The maximal accuracy of stable difference schemes for the wave equation

We consider three time-level difference schemes, symmetric in time and space, for the solution of the wave equation, u_tt =c²u_xx, given by {Mathematical expression} It has already been proved that the maximal order of accuracy p of such schemes is given by p ≤ 2(s + S). In this paper we show that the requirement of stability does not reduce this maximal order for any choice of the pair (s, S). The result is proved by introducing an order star on the Riemann surface of the algebraic function associated with the scheme. Furthermore, Padé schemes, with S = 0, s > 0, and s = 0, S > 0, are proved to be stable for 0 < μ < 1, where μ is the Courant number. These schemes can be implemented with high-order absorbing boundary conditions without reducing the range of μ for which stable solutions are obtained.