(untitled) (Journal of Computational Mathematics, 1984)
The correction procedure has been discussed by L. Fox and V. Pereyra for accelerating the convergence of a certain approximate solution. Its theoretical basis is the existence of an asymptotic expansion for the error of discretization proved by Filippov and Rybinskii and Stetter: u-u_h=h^2 v+O(h^4), where u is the solution of the original differential equation, u_h the solution of the approximate finite difference equation with parameter h and v the solution of a correction differential equation independent of h. Stetter et al. used the extrapolation procedure to eliminate the auxiliary function v while Pereyra et al. used some special procedure to solve v approximately. In the present paper we will present a difference procedure for solving v easily.