On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

We consider a 1D Schrodinger equation with variable coefficients on the half-axis. We study a family of two-level symmetric finite-difference schemes with a three-point parameter dependent averaging in space. This family includes a number of particular schemes. The schemes are coupled to an approximate transparent boundary condition (TBC). We prove two stability bounds with respect to initial data and a free term in the main equation, under suitable conditions on an operator of the approximate TBC. We also consider the family of schemes on an infinite mesh in space. We derive and analyze the discrete TBC allowing to restrict these schemes to a finite mesh and prove the stability conditions for it. Numerical examples are also included.

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