Uniformly convergent numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Bakhvalov-Shishkin mesh Online publication date: Thu, 06-Sep-2018
by Abhishek Das; Srinivasan Natesan
International Journal of Mathematical Modelling and Numerical Optimisation (IJMMNO), Vol. 8, No. 4, 2018
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.
Online publication date: Thu, 06-Sep-2018
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