Stability concepts and their applications

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 Kstability notions guarantee the convergence. Moreover, by using the Nstability we prove the convergence of the centralized Crank--Nicolsonmethod for the periodic initial-value transport equation. The K-stability is applied for the investigation of the forward Euler method and the $\theta$-method for the Cauchy problem with lipschitzian right side. Stability concepts and their applications Imre Fekete, István Faragó Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117 Budapest, Pázmány P. s. 1/C, Hungary MTA-ELTE Numerical Analysis and Large Networks Research Group, H-1117 Budapest, Pázmány P. s. 1/C, Hungary Abstract 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.