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In this article, we have considered a time-dependent two-parameter singularlyperturbed parabolic problem with discontinuous convection coefficient andsource term. The problem contains the parameters $\epsilon$ and $\mu$multiplying the diffusion and convection coefficients, respectively. A boundarylayer develops on both sides of the boundaries as a result of these parameters.An interior layer forms near the point of discontinuity due to thediscontinuity in the convection and source term. The width of the interior andboundary layers depends on the ratio of the perturbation parameters. We discussthe problem for ratio $\displaystyle\frac{\mu^2}{\epsilon}$. We used an upwindfinite difference approach on a Shishkin-Bakhvalov mesh in the space and theCrank-Nicolson method in time on uniform mesh. At the point of discontinuity, athree-point formula was used. This method is uniformly convergent with secondorder in time and first order in space. Shishkin-Bakhvalov mesh providesfirst-order convergence; unlike the Shishkin mesh, where a logarithmic factordeteriorates the order of convergence. Some test examples are given to validatethe results presented.

Parameter uniform numerical method for singularly perturbed two  parameter parabolic problem with discontinuous convection coefficient and  source term