Grid Approximation of a Singularly Perturbed Parabolic Equation with Degenerating Convective Term and Discontinuous Right-Hand Side

Abstract

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by $\varepsilon^2$, where $\varepsilon$ is the perturbation parameter, $\varepsilon\in (0,1]$. For small values of $\varepsilon$, an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition $N^{-1} = o(\varepsilon)$, $N^{-1}_0 = o(1)$, where $N +1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges $\varepsilon$-uniformly at the rate $\mathcal{O}(N^{-1}lnN+N^{-1}_0)$. Numerical experiments confirm the theoretical results.