The Numerov process over a non-uniform grid

The Numerov process is a solution method applicable to some classes of differential equations, that provides an error term of the fifth order in the grid size with a computational cost comparable to that of the finite-difference scheme. In the original formulation of the method, a uniform grid size is required; the paper shows a procedure for extending its applicability to a non-uniform grid in one dimension. The effectiveness of the procedure is tested on a model problem, and comparisons with other methods are carried out. Finally, it is shown how to extend the applicability of the method to a larger class of equations; among these, the mathematical model of semiconductor devices is important in view of its applications to the integrated-circuit technology.