A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers. (2nd January 2020)

• Record Type: Journal Article
• Authors: Majumdar, Anirban; Natesan, Srinivasan
• Is Part Of: International journal of mathematical modelling and numerical optimisation. Volume 10:Number 1(2019)
• Page Start: 68
• Page End: 101
• Publication Date: 2020-01-02
• Subjects: singularly perturbed convection-diffusion problem -- interior layer -- piecewise-uniform Shishkin Mesh -- finite difference scheme -- uniform convergence; Mathematical models -- Periodicals; Mathematical optimization -- Periodicals; 519.605
• Languages: English
• ISSNs: 2040-3607

Abstract

In this paper, we study the numerical solutions of singularly perturbed convection-diffusion two-point BVP as well as one-dimensional parabolic convection-diffusion IBVP with discontinuous convection coefficient (positive throughout the domain) and source term. The analytical solutions of these kind of problems exhibit a boundary layer near x = 0 and a weak interior layer near x = ξ. We discretise the spatial domain by the piecewise-uniform Shishkin mesh and the temporal domain by a uniform mesh. To approximate the spatial derivatives, we apply the hybrid finite difference scheme. The implicit-Euler scheme is used for discretising the temporal derivative. For the time independent problem, we derive that the proposed hybrid scheme is ε-uniformly convergent of almost second-order and for the time dependent problem, we also prove that the proposed scheme is ε-uniformly convergent of almost second-order in space and first-order in time. To validate the theoretical estimates, some numerical results are presented.