A fast singly diagonally implicit runge-kutta method for solving 1D unsteady convection-diffusion equations

In this article, a fast singly diagonally implicit Runge-Kutta method is designed to solve unsteady one-dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three-stage singly diagonally implicit Runge-Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth-order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature.