A generalized Numerov method for linear second-order differential equations involving a first derivative term

TL;DR

This work tackles the challenge of solving linear second-order differential equations with a first-derivative term, which arise in self-consistent Skyrme-Hartree-Fock calculations due to an effective mass. It introduces the Generalized Linear Numerov Method (GLNM), deriving a sixth-order accurate three-point recurrence and a practical derivative evaluation that maintain Numerov precision while accommodating $g(x)$ in the equation. The method is applied to radial HF problems by outward/inward integration and a matching condition to extract eigenvalues, with boundary values obtained from analytic expansions and asymptotic functions, and then used iteratively for self-consistency. Implementation in a HF code and validation on doubly magic nuclei across Skyrme parametrizations demonstrate accurate reproduction of known results and improved efficiency by avoiding interpolation of $g$ and $f$ between grid points.

Abstract

The Numerov method for linear second-order differential equations is generalized to include equations containing a first derivative term. The method presented has the same degree of accuracy as the ordinary Numerov sixth-order method. A general scheme of the application to the numerical solution of the Hartree-Fock equations is considered.