A stability analysis of convolution quadratures for Abel-Volterra integral equations. (English) Zbl 0587.65090

This article investigates stability properties of convolution quadrature methods for the weakly singular Abel-Volterra integral equation \[ y(t)=f(t)+(\lambda /\Gamma (\alpha))\int^{t}_{0}(t-s)^{\alpha - 1}y(s)ds,\quad 0<\alpha \leq 1. \] The classical stability concepts for ordinary differential equations (A-stability, A(\(\theta)\)-stability, stability region) are extended to the above integral equation.
The main result is a nice characterization of the stability region of a method in terms of its quadrature weights. It is used to extend the famous second Dahlquist barrier (for multistep methods and ODE’s) to weakly singular integral equations. Special attention is payed to the implicit Adams product quadrature rules, whose A(\(\pi\) /2)-stability is studied in detail.
Reviewer: E.Hairer


65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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