An inequality for stability control of a Ceschino’s scheme of second order of accuracy is constructed. A numerical formula of order one is developed that is based on the stages of the this method and its stability interval is extended to 32. On a base of L-stable (2,1)-scheme and a numerical Ceschino’s formula, an algorithm of alternating structure, in which an efficient numerical formula is chosen on an every step by a stability criterion, is constructed. The algorithm is intended for solving stiff and non-stiff problems.
