An efficient numerical method for 1D singularly perturbed parabolic convection-diffusion systems with repulsive interior turning points

In this work, we propose and study a numerical method to solve efficiently one-dimensional parabolic singularly perturbed systems of convection-diffusion type, for which the convection coefficient is zero at an interior point of the spatial domain. We focus our attention on the case of having the same diffusion parameter in both equations; as well we assume adequate signs on the convective coefficients in order to the interior turning point is of repulsive type. Under these conditions, if the data of the problem are composed by continuous functions, the exact evolutionary solution, in general, has regular boundary layers at the end points of the spatial domain. To solve this type of problems, we combine the fractional implicit Euler method and the classical upwind scheme, defined on a special mesh of Shishkin type. The resulting numerical method reach uniform convergence of first order in time and almost first order in space. Numerical results obtained for different test problems are shown which corroborate in practice the uniform convergence of the numerical algorithm and also their computational efficiency in comparison with classical numerical methods used for the same type of problems.