Improving the accuracy of finite difference methods for solving boundary value ordinary differential equations

Abstract

Abstract.It is shown that by introducing one or three interpolated values in each subinterval the local truncation error of finite difference (box, gap and deferred correction) methods can be decreased by two to four orders of magnitude when solving two point boundary value ordinary differential equations.

An adaptive implementation of interpolation methods for boundary value ordinary differential equations

“…This problem leads to relatively less accurate results than the other problems [1,[5][6] due to the large values~of the derivatives of f. It is pointed out in [1] that for this problem the Trapezoidal Rule method required 21° mesh points for 10 -6 accuracy, and the deferred correction method needed 65 mesh points with 7 corrections. All of our computations were performed in double precision on a UNIVAC 1100 using FORTRAN (ASC II) Level 9R1.…”

“…The last section contains the results of some numerical experiments on a difficult test problem. This paper makes use of some of our results given in a recent paper in this journal [5]. The values of f~-3/4, f~-1/2 and f~-1/4 are computed by the appropriate interpolation formulas given in [5], such that (2.1) and (2.2) are, respectively, O(h s) and O(h7).…”

“…This paper makes use of some of our results given in a recent paper in this journal [5]. The values of f~-3/4, f~-1/2 and f~-1/4 are computed by the appropriate interpolation formulas given in [5], such that (2.1) and (2.2) are, respectively, O(h s) and O(h7). Let e be a given local error tolerance, and y an approximate solution vector for (1.1, 1.2).…”