A Richardson scheme of an increased order of accuracy for a semilinear singularly perturbed elliptic convection-diffusion equation

G. I. Shishkin ; L. P. Shishkina

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 3, pp. 458-478

Résumé

The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges $\varepsilon$-uniformly with an order at most almost one. The Richardson technique is used to construct a nonlinear scheme that converges $\varepsilon$-uniformly with an improved order, namely, at the rate $O(N_1^{-2}\ln_1^2N+N_2^{-2})$, where $N_1+1$ and $N_2+1$ are the number of grid nodes along the $x_1$-axis and per unit interval of the $x_2$-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct a linearized iterative Richardson scheme converging $\varepsilon$-uniformly with an improved order. Both the basic and improved iterative schemes converge $\varepsilon$-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration step at which the same $\varepsilon$-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown that no Richardson schemes exist for the convection-diffusion boundary value problem converging $\varepsilon$-uniformly with an order greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based.