Uniform \(l^1\) behavior in the Crank-Nicolson methods for a linear Volterra equation with convex kernel
type: Item
instance of: scholarly article
title: Uniform \(l^1\) behavior in the Crank-Nicolson methods for a linear Volterra equation with convex kernel
zbMATH Open document ID: 1317.65260
author: Da Xu
published in: Calcolo
publication date: 2015-03-23T00:00:00Z
review text: The author studies the time discretization of the Cauchy problem \[ u_t+\int _0^t\beta(t-s)Lu(s)ds=0,\, t>0, \,u(0)=u_0, \] where \(L\) is a self-adjoint densely defined linear operator on the Hilbert space having a complete eigensystem \({\{\lambda_m,\varphi_m\}}_{m=1}^\infty\). The kernel \(\beta\in C(0,\infty)\cap L^1(0,1)\) is real-valued, nonnegative, nonincreasing and convex, \(-\beta '\) is also convex. The discretization in time is based on using the Crank-Nicolson method and on the trapezoidal rule. The integral term is also approximated by the trapezoidal rule. Uniform error estimates are also studied and some numerical examples are given.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45A05
zbMATH DE Number: 6418667
zbMATH Keywords: linear Volterra integro-differential equation
zbMATH Keywords: Crank-Nicolson method
zbMATH Keywords: \(l_1\) stability
zbMATH Keywords: uniform convergence
zbMATH Keywords: Cauchy problem
zbMATH Keywords: error estimate
zbMATH Keywords: numerical example
reviewed by: Josef Kofroň
MaRDI profile type: Publication
full work available at URL: s10092-012-0075-z
OpenAlex ID: W2019416994
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DOI: 10.1007/S10092-012-0075-Z
description: scientific article; zbMATH DE number 6418667