Grid approximation for a singularly perturbed parabolic reaction-diffusion equation with a moving concentrated source

On an axis $\mathbb R$, we consider an initial value problem for a singularly perturbed parabolic reactiondiffusion equation in the presence of a moving concentrated source. Classical finite difference schemes for such problem converge only under the condition $\varepsilon\gg N^{-1}+N_0^{-1}$, where $\varepsilon$ is the singular perturbation parameter, the values $N$ and $N_0$ define the number of nodes in the grids with respect to $x$ (on a segment of unit length) and $t$. We study schemes on meshes which are locally refined in a neighbourhood of the set $\gamma^*$, that is, the trajectory of the moving source. It is shown that there are no schemes convergent $\varepsilon$-uniformly, in particular, for $\varepsilon=\mathscr O(N^{-2}+N_0^{-2})$, in the class of schemes based on classical approximations of the problem on “piecewise uniform” rectangular meshes which are locally condensing with respect to both $x$ and $t$. Using stencils with nonorthogonal (in $x$ and $t$) arms in the nearest neighbourhood of the set $\gamma^*$ and meshes condensing, along $x$, in the neighbourhood of $\gamma^*$, we construct schemes that converge euniformly with the rate $\mathscr O(N^{-k}\ln^kM+N_0^{-1})$, $k=1,2$.