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Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection–diffusion problems

  • Kaushik Mukherjee
  • Srinivasan NatesanEmail author
Original Research
  • 82 Downloads

Abstract

This article focuses on developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection–diffusion initial-boundary value problems exhibiting a regular boundary layer. For approximating the time derivative, we use the Peaceman–Rachford alternating direction implicit method on uniform mesh and for the spatial discretization, a hybrid finite difference scheme is proposed on a special rectangular mesh which is tensor-product of piecewise-uniform Shishkin meshes in the spatial directions. We prove that the numerical scheme converges uniformly with respect to the perturbation parameter \(\varepsilon \) and also attains almost second-order spatial accuracy in the discrete supremum norm. Finally, numerical results are presented to validate the theoretical results. In addition to this, numerical experiments are conducted to demonstrate the effect of the time-dependent boundary conditions in the order of convergence numerically by introducing the classical evaluation of the boundary data; and also the improvement in the spatial order of accuracy of the present method by considering the Bakhvalov–Shishkin mesh in the spatial directions.

Keywords

Singularly perturbed parabolic problem Regular boundary layer Numerical scheme Alternating directions Piecewise-uniform Shishkin mesh Bakhvalov–Shishkin mesh uniform convergence 

Mathematics Subject Classification

65M06 65M12 CR G1.8 

Notes

Acknowledgements

The authors wish to acknowledge the referees for their valuable comments and suggestions, which helped to improve the presentation.

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Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Space Science and TechnologyThiruvananthapuramIndia
  2. 2.Department of MathematicsIndian Institute of Technology GuwahatiGuwahatiIndia

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