  In this article we develop a numerical scheme to deal with interfaces between
touching numerical grids when solving Schr\"o{}dinger equation. In order to
pass the information among grids we use the values of the fields only at the
contact point between them. Surprisingly we obtain a convergent methods which
is third order accurate with respect to the spatial resolution. In test cases,
at the minimal resolution needed to describe correctly the waves, the error of
this approximation is similar to that of a homogeneous (centered differences
everywhere) scheme with three points stencil, that is a sixth order finite
difference operator. The semi-discrete approximation preserves the norm and
uses standard finite difference operators satisfying summation by parts. For
the time integrator we use a semi-implicit IMEX Runge Kutta method.
