Grid approximation of singularly perturbed parabolic convection-diffusion equations with a piecewise-smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter $\varepsilon$ that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point $x_0$. For small values of $\varepsilon$ a boundary layer with the typical width of $\varepsilon$ appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point $(x_0,0)$, a transient (moving in time) layer with the typical width of $\varepsilon^{1/2}$ appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem $\varepsilon$-uniformly on the entire set $\bar G$, approximate the diffusion flow (i.e., the product $\varepsilon(\partial/\partial x)u(x,t))$ on the set $\bar G^*=\bar G\setminus\{(x_0,0)\}$, and approximate the derivative $(\partial/\partial x)u(x,t)$ on the same set outside the $m$-neighborhood of the boundary layer. The approximation of the derivatives $\varepsilon^2(\partial^2/\partial x^2)u(x,t))$ and $(\partial/\partial t)u(x, t)$ on the set $\bar G^*$ is also examined.