Extrapolation method for solving two-dimensional volterral integral equations of the second kind

In this paper, we propose a numerical quadrature method for solving two-dimensional linear and nonlinear Volterra integral equations. Firstly, we generalize one-dimensional quadrature formula to two-dimensional case and its corresponding error asymptotic expansion. Based on the quadrature formula and the error expansion, we next construct an iterative scheme and extrapolation algorithm. The numerical solution of any point can be calculated by iterative scheme, and the error accuracy and convergence order of the numerical solution are further improved by extrapolation algorithm. Using the extrapolation algorithm, we can improve the convergence order from O(h(0)(2)) to O(h(0)(3)) or even O(h(0)(4)). Since, the numerical solution of each point is obtained by assignment operation and iteration, the computational complexity can be greatly reduced. Finally, four numerical examples are given to illustrate the effectiveness of the method. (C) 2019 Elsevier Inc. All rights reserved.