u(x) is oscillatory as |x| --> [infinity] (with an amplitude [alpha] which is proportional to exp(-q/[var epsilon]) for some constant q). Pade approximants and the Chebyshev method do not converge, but instead have an accuracy which is limited to O([alpha]). When a special "radiation function" is added to the spectral basis, however, it is possible to obtain arbitrarily high accuracy. The numerically computed coefficient of the radiation function is an accurate approximation to the amplitude of the asymptotic radiation, [alpha]([var epsilon]).