A parameter uniform higher order scheme for 2D singularly perturbed parabolic convection–diffusion problem with turning point

Abstract

In this article, we construct and analyze a higher order numerical method for a class of two dimensional parabolic singularly perturbed problem (PSPP) of convection–diffusion (C–D) type for the case when the convection coefficient is vanishing inside the domain. The asymptotic behavior of the exact solution is studied for the considered problem. Peaceman–Rachford scheme on a uniform mesh is used for time discretization and a hybrid scheme on the Bakhvalov–Shishkin mesh is applied for spatial discretization. The convergence analysis shows that the proposed scheme is uniformly convergent with respect to parameter ɛ. It is established that the hybrid scheme on the Bakhvalov–Shishkin mesh has second order of convergence despite the use of the standard Shishkin mesh which leads to order reduction due to the presence of a logarithmic term. The numerical results corroborate the theoretical expectations and show high accuracy of the proposed scheme over the hybrid scheme on a standard Shishkin mesh. Also, the hybrid scheme is compared with the upwind scheme through the numerical results.