Numerical Solution of the Modified Bessel Equation
This provides an efficient and accurate numerical method for solving the Poisson equation in cylindrical coordinates, which is useful for computational physics and engineering applications.
The authors developed a Green's function-based solver for the modified Bessel equation to solve the Poisson equation in cylindrical geometries, achieving machine precision error and linear computational time with respect to the number of nodes.
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.