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

摘要

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 ε_1 and ε_2 multiplying, respectively, the second and first derivatives with respect to the space variable. Depending on the relation between the parameters, the differential equation can be either of reaction–diffusion type or of convection–diffusion 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 fluid out of the boundary layer (model problems of this type appear in the mathematical modelling of heat transfer processes for flow past a flat 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 finite difference scheme on piecewise uniform meshes which generates numerical solutions converging ε-uniformly with order O(N~(-1) ln N + N_0~(-1)), 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 ε_1 → 0, the character of the boundary layer depends essentially on the vector-valued parameter ε = (ε_1, ε_2). This prevents us from constructing an ε-uniformly convergent scheme having a transition parameter which is independent of the parameter ε_2.