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KRM

We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large $x$.
We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it.
The method is coupled to an approximate transparent boundary condition (TBC).
We prove uniform in time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC.
We also consider the corresponding method on an infinite mesh on the half-axis.
We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh.
The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution.
The stability conditions are justified for it.
The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials.

KRM

We consider a 1D Schrödinger 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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