We study the convergence and performance of iterative methods with the fourth-order compact discretization schemes for the one and two dimensional convection-diffusion equations. For the one dimensional problem, we investigate the symmetrizability of the coefficient matrix and derive analytical formula for the spectral radius of the point Jacobi iteration matrix. For the two dimensional problem, we conduct Fourier analysis to determine the error reduction factors of several basic iterative methods and comment on their potential use as the smoothers for the multi-level methods. Finally, we perform numerical experiments to verify our Fourier analysis results.
Key words: Convection-diffusion equation, iterative methods, fourth-order compact discretization schemes.