A Quasi-Optimal Spectral Solver for the Heat and Poisson Equations in a Closed Cylinder

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.