Strongly stable one-step methods for stiff Volterra integral equations of the second kind
We propose some implicit methods for stiff Volterra integral equations of the second kind. The methods are constructed on the integro-differential equation obtained by differentiation of the Volterra equation. The numerical schemes are derived using a class of A-stable and L-stable methods for ordinary differential equations associated with the Gregory quadrature formula. Related to the test equation: \(y(t)=1+\int^{t}_{0}[\lambda +\mu (t-s)]y(s)ds\), \(\lambda,\mu <0\) we give the definition of \({\mathcal A}\)-stability and \({\mathcal L}\)-stability for the proposed numerical methods as natural extension of the A-stability and L-stability for the schemes for solving ordinary differential equations. We show how to choose the parameters of the methods in order to obtain \({\mathcal A}\)-stable and \({\mathcal L}\)-stable schemes. Because of these properties the proposed schemes are particularly advantageous in the case of stiff Volterra integral equations.
stiff Volterra integral equations of the second kind
scientific article; zbMATH DE number 4030935