Numerical solution of a quasilinear parabolic equation with a boundary layer

To solve a quasilinear parabolic equation with small parameter multiplying the derivatives with respect to the spatial variables, a numerical method is constructed with an estimate of the error, which is uniform with respect to the parameter. The construction of a nonlinear difference scheme is based on the method of straight lines and on the application of exact systems to one-dimensional problems. The computational mesh is chosen so that its density increases in a suitable way in the neighbourhood of the boundary. We propose that the nonlinear scheme be solved by an iterative algorithm, which converges uniformly with respect to the small parameter.