Convergence and stability of quadrature methods applied to Volterra equations with delay

A class of quadrature methods for the numerical solution of equations of the form $y(t)=f(t)+∫t-τtK(t,s,y(s))ds,t∈(0,∞);y(t)=ψ(t),t∈[-τ,0]$ is proposed and the convergence and stability of the methods are analyzed. The quadrature methods provide numerical approximations which converge under mild assumptions to the true solution on every subset [0 T] of R₊ at a rate determined by the local truncation error. The dominant form of the error is established, and the weak stability of a class of formulae follows. Stability restrictions on the choice of discretization parameter are investigated in the case of some linear test equations. Various results are compared and contrasted with those which obtain in the treatment of other evolutionary problems including classical Volterra equations.