Convergence of discrete-time relaxation methods based on Volterra backward differentiation formulas
type: Item
instance of: scholarly article
title: Convergence of discrete-time relaxation methods based on Volterra backward differentiation formulas
zbMATH Open document ID: 0952.65108
author: M. R. Crisci
author: Elvira Russo
author: Antonia Vecchio
published in: Ricerche di Matematica
publication date: 2001-01-02T00:00:00Z
review text: The author proposes the discrete-time relaxation methods based on indirect backward differentiation methods for the numerical solution of large system of second kind Volterra integral equations (VIEs). A necessary and sufficient condition for the convergence of the iterative process in the case of a system of linear VIEs is obtained. Some more practical sufficient conditions not involving the eigenvalues of the convergence matrix are proved. The case of a linear convolution system is considered and a sufficient condition for the convergence of the waveform relaxation backward differentiation (WRBDF) method to the underlying method is furnished. This condition depends on the logarithmic norm of the chosen splitting and is independent of the stepsize and of the window. The case of parallel splittings is analyzed. A theorem for the convergence of the methods based on Jacobi splitting whose hypotheses require some condition on the kernel which are very easy to check is proved. A useful criterion for the choice of the Richardson splitting leading to convergent WRBDF method is obtained.
reviewed by: Nikolay Yakovlevich Tikhonenko
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45F05
Mathematics Subject Classification ID: 65Y05
zbMATH DE Number: 1462409
zbMATH Keywords: system of linear Volterra integral equations
zbMATH Keywords: discrete-time relaxation
zbMATH Keywords: convergence
zbMATH Keywords: linear convolution system
zbMATH Keywords: linear convolution kernel
zbMATH Keywords: waveform relaxation backward differentiation method
zbMATH Keywords: parallel splittings
zbMATH Keywords: Jacobi splitting
zbMATH Keywords: Richardson splitting
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1462409