Numerical treatment of interfaces in Quantum Mechanics

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.