Node Selection for Two-Point Boundary-Value Problems

The solutions of two-point boundary-value problems often have boundary layers, narrow regions of sharp variation, that can occur in any part of the interval between the points. A finite difference method of numerical solution will generally require more closely spaced nodes in the boundary layers than elsewhere. An automatic method is needed for achieving the irregular spacing when the location of the boundary layer is not known in advance. Several automatic node-insertion or node-movement methods have been proposed. A new node-movement method is presented that is optimal under the criterion of producing the least sum of squares of the truncation errors at the nodes. For the Keller box scheme applied to a system of N coupled first-order differential equations this truncation-error minimizing (TEM) method increases the system size to N+6 equations. The campylotropic coordinate transformation method and other published methods based on heuristically derived monitor functions are node-movement methods that involve systems of only N+1 or N+2 first order equations. A comparison is made of the accuracies of several such methods and the TEM method in the solution of a standard problem.