A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers Online publication date: Fri, 03-Jan-2020
by Anirban Majumdar; Srinivasan Natesan
International Journal of Mathematical Modelling and Numerical Optimisation (IJMMNO), Vol. 10, No. 1, 2020
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.
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