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：我们提出了一种基于傅里叶-贝塞尔函数的谱求解器，用于求解齐次狄利克雷边界条件下极坐标中拉普拉斯算子的柯西问题。通过沿方位角方向使用快速傅里叶变换分离角模态，再对各模态沿径向进行离散汉克尔变换以获得谱系数。两种变换通过数值插值和基函数插值实现衔接。我们分析了离散汉克尔变换的边界依赖误差界：最坏情况为$O(N^{-3/2})$，决定方法整体精度；最佳情况达$O(e^{-N})$，此时数值插值主导误差。算法复杂度为$O(N^3)$。利用贝塞尔函数是拉普拉斯算子本征函数的特性，我们求解所有时刻的线性方程组。对于非线性方程，采用时间分裂法进行积分。通过二维波动方程（线性问题）以及两个非线性问题——时间依赖泊肃叶流和圆盘上的玻色-爱因斯坦凝聚流——验证了方法的有效性。