Simulations of Multiscale Schroedinger Equations with Multiscale Splitting Approaches: Theory and Application
This work addresses the computational challenge of solving stiff multiscale Schrödinger equations for researchers in quantum simulation, but the results are preliminary and incremental.
The paper presents a multiscale splitting approach for solving multiscale Schrödinger equations with highly oscillating potentials, enabling larger time steps for the diffusion part and reducing computational time. Numerical experiments confirm the benefit of the extrapolated splitting method.
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.