Accuracy of the Numerical Solution of a Two-Point Boundary Value Problem by the Orthogonal Collocation Method

Accuracies of the numerical solution of a two-point boundary value problem by the orthogonal collocation method are examined. The collocation constants for first and second derivatives are calculated by the matrix operation method (Method 1) and the method based on Lagrange’s interpolation formula (Method 2). Comparison of the calculated results shows that Method 1 produces marked loss-of-significance errors with increasing number of collocation points, N. These values are used to solve the two-point boundary value problem concerning an immobilized enzyme reaction, and then to calculate the effectiveness factor. These numerical solutions are compared with those of the previously proposed method that provides a numerical solution whose accuracy is almost the same as machine accuracy. It is found that Method 1 provides a numerical solution of significantly low accuracy under the condition of steep concentration gradient and this is never improved, even if N is increased, while Method 2 gives highly accurate values by simply increasing N. These tendencies are shown to depend strongly on the accuracies of the collocation constants.