Parameter uniform numerical method for singularly perturbed two parameter parabolic problem with discontinuous convection coefficient and source term
Abstract
In this article, we have considered a timedependent twoparameter singularly perturbed parabolic problem with discontinuous convection coefficient and source term. The problem contains the parameters $\epsilon$ and $\mu$ multiplying the diffusion and convection coefficients, respectively. A boundary layer develops on both sides of the boundaries as a result of these parameters. An interior layer forms near the point of discontinuity due to the discontinuity in the convection and source term. The width of the interior and boundary layers depends on the ratio of the perturbation parameters. We discuss the problem for ratio $\displaystyle\frac{\mu^2}{\epsilon}$. We used an upwind finite difference approach on a ShishkinBakhvalov mesh in the space and the CrankNicolson method in time on uniform mesh. At the point of discontinuity, a threepoint formula was used. This method is uniformly convergent with second order in time and first order in space. ShishkinBakhvalov mesh provides firstorder convergence; unlike the Shishkin mesh, where a logarithmic factor deteriorates the order of convergence. Some test examples are given to validate the results presented.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 DOI:
 10.48550/arXiv.2208.04109
 arXiv:
 arXiv:2208.04109
 Bibcode:
 2022arXiv220804109R
 Keywords:

 Mathematics  Numerical Analysis;
 65M06;
 65M12;
 65M15