We consider a singularly perturbed parabolic problem in two dimensions containing a discontinuous convection coefficient and a source term. In this problem the diffusion and convection coefficients are multiplied with the parameters \(0<\epsilon \ll 1\) and \(0\le \mu \ll 1,\) respectively. The solution of this problem possess both boundary and interior layers. We discuss the problem for ratio \(\displaystyle \frac{\mu ^2}{\epsilon } \rightarrow 0\) as \(\epsilon \rightarrow 0\) . In the space variable, an upwind difference scheme on a non-uniform mesh is used. In the time variable, the Crank–Nicolson method is applied on a uniform mesh. At the point of discontinuity, a three-point formula is used. When \(\epsilon \le N^{-1}\) , Shishkin–Bakhvalov mesh is used in space and for \(N^{-1} \le \epsilon \) , Bakhvalov type mesh is used. The method is first order parameter uniform convergent in space and second order in time in the supremum norm with respect to the singular perturbation parameters. This modification in mesh when \(\epsilon \le N^{-1}\) and \(\epsilon \ge N^{-1}\) provides first-order convergence for all values of N. Numerical investigations also confirm these theoretical results.
