Fekete I.,Eötvös Loránd University |
Farago I.,MTA ELTE Numerical Analysis and Large Networks Research Group
Computers and Mathematics with Applications | Year: 2014
The stability is one of the most basic requirement for the numerical model, which is mostly elaborated for the linear problems. In this paper we analyze the stability notions for the nonlinear problems. We show that, in case of consistency, both the N-stability and K-stability notions guarantee the convergence. Moreover, by using the N-stability we prove the convergence of the centralized Crank-Nicolson-method for the periodic initial-value transport equation. The K-stability is applied for the investigation of the forward Euler method and the θ-method for the Cauchy problem with Lipschitzian right side.
Csomos P.,MTA ELTE Numerical Analysis and Large Networks Research Group |
Farago I.,MTA ELTE Numerical Analysis and Large Networks Research Group |
Farago I.,Eötvös Loránd University |
Fekete I.,MTA ELTE Numerical Analysis and Large Networks Research Group |
Fekete I.,Eötvös Loránd University
Computers and Mathematics with Applications | Year: 2015
The paper deals with discretisation methods for nonlinear operator equations written as abstract nonlinear evolution equations. Brezis and Pazy showed that the solution of such problems is given by nonlinear semigroups whose theory was founded by Crandall and Liggett. By using the approximation theorem of Brezis and Pazy, we show the N-stability of the abstract nonlinear discrete problem for the implicit Euler method. Motivated by the rational approximation methods for linear semigroups, we propose a more general time discretisation method and prove its N-stability as well. © 2015 Elsevier Ltd.