A decomposition method for singularly perturbed parabolic convectiondiffusion equations with discontinuous initial conditions

Grid approximations of the Cauchy one-dimensional problem for singularly perturbed parabolic equations are considered. The limit equation (with $\varepsilon=0$, where $\varepsilon$ is the perturbation parameter multiplying the highest derivative) contains the derivative with respect to the spatial variable (convective term). The initial condition has a discontinuity of the first kind. The solution of this problem has singularities in a neighbourhood of the discontinuity for fixed values of the parameter $\varepsilon$, and also a transient layer for small values of $\varepsilon$. We construct special finite difference schemes which $\varepsilon$ uniformly converge on the whole domain. For this we use the domain and solution decomposition technique. The singular solution, generated by the discontinuity of the initial function, is split off and represented in the explicit form in the nearest neighbourhood of the discontinuity. In the neighbourhood of the transient layer, we employ the meshes condensing by a special way.