Applied Mathematics and Computation 317 (2018) 223–233 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Numerical approximation of 2D time dependent singularly perturbed convection–diffusion problems with attractive or repulsive turning points C. Clavero a,∗ , J. Vigo-Aguiar b a Department of Applied Mathematics and IUMA, University of Zaragoza, Spain b Department of Applied Mathematics, University of Salamanca, Spain a r t i c l e i n f o Keywords: 2D parabolic convection–diffusion problems Turning points Fractional Euler method Finite differences scheme Special meshes Uniform convergence a b s t r a c t 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 repul- sive 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. © 2017 Elsevier Inc. All rights reserved. 1. Introduction In this paper, we consider 2D parabolic convection–diffusion problems given by Lu ≡ ∂ u ∂ t + (L 1,ε (t ) + L 2,ε (t ))u = f , in  × (0, T ], u(x, y, 0) = ϕ (x, y), in , u(x, y, t ) = 0, in ∂  × [0, T ], (1) where  ≡ (0, 1) 2 and the differential operators L i,ε , i = 1, 2, are defined by L 1,ε (t ) ≡− ε ∂ 2 ∂ x 2 + v 1 (x, y, t ) ∂ ∂ x + k 1 (x, y, t ), L 2,ε (t ) ≡− ε ∂ 2 ∂ y 2 + v 2 (x, y, t ) ∂ ∂ y + k 2 (x, y, t ), (2) respectively. ∗ Corresponding author. E-mail addresses: clavero@unizar.es (C. Clavero), jvigo@usal.es (J. Vigo-Aguiar). http://dx.doi.org/10.1016/j.amc.2017.08.059 0096-3003/© 2017 Elsevier Inc. All rights reserved.