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.

翻译：本文提出一种基于傅里叶-贝塞尔函数的谱方法，用于求解具有齐次狄利克雷边界条件的极坐标拉普拉斯算子型柯西问题。我们沿方位角方向采用快速傅里叶变换（FFT）分离角模态，随后对每个模态沿径向方向实施离散汉克尔变换（DHT）以获得谱系数。两种变换通过数值插值与基数插值相衔接。我们分析了边界依赖性误差界：最坏情形为$\sim N^{-3/2}$（决定方法精度），最优情形为$\sim e^{-N}$（此时数值插值主导误差）。算法复杂度为$O[N^3]$。利用贝塞尔函数作为拉普拉斯算子本征函数的特性，我们可求解所有时间步的线性方程；对于非线性方程，则采用时间分裂法积分求解。通过线性二维波动方程、含时泊肃叶流动及圆盘上玻色-爱因斯坦凝聚体流动两个非线性算例，验证了该方法的有效性。