Technical Report 1999-014

The Godunov-Ryabenkii condition: The beginning of a new stability theory

Bertil Gustafsson

November 1999


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.

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