Fourie-Chebyshev Tau Method for Poisson-type Equations in Cartesian and Polar Coordinates

In this paper, we present a Fourier-Chebyshev tau method (FCTM) with quasi-optimal computational complexity to solve the Poisson-type equations in Cartesian and polar coordinates. In polar coordinates, the problem is solved directly without mapping, while the singularity at the pole is avoided by using Chebyshev-Gauss-Labatto points, and the nonlinear terms are eliminated by introducing the Sturm-Liouville type equation. Several Poisson-type equations have been solved and compared with previous literature, the incompressible Navier-Stokes equations have also been solved. Numerical results show the accuracy and efficiency of the present method. On our personal computer, without parallelization, the method only takes 6.61 seconds to solve a Poisson-type equation with grid number 8192×8192, which is nearly 6 times faster than the spectral collocation method.