Grid approximation of singularly perturbed parabolic equations in the presence of weak and strong transient layers induced by a discontinuous right-hand side

The initial value problem on a line for singularly perturbed parabolic equations with convective terms is investigated. The first-and the second-order space derivatives are multiplied by the parameters $\varepsilon_1$ and $\varepsilon_2$, respectively, which may take arbitrarily small values. The right-hand side of the equations has a discontinuity of the first kind on the set $\bar\gamma=[x=0]\times[0,T]$. Depending on the relation between the parameters, the appearing transient layers can be parabolic or regular, and the “intensity” of the layer (the maximum of the singular component) on the left and on the right of $\bar\gamma$ can be substantially different. If the parameter $\varepsilon_2$ at the convective term is finite, the transient layer is weak. For the initial value problems under consideration, the condensing grid method is used to construct finite difference schemes whose solutions converge (in the discrete maximum norm) to the exact solution uniformly with respect to $\varepsilon_1$ and $\varepsilon_2$ (when $\varepsilon_2$ is finite and, therefore, the transient layers are weak, no condensing grids are required).