Numerical approximations for the heat equation and the advection- diffusion equation in the cylinder are examined. The space discretization is based on the pseudospectral Chebyshev method, which is a collocation method at the Chebyshev Gaussian nodes. This method allows one to achieve high accuracy for smooth solutions. Implementation can be made efficiently using the fast Fourier transform. The pseudospectral solution is advanced in time using the finite difference \(\theta\)-method for the diffusion term. For the fully discrete problems unconditional stability and convergence in the norms of the weighted Sobolev spaces is established. Optimal order of convergence, depending on the time step \(\Delta\) t, the degree N of the polynomial which approximates the solution in space, and of the Sobolev-regularity of the solution, is found.