On the finite difference approximation to the convection-diffusion equation

In this paper, the system of ordinary differential equations arisen from discretizing the convection-diffusion equation with respect to the space variable to compute its approximate solution along a time level is considered. This system involves in computing ekAy for some vector y, where k is the time step-size and A is a large tridiagonal Toeplitz matrix. The common ways to compute an approximate solution of the convection-diffusion equation are based on replacing ekAy by an its approximation. We give an explicit expression for the exact value of ekAy and then the numerical results of our method with that of some well-known methods are compared.