使用离散汉克尔变换求解极坐标下柯西问题的谱求解器
Spectral solver for Cauchy problems in polar coordinates using discrete Hankel transforms
AI总结：
本研究提出基于傅里叶-贝塞尔的极坐标柯西问题谱求解器，结合FFT与离散汉克尔变换，分析误差界与复杂度，通过线性及非线性算例验证方法有效性。
摘要：
我们提出了一种基于Fourier-Bessel（傅里叶-贝塞尔）的谱求解器，用于求解齐次Dirichlet（狄利克雷）边界条件下、极坐标中含拉普拉斯算子的柯西问题。我们在方位角方向使用FFT（快速傅里叶变换）分离角模式，随后对每个模式沿径向执行discrete Hankel transform（DHT，离散汉克尔变换）以得到谱系数，两类变换通过数值插值与基数插值衔接。我们分析了DHT的边界相关误差界：最坏情况约为$N^{-3/2}$，此时该误差主导方法精度；最好情况约为$e^{-N}$，此时数值插值误差占主导，方法的复杂度为$O[N^3]$。利用贝塞尔函数是拉普拉斯算子本征函数的特性，我们可求解任意时刻的线性方程；对于非线性方程，则采用时间分裂法对解进行积分。我们通过二维波动方程（线性问题）、含时泊肃叶流动以及圆盘上玻色-爱因斯坦凝聚体的流动这两个非线性问题给出算例并验证了方法的有效性。
英文摘要：
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.
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