Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not $\varepsilon$-uniformly well conditioned or $\varepsilon$-uniformly stable to perturbations of the data of the grid problem (here, $\varepsilon$ is a perturbation parameter, $\varepsilon\in(0,1]$). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges $\varepsilon$-uniformly in the maximum norm at an $O(N^{-1}\ln N+N_0^{-1})$ rate, where $N+1$ and $N_0+1$ are the numbers of grid nodes in $x$ and $t$, respectively. This scheme is $\varepsilon$-uniformly well conditioned and $\varepsilon$-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order $O(\delta^{-2}\ln\delta^{-1}+\delta_0^{-1})$; i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, $\delta=N^{-1}\ln N$ and $\delta_0=N_0^{-1}$ are the accuracies of the discrete solution in $x$ and $t$, respectively.