Title: Uniformly convergent numerical method for singularly perturbed 2D delay parabolic convection-diffusion problems on Bakhvalov-Shishkin mesh
Authors: Abhishek Das; Srinivasan Natesan
Addresses: Faculty of Science and Technology, ICFAI University, Tripura Campus, Agartala, 799210, India ' Department of Mathematics, Indian Institute of Technology, Guwahati, 781039, India
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.
Keywords: singularly perturbed 2D delay parabolic convection-diffusion problems; boundary layers; finite difference scheme; Bakhvalov-Shishkin mesh; uniform convergence.
International Journal of Mathematical Modelling and Numerical Optimisation, 2018 Vol.8 No.4, pp.305 - 330
Received: 10 Apr 2017
Accepted: 21 Aug 2017
Published online: 31 Aug 2018 *