A numerical approach to approximate the solution of a quasilinear singularly perturbed parabolic convection diffusion problem having a non-smooth source term

The objective of the present paper is to solve a one-dimensional quasilinear parabolic singularly perturbed problem with a discontinuous source term. Due to the presence of such a discontinuity, an interior layer exists at the location of the discontinuity. The problem is solved by discretizing the spatial variable on a piecewise uniform Shishkin mesh using the standard upwind approach, while the backward Euler scheme is employed on a uniform mesh to discretize the time variable. The method is $ \varepsilon $-uniformly convergent, providing first-order convergence in the time domain and almost first-order convergence in the spatial variable. To validate the theoretical findings, the scheme was tested by numerically solving two examples.