On an eighth order formula for solving a Schrödinger equation (Q1282392)

The goal of the note is to compare the classical fourth order Numerov method with another more recent eighth order method introduced by \textit{A. C. Allison, A. D. Raptis} and \textit{T. E. Simos} [J. Comput. Phys. 97, No. 1, 240-248 (1991; Zbl 0753.65064)] for the equation \[ d^2y(r)/dr^2+(c_6r^{-6}-c_{12}r^{-12}+k^2)y(r)=0 \] with boundary conditions \[ y(0)=0,\qquad y(r)\to A(\sin kr+\tan \delta \cos kr) \text{ as } r\to \infty, \] where \(c_6\), \(c_{12}\), \(k\), \(A\), \(\delta\) are constants. It is concluded that caution is needed when applying any finite difference method to solve a differential equation with singularities. As might be expected, the numerical tests show that the eighth order formula is more accurate than Numerov's method, but also that the Allison's formula is flattered by the above mentioned equation example (proposed also in Allison's paper) because the wave function grows significantly just when the local truncation error decreases acceptably.