Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 5, pp. 813-830

Résumé

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an $\varepsilon$-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an $\varepsilon$-uniform bound $O(\delta^{-2}\ln\delta_1^{-1}+\delta_0^{-1})$ is obtained, where $\delta_1$ and $\delta_0$ are the error components due to the approximation of the derivatives with respect to $x$ and $t$, respectively. Thus, this scheme is $\varepsilon$-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate $O(\varepsilon^{-1}\delta_1^{-2}+\delta_0^{-1})$; this scheme is not $\varepsilon$-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as $\varepsilon\to0$, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of $O(\varepsilon^{-1}\delta_1^{-2}+\delta_0^{-1})$. Neither the matrix of the $\varepsilon$-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is $\varepsilon$-uniformly well-conditioned.