@TechReport{ it:1999-014,
author = {Bertil Gustafsson},
title = {The {G}odunov-{R}yabenkii condition: The beginning of a
new stability theory},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {1999},
number = {1999-014},
abstract = {The analysis of difference methods for initial-boundary
value problems was difficult during the first years of the
development of computational methods for PDE. The Fourier
analysis was available, but of course not sufficient for
nonperiodic boundary conditions. The only other available
practical tool was an eigenvalue analysis of the evolution
difference operator Q. Actually, there were definitions
presented, that defined an approximation as stable if the
eigenvalues of Q were inside the unit circle for a fixed
step-size h.
In the paper ``Special criteria for stability for
boundary-value problems for non-self-adjoint difference
equations'' by S.K. Godunov and V.S. Ryabenkii in 1963, the
authors presented an analysis of a simple difference scheme
that clearly demonstrated the shortcomings of the
eigenvalue analysis. They also gave a new definition 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
first step towards a better understanding of
initialboundary value problems. The theory was later
developed in a more general manner by Kreiss and others,
leading to necessary and sufficient conditions for
stability.
In this paper we shall present the contribution by Godunov
and Ryabenkii, and show the connection to the general
Kreiss theory.},
month = nov
}