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A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data.

Nirmali Roy1, Anuradha Jha1

  • 1Indian Institute of Information Technology Guwahati, Bongora, 781015, India.

Methodsx
|January 23, 2023
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Summary

This study presents a numerical method for singularly perturbed reaction-convection-diffusion problems with discontinuous coefficients. The approach achieves first-order uniform convergence using an upwind difference scheme on a Shishkin-Bakhvalov mesh.

Keywords:
Boundary layersFinite difference methodInterior layersShishkin-Bakhvalov mesh

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Singularly perturbed problems are crucial in modeling phenomena with multiple scales.
  • Discontinuities in coefficients pose significant challenges for standard numerical methods.
  • Efficient numerical solutions are needed for reaction-convection-diffusion equations.

Purpose of the Study:

  • To develop and analyze a numerical method for one-dimensional singularly perturbed reaction-convection-diffusion problems.
  • To address the challenges posed by discontinuous convection coefficients and source terms.
  • To achieve uniform convergence with respect to perturbation parameters.

Main Methods:

  • Utilized the upwind difference method for numerical solution.
  • Employed a Shishkin-Bakhvalov mesh, graded in the layer region and uniform elsewhere.
  • Applied a three-point difference scheme at the point of discontinuity.

Main Results:

  • Demonstrated first-order uniform convergence in the supremum norm.
  • Numerical results confirmed the theoretical convergence rate.
  • The Shishkin-Bakhvalov mesh yielded better convergence orders than the standard Shishkin mesh.

Conclusions:

  • The proposed numerical method is effective for the studied problem class.
  • The Shishkin-Bakhvalov mesh is advantageous for handling discontinuities.
  • The method provides a reliable approach for singularly perturbed problems with discontinuous coefficients.