Stability analysis of runge-kutta methods for volterra integral equations of the second kind

Stability analysis of Volterra-Runge-Kutta methods based on the basic test equation of the form. y(t)=1+λ∫0ty(s) ds (t≥0),where λ is a complex parameter, and on the convolution test equation. y(t)=1+∫0t[λ+σ(t-s)]y(s)ds (t≥0),where λ and σ are real parameters, is presented. General stability conditions are derived and applied to construct numerical methods with good stability properties. In particular, a family of second-order V^o-stable Volterra-Runge-Kutta methods is obtained. No V^o-stable methods of order greater than one have been presented previously in the literature.