Uniformly convergent numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Bakhvalov-Shishkin mesh

Abstract

In this paper, we consider a class of singularly perturbed 2D delay parabolic convection-diffusion initial-boundary-value problems. To solve this problem numerically, we use a modified Shishkin mesh (Bakhvalov-Shishkin mesh) for the discretisation of the domain in the spatial directions and uniform mesh in the temporal direction. The time derivative is discretised by the implicit-Euler scheme and the spatial derivatives are discretised by the upwind finite difference scheme. We derive some conditions on the mesh-generating functions which are useful for the convergence of the method, uniformly with respect to the perturbation parameter. We prove that the proposed scheme on the Bakhvalov-Shishkin mesh is first-order convergent in the discrete supremum norm, which is optimal and does not require any extra computational effort compared to the standard Shishkin mesh. Numerical experiments verify the theoretical results.