Solution to the Modified Helmholtz Equation for Arbitrary Periodic Charge Densities

Frontiers in Physics, vol. 8

Abstract

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.

Authors 4

1. Miriam Winkelmann corresponding Aachen Physics Department

   Forschungszentrum Jülich · Jülich Aachen Research Alliance · Jülich Supercomputing Centre · RWTH Aachen University

   Affiliation as printed

   Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   JARA-CSD, Jülich, Germany

   Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany

   Peter Grünberg Institute, Forschungszentrum Jülich, Jülich, Germany

   Physics Department, RWTH-Aachen University, Aachen, Germany

   Institute for Advanced Simulation, Forschungszentrum J�lich, Germany

   JARA-CSD, Germany

   J�lich Supercomputing Centre, Forschungszentrum J�lich, Germany

   Peter Gr�nberg Institute, Forschungszentrum J�lich, Germany

   Physics Department, RWTH-Aachen University, Germany

2. Edoardo Di Napoli

   Forschungszentrum Jülich · Jülich Aachen Research Alliance · Jülich Supercomputing Centre

   Affiliation as printed

   Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   JARA-CSD, Jülich, Germany

   Jülich Supercomputing Centre, Forschungszentrum Jülich, Jülich, Germany

   Institute for Advanced Simulation, Forschungszentrum J�lich, Germany

   JARA-CSD, Germany

   J�lich Supercomputing Centre, Forschungszentrum J�lich, Germany

3. Daniel Wortmann

   Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   JARA-CSD, Jülich, Germany

   Peter Grünberg Institute, Forschungszentrum Jülich, Jülich, Germany

   Institute for Advanced Simulation, Forschungszentrum J�lich, Germany

   JARA-CSD, Germany

   Peter Gr�nberg Institute, Forschungszentrum J�lich, Germany

4. Stefan Blügel

   Forschungszentrum Jülich · Jülich Aachen Research Alliance

   Affiliation as printed

   Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

   JARA-CSD, Jülich, Germany

   Peter Grünberg Institute, Forschungszentrum Jülich, Jülich, Germany

   Institute for Advanced Simulation, Forschungszentrum J�lich, Germany

   JARA-CSD, Germany

   Peter Gr�nberg Institute, Forschungszentrum J�lich, Germany