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 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 O(N 4/3 ) (in previous Chebyshev-based methods) to 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 tr ansforms; th is is cr itical in ap plications wi th no n-linear fo rcing, wh ich we di scuss in the context of the incompressible Navier-Stokes equations.Lastly, we provide numerical simulations of the heat equation, demonstrating significant speed-ups over traditional spectral collocation methods and finite d ifference methods.