Numerical Solution of the Modified Bessel Equation

TL;DR

This paper presents a Green's function based numerical solver for the modified Bessel equation, optimized for cylindrical Poisson problems, achieving high accuracy and linear computational scaling.

Contribution

It introduces a novel solver using Discrete Hankel Transform and specialized Bessel function computation to handle large order problems efficiently.

Findings

Error reaches machine precision

Computational time is linearly proportional to number of nodes

Effective for large azimuthal grid problems

Abstract

A Green's function based solver for the modified Bessel equation has been developed with the primary motivation of solving the Poisson equation in cylindrical geometries. The method is implemented using a Discrete Hankel Transform and a Green's function based on the modified Bessel functions of the first and second kind. The computation of these Bessel functions has been implemented to avoid scaling problems due to their exponential and singular behavior, allowing the method to be used for large order problems, as would arise in solving the Poisson equation with a dense azimuthal grid. The method has been tested on monotonically decaying and oscillatory inputs, checking for errors due to interpolation and/or aliasing. The error has been found to reach machine precision and to have computational time linearly proportional to the number of nodes.