Grid approximations of singularly perturbed systems for parabolic convection-diffusion equations with counterflow

G. I. Shishkin

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 1 (1998) no. 3, pp. 281-297

Résumé

The first boundary value problem is considered on a strip for a system of two singularly perturbed parabolic equations. The perturbation parameters multiplying the highest derivatives of each of the equations are mutually independent and can take arbitrary values from the interval $(0,1]$. When these parameters equal zero, the system of parabolic equations degenerates into a system of hyperbolic first order equations coupled by the reaction terms. The convective terms (i.e., their components orthogonal to the boundaries of the strip) that are involved in the different equations have the opposite directions (convection with counterflow). This case brings us to the appearance of boundary layers in the neighbourhood of both boundaries of the strip. For this boundary value problem, the difference schemes that converge uniformly with respect to the parameters are constructed here using the condensing mesh method. We also consider the construction of parameter uniform convergent difference schemes for a system of singularly perturbed elliptic equations that degenerate into first order equations if the parameter equals zero.