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

TL;DR

This paper develops a uniform numerical method for a complex two-parameter singularly perturbed parabolic problem with discontinuities, achieving reliable convergence on specialized meshes and addressing boundary and interior layers.

Contribution

It introduces a parameter-uniform finite difference scheme on a Shishkin-Bakhvalov mesh for a challenging two-parameter problem with discontinuities, ensuring second-order temporal and first-order spatial convergence.

Findings

Method achieves uniform convergence despite boundary and interior layers.

Shishkin-Bakhvalov mesh improves convergence over traditional Shishkin mesh.

Numerical tests validate the theoretical convergence rates.

Abstract

In this article, we have considered a time-dependent two-parameter 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 Shishkin-Bakhvalov mesh in the space and the Crank-Nicolson method in time on uniform mesh. At the point of discontinuity, a three-point formula was used. This method is uniformly convergent with second order in time and first order in space. Shishkin-Bakhvalov mesh provides first-order convergence; unlike the Shishkin mesh, where a logarithmic factor deteriorates the order of convergence. Some test examples are given to validate the results presented.