Asymptotic Error Expansion of the Iterated Superconvergent Nyström Method for Urysohn Integral Equations

Abstract

For a nonlinear Urysohn integral equation with smooth kernel, a superconvergent Nyström method proposed recently in Allouch et al. (BIT Numer Math 258:10777-10790, 2017), based on an interpolatory projection onto a space of piecewise polynomials of degree ≤ r , is shown to have convergence of order 4r for its iterated version. In this paper, a general theorem dealing with asymptotic error expansion for the iterated superconvergent Nyström solution is proved. The Richardson extrapolation is then applied to obtain a more accurate solution of order 4r + 2. Numerical results are given to illustrate this improvement of the convergence order.