A problem for grid approximation of the diffusion flow in numerical modelling of pollution transport

Boundary value problems for quasi-linear parabolic equations with Dirichlet boundary condition are considered on a segment and a rectangular domain. Highest derivatives of these equations are multiplied by a parameter which can get any value in the interval $(0,1]$. When the parameter is equal to zero the parabolic equations are reduced to the first order equations which don't contain spatial derivatives. For such problems the solution of classical difference scheme and approximation of the diffusion flow don't converge uniformly with respect to the parameter. Relative error of the diffusion flow increases without bound when the parameter tends to zero. With using of special condensing grids we construct the difference schemes which allow to approximate the solution and the normalized diffusion flow uniformly with respect to the parameter.