Difference scheme for a singularly perturbed parabolic convection–diffusion equation in the presence of perturbations

An initial–boundary value problem is considered for a singularly perturbed parabolic convection–diffusion equation with a perturbation parameter $\varepsilon$ $(\varepsilon\in(0, 1])$ multiplying the highest order derivative. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform mesh is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. The scheme does not converge $\varepsilon$-uniformly in the maximum norm as the number of its grid nodes is increased. When the solution of the difference scheme converges, which occurs if $N^{-1}\ll\varepsilon$ and $N_0^{-1}\ll1$, where $N$ and $N_0$ are the numbers of grid intervals in $x$ and $t$, respectively, the scheme is not $\varepsilon$-uniformly well conditioned or stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions on the “parameters” of the difference scheme and of the computer (namely, on $\varepsilon$, $N$, $N_0$, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions. Additionally, the conditions are obtained under which the perturbed numerical solution has the same order of convergence as the solution of the unperturbed standard difference scheme.