Notes on the basic notions in nonlinear numerical analysis ∗
نویسندگان
چکیده
In this paper we investigate the numerical solution of non-linear equations in an abstract (Banach space) setting. The main result is that the convergence can be guaranteed by two, directly checkable conditions (namely, by the consistency and the stability). We show that these conditions together are a sufficient, but not necessary condition for the convergence. Our theoretical results are demonstrated on the numerical solution of a Cauchy problem for ordinary differential equations by means of the explicit Euler method.
منابع مشابه
Qualitative Uncertainty Orderings Revised
In recent decades, qualitative approaches to probabilistic uncertainty have been receiving wider and wider attention. We propose a new characterization of some of the most adopted partial preference orders by providing an uniform axiomatic treatment of a variety of qualitative uncertainty notions. We prove a representation result that connects qualitative notions of partial uncertainty to their...
متن کاملStability 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--...
متن کاملNonlinear Numerical Integration Scheme in Strain Space Plasticity
Strains are applied to the integration procedure in nonlinear increments todecrease the errors arising from the linearization of plastic equations. Two deformationvectors are used to achieve this. The first vector is based on the deformations obtained bythe first iteration of the equilibrium step, and the second is acquired from the sum of thesucceeding iterations. By applying these vectors and...
متن کاملBasic Control for the Viscous Moore--Greitzer Partial Differential Equation
The notions of basic controllability and basic control are deened. A quadratic optimal control of the linearized viscous Moore-Greitzer equation is presented and it is connrmed that stall is uncontrollable in this model. A basic control is constructed for the nonlinear viscous Moore-Greitzer equation which can control both surge and stall. Some extensions of this construction are discussed. Num...
متن کاملStability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems
The aim of these notes is to present some results on the stability of finite difference approximations of hyperbolic initial boundary value problems. We first recall some basic notions of stability for the discretized Cauchy problem in one space dimension. Special attention is paid to situations where stability of the finite difference scheme is characterized by the so-called von Neumann condit...
متن کامل