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:本文介绍了一种基于傅里叶-贝塞尔变换的离散汉克尔变换谱解法,用于处理极坐标下具有拉普拉斯算子的Cauchy问题。研究对象在同质Dirichlet边界条件下。我们在方位角方向上使用FFT来分离角度模式,然后在径向上对每个模式执行离散汉克尔变换(DHT)以获得谱系数。两种变换通过数值插值和一个基于kardinal插值结合起来。该文分析了DHT的边界相关误差界限; 最坏情况是 $\sim N^{-3/2}$,它是这种方法的主导因素,然后最优的 $\sim e^{-N}$,这时数值插值起主导作用。算法的时间复杂度是$O[N^3]$。 由于贝塞尔函数是拉普拉斯算子的本征函数,我们能够求解所有时刻的线性方程组。对于非线性方程,我们使用时间分裂方法来积分解。我们在二维波动方程上进行了验证,该方程是线性的,以及在两个非线性问题上,即时变的Poiseuille流和旋转圆盘上的玻色-爱因斯坦凝聚流。