Runge-Kutta convolution quadrature and FEM-BEM coupling for the time dependent linear Schr\"odinger equation

TL;DR

This paper introduces a numerical method combining Runge-Kutta time-stepping, finite element spatial discretization, and convolution quadrature boundary elements to solve the time-dependent linear Schrödinger equation on unbounded domains, with proven stability and convergence.

Contribution

It presents a novel integrated scheme for Schrödinger equations on unbounded domains, analyzing stability and convergence of the combined FEM-BEM approach.

Findings

The scheme is stable under certain conditions.

Convergence rates are established theoretically.

Numerical experiments confirm theoretical predictions.

Abstract

We propose a numerical scheme to solve the time dependent linear Schr\"odinger equation. The discretization is carried out by combining a Runge-Kutta time-stepping scheme with a finite element discretization in space. Since the Schr\"odinger equation is posed on the whole space $\R^d$ we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.