Kh * Rgxqry 5 \ Dehqnll Frqglwlrq 7 Kh Ehjlqqlqj Ri D Qhz Vwdelolw \ Wkhru \ %

The analysis of diierence methods for initial-boundary value problems was dif-cult during the rst years of the development of computational methods for PDE. The Fourier analysis was available, but of course not suucient for non-periodic boundary conditions. The only other available practical tool was an eigenvalue analysis of the evolution diierence operator Q. Actually, t h e r e w ere deenitions presented, that deened an approximation as stable if the eigenvalues of Q were inside the unit circle for a xed step-size h. In the paper "Special criteria for stability for boundary-value problems for non-self-adjoint diierence equations" by S.K. Godunov and V.S. Ryabenkii in 1963, the authors presented an analysis of a simple diierence scheme that clearly demonstrated the shortcomings of the eigenvalue analysis. They also gave a new deenition of the spectrum of a family of operators, and stated a new necessary stability criterion. This criterion later became known as the Godunov-Ryabenkii condition, and it was the rst step towards a better understanding of initial-boundary value problems. The theory was later developed in a more general manner by Kreiss and others, leading to necessary and suucient conditions for stability. In this paper we shall present the contribution by Godunov and Ryabenkii, and show the connection to the general Kreiss theory.