A TWO-STEP PREDICTOR-CORRECTOR ADAMS METHOD FOR SECOND-KIND VOLTERRA INTEGRAL EQUATIONS

Source abstract

This paper analyzes a two-step predictor-corrector scheme for linear and nonlinear Volterra integral equations of the second-kind. On a uniform mesh, the history integral is approximated by the composite trapezoidal rule. The last subinterval is first estimated by a two-step Adams-Bashforth extrapolation and then closed by a trapezoidal Adams-Moulton-type evaluation in which the diagonal kernel is computed at the predicted value. The resulting predict-evaluatecorrect-evaluate (PECE) scheme is explicit, since the predicted value is used in the corrector and no nonlinear equation has to be solved. Under smoothness and Lipschitz assumptions, a discrete Gronwall argument gives a uniform error bound of order two. Three test equations are considered: a convolution equation with solution $\cosh (t)$, a linear equation with solution $\exp (t)$, and a nonlinear equation with solution $1 /(1-t)$. For the first problem, the maximum error decreases from $4.89 \times 10^{-4}$ at $h=0.1$ to $7.65 \times 10^{-6}$ at $h=$ 0.0125, with observed orders tending to 2. The linear and nonlinear tests show the same asymptotic behaviour, including cases for which the diagonal kernel is nonzero. These calculations support the theoretical estimate and clarify the role of the predicted value in the corrector.