An efficient solution for Dirac equation in 3D lattice space with the conjugate gradient method

TL;DR

This paper introduces a preconditioned conjugate gradient method with a filtering function (PCG-F) for efficiently solving the Dirac equation in 3D lattice space, improving convergence speed especially for deformed potentials.

Contribution

The paper presents a novel PCG-F method that avoids variational collapse and accelerates convergence in solving the Dirac equation in 3D lattice space.

Findings

PCG-F method converges faster than existing methods.

High accuracy solutions for spherical and deformed potentials.

Potential application to relativistic Hartree-Bogoliubov equations.

Abstract

An efficient method, preconditioned conjugate gradient method with a filtering function (PCG-F), is proposed for solving iteratively the Dirac equation in 3D lattice space for nuclear systems. The filtering function is adopted to avoid the variational collapsed problem and a momentum-dependent preconditioner is introduced to promote the efficiency of the iteration. The PCG-F method is demonstrated in solving the Dirac equation with given spherical and deformed Woods-Saxon potentials. The solutions given by the inverse Hamiltonian method in 3D lattice space and the shooting method in radial coordinate space are reproduced with a high accuracy. In comparison with the existing inverse Hamiltonian method, the present PCG-F method is much faster in the convergence of the iteration, in particular for deformed potentials. It may also provide a promising way to solve the relativistic Hartree-Bogoliubov equation iteratively in the future.