Abstract
In this paper we numerically compute the long-time evolution of the nonsingular Saffman-Taylor finger (NSTF) with negligible surface tension using a conformal mapping method. Confirming the one-step theory prediction of the NSTF, we resolve one of the Saffman-Taylor riddles successfully. At the same time, we find a successive averaging operator that not only can reduce residual errors in summing up the Fourier series and speed up calculation greatly but also can be easily applied. Its convergence is proved theoretically.
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