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

Abstract

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 ɛ-uniformly well conditioned or ɛ-uniformly stable to perturbations of the data of the grid problem (here, ɛ is a perturbation parameter, ɛ ∈ (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 ɛ-uniformly in the maximum norm at an O(N−1lnN + N0−1) rate, where N + 1 and N0 + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ɛ-uniformly well conditioned and ɛ-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ−2lnδ−1 + δ0−1); 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, δ = N−1lnN and δ0 = N0−1 are the accuracies of the discrete solution in x and t, respectively.