Chebyshev domain truncation is inferior to fourier domain truncation for solving problems on an infinite interval

Abstract

“Domain truncation” is the simple strategy of solving problems onyε [-∞, ∞] by using a large but finite computational interval, [− L, L] Sinceu(y) is not a periodic function, spectral methods have usually employed a basis of Chebyshev polynomials,T _n(y/L). In this note, we show that becauseu(±L) must be very, very small if domain truncation is to succeed, it is always more efficient to apply a Fourier expansion instead. Roughly speaking, it requires about 100 Chebyshev polynomials to achieve the same accuracy as 64 Fourier terms. The Fourier expansion of a rapidly decaying but nonperiodic function on a large interval is also a dramatic illustration of the care that is necessary in applying asymptotic coefficient analysis. The behavior of the Fourier coefficients in the limitn→∞ for fixed intervalL isnever relevant or significant in this application.