Piecewise-quasilinearization techniques for singularly perturbed Volterra integro-differential equations

Abstract

Three families of iterative quasilinear methods for the numerical solution of singularly perturbed Volterra integro-differential equations are presented. Two of these methods are based on the piecewise integration of the differential equations and the use of quadrature rules, while the third one employs a locally frozen approximation for the coefficients and right-hand side of the differential equation. It is shown that these quasilinear methods are first-order accurate for step sizes equal to or larger than the initial layer thickness, and are less accurate than exponentially fitted techniques whose coefficients are selected so that the finite difference equations have truncation errors of second- and third-order, even though they exhibit only first-order accuracy for initial layer thicknesses equal to or smaller than the step size. It is also shown that the quasilinear methods presented in this paper are of exponential type and provide more accurate results than the backward Euler and trapezoidal methods for step sizes larger than the initial layer thickness, and the quasilinear frozen method is less accurate than quasilinear techniques that make use of an integrating factor and numerical quadratures.