Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator

TL;DR

This paper introduces a new class of fast, high-order accurate elliptic solvers in cylindrical coordinates, combining integral and Fourier methods, with applications to Coulomb collision operators in plasma physics.

Contribution

It presents a novel approach that integrates integral equation and Fourier methods for efficient elliptic PDE solving in cylindrical coordinates, specifically targeting Coulomb collision computations.

Findings

Achieves high-order accuracy in potential and derivatives.

Effectively handles singularities with specialized quadratures.

Demonstrates applicability to Coulomb collision operator evaluation.

Abstract

In this paper, we describe a new class of fast solvers for separable elliptic partial differential equations in cylindrical coordinates $(r,\theta,z)$ with free-space radiation conditions. By combining integral equation methods in the radial variable $r$ with Fourier methods in $\theta$ and $z$, we show that high-order accuracy can be achieved in both the governing potential and its derivatives. A weak singularity arises in the Fourier transform with respect to $z$ that is handled with special purpose quadratures. We show how these solvers can be applied to the evaluation of the Coulomb collision operator in kinetic models of ionized gases.