The convection-diffusion equation is of primary importance in understanding transport phenom-ena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in both time and space variables. A procedure was given in detail to solve the one-dimensional unsteady convec-tion-diffusion equation through a combination of Runge-Kutta methods and compact difference schemes. The combination method has fourth-order accuracy in both time and space variables. Numerical experiments were conducted and the results were compared with those obtained by the Crank-Nicolson method in order to check the accuracy of the combination method. The anal-ysis results indicated that the combination method is numerically stable at low wave numbers and small CFL numbers. The combination method has higher accuracy than the Crank-Nicolson method.