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

On the Richardson Extrapolation in Time of Finite Element Method with Discrete TBCs for the Cauchy Problem for the 1D Schr\"odinger Equation