  We consider the Cauchy problem for the 1D generalized Schr\"odinger equation
on the whole axis. To solve it, any order finite element in space and the
Crank-Nicolson in time method with the discrete transparent boundary conditions
(TBCs) has recently been constructed. Now we engage the Richardson
extrapolation to improve significantly the accuracy in time step. To study its
properties, we give results of numerical experiments and enlarged practical
error analysis for three typical examples. The resulting method is able to
provide high precision results in the uniform norm for reasonable computational
costs that is unreachable by more common 2nd order methods in either space or
time step. Comparing our results to the previous ones, we obtain much more
accurate results using much less amount of both elements and time steps.
