Grid approximation of a singularly perturbed boundary value problem modelling heat transfer in the case of 'ow over a 'at plate with suction of the boundary layer

In the present paper we consider a boundary value problem on the semiaxis (0;∞) for a singularly perturbed parabolic equation with the two perturbation parameters 1 and 2 multiplying, respectively, the second and 5rst derivatives with respect to the space variable. Depending on the relation between the parameters, the di7erential equation can be either of reaction–di7usion type or of convection–di7usion type. Correspondingly, the boundary layer can be either parabolic or regular. For this problem we consider the case when the boundary layer can be controlled by continuous suction of the 'uid out of the boundary layer (model problems of this type appear in the mathematical modelling of heat transfer processes for 'ow past a 'at plate). Errors in the approximations generated by standard numerical methods can be unsatisfactorily large for small values of the parameter 1. We construct a monotone 5nite di7erence scheme on piecewise uniform meshes which generates numerical solutions converging -uniformly with order O(N lnN +N 0 ), where N0 is the number of nodes in the time mesh and N is the number of meshpoints on a unit interval of the semiaxis in x. Although the solution of problem has a singularity only for 1 → 0, the character of the boundary layer depends essentially on the vector-valued parameter = ( 1; 2). This prevents us from constructing an -uniformly convergent scheme having a transition parameter which is independent of the parameter 2. c © 2003 Elsevier B.V. All rights reserved. This work has been supported partially by the Russian Foundation for Basic Research under Grant No. 01-01-01022 and by the Enterprise Ireland Research Grant SC-2000-070. ∗Corresponding author. Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, 16 S. Kovalevskaya Street, Ekaterinburg 620219, Russia. E-mail addresses: [email protected], [email protected] (J.J.H. Miller), [email protected], [email protected] (G.I. Shishkin), [email protected] (B. Koren), [email protected] (L.P. Shishkina). 0377-0427/$ see front matter c © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2003.09.026 222 J.J.H. Miller et al. / Journal of Computational and Applied Mathematics 166 (2004) 221–232