We develop a spectral method to solve the heat equation in a closed cylinder, achieving a near-optimal $\mathcal{O}(N\log N)$ complexity and high-order, spectral accuracy. The algorithm relies on a novel 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, which improves the complexity of each solve from cubic in the matrix size (in more traditional methods) to log-linear; overall, this represents an improvement in the heat equation solver from $\mathcal{O}(N^{7/3})$ (in traditional methods) to $\mathcal{O}(N\log N)$. Lastly, we provide numerical simulations demonstrating significant speed-ups over traditional spectral collocation methods and finite difference methods, and we provide a framework by which this heat equation solver could be applied to the incompressible Navier--Stokes equations. For the latter, we decompose the equations using a poloidal-toroidal (PT) decomposition, turning them into heat equations with nonlinear forcing from the advection term; by using implicit-explicit methods to integrate these, we can achieve the same $\mathcal{O}(N\log N)$ complexity and spectral accuracy achieved here in the heat equation.
翻译:我们开发了一个光谱方法来解决封闭气瓶中的热方程, 实现近最佳 $\ mathcal{O}( N\log N) 的复杂度和高序、 光谱精度。 算法依赖于气瓶的新奇Chebyshev- Chebyshev- Fourier( CCF) 离散化, 这个方法很容易实施, 并且将热方程分解成一个较小、 稀疏的 Sylvester 方程的集合。 最后, 我们提供数字模拟, 表明超越传统光谱相向内隐含( ADI) 方法的快速增速, 这个方法可以提高每立方格的复杂性( 以更传统的方法) 到日志线; 总的来说, 这代表着热方程求解的改进器从 $\ mathcal{O} ( N\ 7/3} ( ) 美元( 在传统方法中) 到 $ mathclimal decomplain 等式的变异式, 也可以将这种变异式的变换成 等式。