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A uniformly convergent quadratic B-spline based scheme for singularly perturbed degenerate parabolic problems

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  • Singh, Satpal
  • Kumar, Devendra
  • Ramos, Higinio

Abstract

In this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerate parabolic problems. The degenerative nature of the problem is due to the coefficient b(x,t)=b0(x,t)xp,p≥1 of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layer in the neighborhood of the left end side of the domain. The problem is semi-discretized using the Crank–Nicolson scheme, and then the quadratic spline basis functions are used to discretize the semi-discrete problem. A priori bounds for the solution (and its derivatives) of the continuous problem are given, which are necessary to analyze the error. A rigorous error analysis shows that the proposed method is boundary layer resolving and second-order parameter uniformly convergent. Some numerical experiments have been devised to support the theoretical findings and the effectiveness of the proposed scheme.

Suggested Citation

  • Singh, Satpal & Kumar, Devendra & Ramos, Higinio, 2022. "A uniformly convergent quadratic B-spline based scheme for singularly perturbed degenerate parabolic problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 195(C), pages 88-106.
    • Handle: RePEc:eee:matcom:v:195:y:2022:i:c:p:88-106
    DOI: 10.1016/j.matcom.2021.12.026
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    References listed on IDEAS

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    1. S. Natesan & M. Ramanujam, 1998. "Initial-Value Technique for Singularly-Perturbed Turning-Point Problems Exhibiting Twin Boundary Layers," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 37-52, October.
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    Cited by:

    1. Singh, Satpal & Kumar, Devendra, 2023. "Parameter uniform numerical method for a system of singularly perturbed parabolic convection–diffusion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 360-381.

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