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An initial-boundary value problem for the 1D self-adjoint parabolic equationon the half-axis is solved. We study a broad family of two-levelfinite-difference schemes with two parameters related to averagings both intime and space. Stability in two norms is proved by the energy method. Alsodiscrete transparent boundary conditions are rigorously derived for schemes byapplying the method of reproducing functions. Results of numerical experimentsare included as well.

On a family of finite-difference schemes with discrete transparent boundary conditions for a parabolic equation on the half-axis