Finite difference simulation of fluid flows under astrophysical conditions is often complicated by factors such as complex gas physics, the occurrence of dynamics at widely differing length scales, and the necessity of using implicit difference equations. This report describes a simple, general, and efficient algorithm for solving one-dimensional, spherically symmetric problems using a variation of the ICED-ALE method. A computer code named VEGA has been written based on this algorithm, and the early stages of the collapse of a one-solar mass protostellar cloud are presented as a sample solution. (auth)