Title: A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers
Authors: Anirban Majumdar; Srinivasan Natesan
Addresses: Department of Science and Humanities, National Institute of Technology Nagaland, Chumukedima, Nagaland, 797103, India ' Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India
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.
Keywords: singularly perturbed convection-diffusion problem; interior layer; piecewise-uniform Shishkin Mesh; finite difference scheme; uniform convergence.
International Journal of Mathematical Modelling and Numerical Optimisation, 2020 Vol.10 No.1, pp.68 - 101
Received: 12 Nov 2018
Accepted: 29 Apr 2019
Published online: 02 Jan 2020 *