Order stars and the maximal accuracy of stable difference schemes for the wave equation

The wave equation utt = c²uxx is solved by means of a three-time-level difference scheme, symmetric in time and space, of the form It was proved by Renaut that the maximal order of accuracy p of such a scheme is given by p 5 ≤ (s + S). In this paper we show that the requirement of stability does not reduce this maximal order. Our result is proved for S ≤ s by introducing an order star on the Riemann surface of the algebraic function associated with the scheme. 1980 Mathematics Subject Classification (1985 Revision). 65M10.