P-stable eighth algebraic order methods for the numerical solution of the Schrödinger equation

A P-stable method of algebraic order eight for the approximate numerical integration of the Schrödinger equation is developed in this paper. Since the method is P-stable (i.e. its interval of periodicity is equal to (0, ∞)), large step sizes for the numerical integration can be used. Based on this new method and on a sixth algebraic order P-stable method developed by Simos (Phys. Scripta 55 (1997) 644–650), a new variable step method is obtained. Numerical results presented for the phase-shift problem of the radial Schrödinger equation and for the coupled differential equations arising from the Schrödinger equation show the efficiency of the developed method.