Spectral solver for Cauchy problems in polar coordinates using discrete Hankel transforms

TL;DR

This paper presents a spectral solver based on Fourier-Bessel transforms for efficiently solving Cauchy problems with Laplacians in polar coordinates, validated on wave and fluid flow equations.

Contribution

The paper introduces a novel spectral method combining FFTs and discrete Hankel transforms for polar coordinate problems, with detailed error analysis and applications to linear and nonlinear PDEs.

Findings

Achieves spectral accuracy with exponential convergence in ideal cases.

Demonstrates effectiveness on wave, Poiseuille flow, and Bose-Einstein condensate equations.

Provides complexity analysis and boundary error bounds for the method.

Abstract

We introduce a Fourier-Bessel-based spectral solver for Cauchy problems featuring Laplacians in polar coordinates under homogeneous Dirichlet boundary conditions. We use FFTs in the azimuthal direction to isolate angular modes, then perform discrete Hankel transform (DHT) on each mode along the radial direction to obtain spectral coefficients. The two transforms are connected via numerical and cardinal interpolations. We analyze the boundary-dependent error bound of DHT; the worst case is $\sim N^{-3/2}$, which governs the method, and the best $\sim e^{-N}$, which then the numerical interpolation governs. The complexity is $O[N^3]$. Taking advantage of Bessel functions being the eigenfunctions of the Laplacian operator, we solve linear equations for all times. For non-linear equations, we use a time-splitting method to integrate the solutions. We show examples and validate the method on the two-dimensional wave equation, which is linear, and on two non-linear problems: a time-dependent Poiseuille flow and the flow of a Bose-Einstein condensate on a disk.