Discrete transparent boundary conditions for parabolic equations

Transparent boundary conditions for linear parabolic equations for a bounded domain can be derived under the assumption that the coefficients are constant on the exterior of that domain. In 1D these boundary conditions are non-local in time (of memory type). For the Crank-Nicolson finite difference scheme, discrete transparent boundary conditions are derived, and the resulting scheme is proved to be stable. The approach is closely related to [1].