  An initial-boundary value problem for the 1D self-adjoint parabolic equation
on the half-axis is solved. We study a broad family of two-level
finite-difference schemes with two parameters related to averagings both in
time and space. Stability in two norms is proved by the energy method. Also
discrete transparent boundary conditions are rigorously derived for schemes by
applying the method of reproducing functions. Results of numerical experiments
are included as well.
