Accuracy Enhancement of Numerical Solution for Ordinary Differential Equations

An accuracy enhancing scheme to ensure the accuracy of the numerical solution of a stiff ordinary differential equation to the machine error level is proposed. The key point is to accurately approximate the numerical solution in terms of one or more polynomials. By properly choosing a fraction of data points to construct the interpolation polynomial, the Newton divided difference interpolation method provides an approximating polynomial for a smooth data string. At all the rest of data points, the differences between the polynomial and original data are approximation error. If the data string is smooth enough, the resulting error can be suppressed down to the machine error range. Consequently, the truncation error of the numerical equation can be precisely estimated so that the solution error can be iteratively removed. Numerical tests of two ordinary differential equations show that their final solutions reach the error limitation of the computing device: one is a stiff equation and the other is a simple equation. In other words, the proposed algorithm successfully promotes a numerical solution to be almost exact without infinitely refining the mesh size.