An almost second order hybrid scheme for the numerical solution of singularly perturbed parabolic turning point problem with interior layer

• Published in: Mathematics and computers in simulation 2021-07, Vol.185, p.733-753
• Main Authors: Yadav, Swati, Rai, Pratima
• Format: Article
• Language: English
• Subjects: Hybrid scheme; Interior boundary layer; Parabolic differential equations; Shishkin mesh; Singular perturbation; Turning point
• ISSN: 0378-4754; 1872-7166
• DOI: 10.1016/j.matcom.2021.01.017

In this article, we consider a one dimensional singularly perturbed parabolic convection–diffusion problem with interior turning point. The convection coefficient of the considered problem is vanishing inside the spatial domain and also, exhibits an interior layer. As a result, the exact solution of the considered problem contains an interior layer. A higher order numerical method is constructed and analyzed for the numerical solution of the considered problem. To discretize the time direction, we have used the classical implicit Euler method on a uniform mesh. Also, a hybrid finite difference scheme is employed on a generalized Shishkin mesh condensing in the interior layer region to discretize the spatial domain. Rigorous analysis is performed to show that the proposed method is ε-uniformly convergent of order almost two. The higher accuracy and convergence rate of the proposed scheme are verified via implementing numerical experiments on two test problems. Comparison is done with the scheme proposed in O’Riordan and Quinn (2015) for the considered class of problems.