论文标题
使用离散的汉克尔变换的光谱求解器用于极性坐标的库奇问题
Spectral solver for Cauchy problems in polar coordinates using discrete Hankel transforms
论文作者
论文摘要
我们引入了一个基于傅立叶的光谱求解器，用于在同质迪里奇（Dirichlet）边界条件下的极地坐标中的拉普拉斯（Laplacians）。我们在方位角使用FFT来分离角模式，然后沿径向方向在每个模式上执行离散的Hankel变换（DHT），以获得光谱系数。这两个转换是通过数值和基本插值连接的。我们分析了DHT的边界依赖性误差结合；最坏的情况是控制该方法的$ \ sim n^{ - 3/2} $，最佳的$ \ sim e^{ - n} $，然后是数值插值控制的。复杂性为$ o [n^3] $。利用Bessel函数是Laplacian操作员的本征函数，我们一直在求解线性方程。对于非线性方程式，我们使用时间分类方法来集成解决方案。我们显示了示例并验证二维波方程的方法，该方程是线性的，以及两个非线性问题：一个时间依赖性的Poiseuille流动以及磁盘上Bose-Einstein冷凝物的流动。
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.
