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

Boyd, J.P., A comparison of numerical and analytical methods for the reduced wave equation with multiple spatial scales, Applied Numerical Mathematics 7 (1991) 453-479. We compare four different techniques for solving the ordinary differential equation u,, + u = I on the unbounded interval, x E [ 00, co], when f( EX) decays rapidly as ) x ( + co. 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 E es 1, the solution has two length scales: the “fast”, O(1) scale of the homogeneous solutions of the differential equation and the “slow”, O(~/E) scale of the forcing function. The four methods are: (1) perturbation series in E (“method of multiple scales”); (2) Padt approximants formed from the c-series; (3) rational Chebyshev pseudospectral algorithm; and (4) the pseudospectral method with a mixed basis that includes a special “radiation function” for the plus sign only. 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 1 -+ 00. Pad& approximants converge and the rational Chebyshev pseudospectral method is very accurate. One might suppose that the numerical method would be ineffective for small E 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 l/.s, as few as twenty basis functions give six decimal place accuracy for a smooth f(x) for all E. When the sign of the differential equation is negative, u(x) is oscillatory as 1 x ( ---f cc (with an amplitude LY which is proportional to exp( q/e) for some constant q). PadC approximants and the Chebyshev method do not converge, but instead have an accuracy which is limited to O(a). 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, a(e).