Numerical treatment of interfaces in Quantum Mechanics

TL;DR

This paper introduces a third-order accurate numerical scheme for solving the Schrödinger equation across interfaces between touching grids, maintaining norm preservation and achieving high accuracy with minimal resolution.

Contribution

A novel numerical method for quantum interface problems that is third order accurate, norm-preserving, and effective at minimal resolution, using contact point information and standard finite differences.

Findings

Achieves third order spatial accuracy across interfaces.

Error comparable to sixth order homogeneous schemes at minimal resolution.

Preserves norm using summation-by-parts finite difference operators.

Abstract

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.