Uniform $l^1$ Behavior of a Time Discretization Method for a Volterra Integrodifferential Equation with Convex Kernel; Stability

Abstract

We study stability of a numerical method in which the backward Euler method is combined with order one convolution quadrature for approximating the integral term of the linear Volterra integrodifferential equation $\mathbf{u}'(t)+\int_{0}^{t}\beta(t-s)\mathbf{Au}(s)\,ds=0$, $t\geq0$, $\mathbf{u}(0)=\mathbf{u}_{0}$, which arises in the theory of linear viscoelasticity. Here $\mathbf{A}$ is a positive self-adjoint densely defined linear operator in a real Hilbert space, and $\beta(t)$ is locally integrable, nonnegative, nonincreasing, convex, and $-\beta'(t)$ is convex. We establish stability of the method under these hypotheses on $\beta(t)$. Thus, the method is stable for a wider class of kernel functions $\beta(t)$ than was previously known. We also extend the class of operators $\mathbf{A}$ for which the method is stable.