2022/10/18 by Rundong Zhou, Zhou, Rundong, Nicolas Grisouard +1
Mathematics · Physics and Astronomy · #Computational Physics (physics.comp-ph) #Electromagnetic Scattering and Analysis #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Quantum Gases (cond-mat.quant-gas) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2210.09736
openalex publication_date 2022/10/18 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28
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 ∼ N-3/2, which governs the method, and the best ∼ e-N, which then the numerical interpolation governs. The complexity is O[N3]. 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.