A high-order exponential scheme for solving 1D unsteady convection-diffusion equations

In this paper, a high-order exponential (HOE) scheme is developed for the solution of the unsteady one-dimensional convection-diffusion equation. The present scheme uses the fourth-order compact exponential difference formula for the spatial discretization and the (2,2) Padé approximation for the temporal discretization. The proposed scheme achieves fourth-order accuracy in temporal and spatial variables and is unconditionally stable. Numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature. The results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems.