Asymptotic error expansion of a collocation-type method for Volterra-Hammerstein integral equations

Abstract

Recently, Kumar and Sloan introduced a new collocation-type method for the numerical solution of Fredholm-Hammerstein integral equations. Brunner applied this method to nonlinear Volterra integral and integro-differential equations and discussed its connection with the iterated collocation method. In the present paper, the asymptotic error expansion of this method for nonlinear Volterra integral equations at mesh points is obtained. We show that when piecewise polynomials of Θin p-1 are used, the approximate solution admits an error expansion in powers of stepsize h, beginning with a term h p . For a special choice of the collocation points, the leading terms in the error expansion for the collocation solution contain only even powers of the stepsize h, beginning with a term h 2 p . Thus Richardson's extrapolation can be performed on the solution; and this will greatly increase the accuracy of the numerical solution.