非线性二维Volterra积分方程的一个高阶数值格式

A high order schema for the numerical solution of the nonlinear two-dimensional Volterra integral equations

对非线性二维Volterra积分方程构造了一个高阶数值格式.block-byblock方法对积分方程来说是一个非常常见的方法,借助经典block-by-block方法的思想,构造了一个所谓的修正block-by-block方法.该方法的优点在于除u(x_1,y),u(x_2,y),u(x,y_1)和u(x,y_2)外,其余的未知量不需要耦合求解,且保存了block-by-block方法好的收敛性.并对此格式的收敛性进行了严格的分析,证明了数值解逼近精确解的阶数是4阶。

This paper presents a general technique to construct high order schemes for the numerical solutions of the second kind nonlinear two-dimensional Volterra integral equations. This technique is based on the so-called block-by-block approach, which is a common method for the integral equations. In this approach, the classical block-by-block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception at u（x1,y）,u（x2,y）,u（x, y1） and u（x, y2）, while preserving the good convergence property of the block-by-block schemes. By using this new approach, a high order schema is constructed for the second kind nonlinear two-dimensional Volterra integral equations. The convergence of the schema is rigorously established. It is proved that the numerical solution conver^es to the ~Y~ ~,~1,.~; ：.L __J ~