Simulations of Multiscale Schroedinger Equations with Multiscale Splitting Approaches: Theory and Application

TL;DR

This paper introduces a multiscale splitting method for solving complex Schrödinger equations with multiple time scales, enabling larger time steps and reducing computational costs through operator splitting and extrapolation techniques.

Contribution

The paper presents a novel multiscale splitting approach for Schrödinger equations that effectively handles highly oscillating potentials and large time-scale differences.

Findings

Extrapolated splitting methods reduce computational time.

Decoupling diffusion and reaction improves solver efficiency.

Numerical experiments confirm the method's effectiveness.

Abstract

In this paper we present a novel multiscale splitting approach to solve multiscale Schroedinger equation, which have large different time-scales. The energy potential is based on highly oscillating functions, which are magnitudes faster than the transport term. We obtain a multiscale problem and a highly stiff problem, while standard solvers need to small time-steps. We propose multiscale solvers, which are based on operator splitting methods and we decouple the diffusion and reaction part of the Schroedinger equation. Such a decomposition allows to apply a large time step for the implicit time-discretization of the diffusion part and small time steps for the explicit and highly oscillating reaction part. With extrapolation steps, we could reduce the computational time in the highly-oscillating time-scale, while we relax into the slow time-scale. We present the numerical analysis of the extrapolated operator splitting method. First numerical experiments verified the benefit of the extrapolated splitting approaches.