T-Stability of General One-Step Methods for Abstract Initial-Value Problems

Abstract

Abstract:In this paper we investigate the T-stability of one-step methods for initial-value problems. The main result is that we extend the classical result (the well-known Euler method) for variable step size explicit and implicit one-step methods. In addition, we give further properties for the theory of T-stability of nonlinear equations in an abstract (Banach space) setting.

Stability concepts and their applications

“…Results of Chapter 3 are based on the Author's papers. Namely, the results of Sections 3.1, 3.2 and 3.3 can be found in paper [28] and [24], respectively.…”

“…A disszertáció az operátoregyenletek stabilitási koncepciójával és azok elméleti numerikus analízisbeli alkalmazási lehetőségeivel foglalkozik. A dolgozat a Szerző megjelent cikkjein [27], [29], [28], [24], elfogadott cikkjén [19] és kéziratán [25] alapszik. A disszertáció öt fejezetből áll.…”

“…This dissertation deals with stability concepts for operator equations and their possible application areas in theoretical numerical analysis. This thesis is based on the Author's papers [27], [24], [29], [28], the accepted paper [19] and the preprint [25]. The thesis consists of five chapters.…”

N-stability of the $\theta$-method for reaction-diffusion problems

“…In the recent years we dealt with the investigation of the numerical solution of nonlinear operator equations in an abstract (Banach space) setting. This work has been summarized in [3,4]. In this paper our primary aim is to describe the discretizated version of the above mentioned problem and verify the N-stability in an abstract setting.…”