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We consider the Cauchy problem for the 1D generalized Schrödinger 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 global Richardson extrapolation in time to derive the high order method both with respect to space and time steps. To study its properties, we comment on its stability and give results of numerical experiments and enlarged practical error analysis for three typical examples. Unlike most common second order (in either space or time step) methods, the proposed method is able to provide high precision results in the uniform norm by using adequate computational costs. It works even in the case of discontinuous potentials and non-smooth solutions far beyond the scope of its standard theory. Comparing our results to the previous ones, we obtain  much more  accurate results using  much less  amount of both elements and time steps.

The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation