An accuracy barrier for stable three-time-level difference schemes for hyperbolic equations

We consider three-time-level difference schemes for the linear constant coefficient advection equation u₁ = cu_x. In 1985 it was conjectured that the barrrier to local order p of schemes which are stable is given by p ≤ 2mm{R, S}. Here R and S denote the number of downwind and upwind points, respectively, in the difference stencil with respect to the characteristic of the differential equation through the update point. Here we prove the conjecture for a class of explicit and implicit schemes of maximal accuracy. In order to prove this result, the existing theory on order stars has to be generalized to the extent where it is applicable to an order star on the Riemann surface of the algebraic function associated with a difference scheme. Proof of the conjecture for all schemes relies on an additional conjecture about the geometry of the order star. We dedicate this paper to the memory of Professor Peter Henrici. With his excellent books on numerics and complex analysis he has helped us all to understand the subjects better.