Computer-algebra program for constructing exponentially fitted methods for solution of the Schrödinger equation

A computer-algebra program, written in Maple, is presented for the production of explicit exponentially fitted methods. By using this program, a family of explicit four-step predictor–corrector exponentially fitted methods is obtained for numerical solution of the Schrödinger equation. The four-step methods considered contain free parameters that allow them to be fitted to exponential functions. These sixth-order methods are very simple and integrate more exponential functions than both the well-known fourth-order Numerov methods and the sixth-order Runge–Kutta methods. Based on this computer-algebra program, a variable-step exponentially fitted method is introduced. Numerical results indicate that the new variable-step method is much more efficient than other well-known methods for numerical solution of the radial Schrödinger equation. © 1998 American Institute of Physics.