Numerical approximation of 2D time dependent singularly perturbed convection-diffusion problems with attractive or repulsive turning points

In this work, we are interested in approximating the solution of 2D parabolic singularly perturbed problems of convection-diffusion type. The convective term of the differential equation, associated to the initial and boundary value problem, is such that each one of its components has an interior simple turning point, which can be of attractive or repulsive type. We describe a numerical method to discretize the continuous problem, which combines the fractional implicit Euler method, defined on a uniform mesh, to discretize in time, and the classical upwind finite difference scheme, defined on a nonuniform mesh of Shishkin type, to discretize in space. The fully discrete algorithm has a low computational cost. From a numerical point of view, we see that the method is efficient and uniformly convergent with respect to the diffusion parameter in both cases when the source term is a continuous function or it has a first kind discontinuity at the turning points. Some numerical results for different test problems are showed; from them, we deduce the good properties of the numerical method.