Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions

The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter $\varepsilon^2$, $\varepsilon\in (0,1]$. For small values of the parameter $\varepsilon$, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width $\varepsilon$), respectively, which have bounded smoothness for fixed values of the parameter $\varepsilon$. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge $\varepsilon$-uniformly with the second order of accuracy in $x$ and the first order of accuracy in $t$, up to logarithmic factors.