论文标题
封闭缸中热和泊松方程的准频谱求解器
A Quasi-Optimal Spectral Solver for the Heat and Poisson Equations in a Closed Cylinder
论文作者
论文摘要
我们开发了一种光谱方法来解决封闭缸中的热方程，从而达到了准最佳$ \ Mathcal {o}（n \ log n）$复杂性和高阶，光谱精度。该算法依赖于Chebyshev-Chebyshev--审议（CCF）的离散化缸，该圆柱体很容易实施，并将热方程式分解为较小，稀疏的Sylvester方程的集合。反过来，在准最佳时间中使用交替方向隐式（ADI）方法来求解这些方程。总体而言，这代表了加热方程求解器的改进，从$ \ MATHCAL {O}（N^{4/3}）$（在以前的Chebyshev方法中）到$ \ Mathcal {O}（n \ log n）$。尽管最近已经开发了基于Legendre的方法来实现相似的计算时间，但我们的Chebyshev离散化允许系数转换速度更快。我们通过概述了求解不可压缩的navier的光谱方法来证明这一点的应用 - 在准最佳时间中将圆柱体中的方程式stokations。最后，我们提供了热方程式的数值模拟，表明了对传统光谱搭配方法和有限差异方法的显着加速。
We develop a spectral method to solve the heat equation in a closed cylinder, achieving a quasi-optimal $\mathcal{O}(N\log N)$ complexity and high-order, spectral accuracy. The algorithm relies on a Chebyshev--Chebyshev--Fourier (CCF) discretization of the cylinder, which is easily implemented and decouples the heat equation into a collection of smaller, sparse Sylvester equations. In turn, each of these equations is solved using the alternating direction implicit (ADI) method in quasi-optimal time; overall, this represents an improvement in the heat equation solver from $\mathcal{O}(N^{4/3})$ (in previous Chebyshev-based methods) to $\mathcal{O}(N\log N)$. While Legendre-based methods have recently been developed to achieve similar computation times, our Chebyshev discretization allows for far faster coefficient transforms; we demonstrate the application of this by outlining a spectral method to solve the incompressible Navier--Stokes equations in the cylinder in quasi-optimal time. Lastly, we provide numerical simulations of the heat equation, demonstrating significant speed-ups over traditional spectral collocation methods and finite difference methods.
