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Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem.

Shegaye Lema Cheru1, Gemechis File Duressa2, Tariku Birabasa Mekonnen3

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Summary

This study introduces a new fitted operator finite difference method for parabolic convection-diffusion-reaction problems. The method achieves second-order accuracy and uniform convergence, outperforming existing techniques.

Keywords:
Crank-NicolsonFitted operatorGaussian quadratureSecond order interpolationSingularly perturbed problems

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Partial differential equations

Background:

  • Parabolic convection-diffusion-reaction problems involve small parameters multiplying diffusion and convection terms.
  • These problems often exhibit boundary layers or internal layers, posing numerical challenges.

Purpose of the Study:

  • To develop and analyze a robust numerical method for solving parabolic convection-diffusion-reaction problems.
  • To ensure the proposed method achieves high accuracy and uniform convergence for problems with singular perturbations.

Main Methods:

  • A fitted operator finite difference method is employed.
  • The Crank-Nicolson method discretizes the time variable, while a two-point Gaussian quadrature rule and second-order interpolation discretize the spatial variables.
  • The fitting factor is determined using singular perturbation theory.

Main Results:

  • The developed numerical scheme is proven to be second-order accurate.
  • Uniform convergence of the scheme is demonstrated.
  • Numerical examples show superior accuracy compared to existing methods.

Conclusions:

  • The proposed fitted operator finite difference method is effective for solving parabolic convection-diffusion-reaction problems.
  • The method provides accurate and uniformly convergent solutions, even for problems with abrupt solution changes.