A Unified Analysis Framework for Uniform Stability of Discretized Volterra Integrodifferential Equations

Abstract

We provide a unified analysis framework for discretized Volterra integrodifferential equations by considering the $ϑ$-type convolution quadrature, where different $ϑ$ corresponds to different schemes. We first derive the long-time $l^∞$ stability of discrete solutions, and then prove a discrete Wiener-Lévy theorem to support the analysis of long-time $l^1$ stability. The methods we adopt include the integral transforms in the Stieltjes sense, the complex analysis techniques, and a linear algebra approach for an indirect estimate of intricate terms. Meanwhile, we relax the commonly-used regularity assumption of the initial data in the literature by novel treatments. Numerical simulations are performed to substantiate the theoretical findings.