Spectral Chebyshev method for two-dimensional convection-diffusion problems

A spectral collocation method based on Chebyshev polynomials for numerical analysis of stationary convection-diffusion problems in a two-dimensional rectangular domain is developed. To implement this discrete model, the stabilized bi-conjugate gradient iterative method with a preconditioner is used. The efficiency of two approaches to constructing the preconditioner is studied. The first is triangular components of the incomplete LU factorization of the system matrix, and the second one is the symmetric part of the matrix of a discrete analog of the differential operator of the problem with constant coefficients. In the both cases, the possibility of eliminating the growth in the number of iterations to convergence with increasing grid size is demonstrated. By means of numerical experiments, it is shown that the use of the finite-difference approximation of differential operators in the preconditioner construction allows one to some reduce the computational costs compared to the direct use of spectral differentiation matrices. The results of numerical experiments show, for the cases of sufficiently smooth solutions of the problem, the proposed spectral technique is many times more efficient than the finite-difference method.