Convergence rates of \(\alpha\)-stable difference methods
A quantitative Lax type theorem is established with the stability and well-posedness conditions replaced by the weaker concepts of alpha stability and alpha well-posedness. Here ''quantitative'' means that in the consistency hypothesis upon the difference method a rate of convergence is prescribed and that in the convergence result a rate is obtained corresponding to the degree of smoothness of the initial value. This is formulated in a general Banach space first and then made more precise by specializing it to Lebesgue- and Besov spaces. The motivation for the concepts of alpha stability and alpha well-posedness comes from their usefulness in characterizing systems of partial differential equations with constant coefficients and the corresponding difference equations. See the authors [Alpha well-posedness, alpha stability, and the matrix theorems of \textit{H.O. Kreiss}, Numer. Math. (to appear)].
scientific article; zbMATH DE number 3856305