An efficient algorithm based on splitting for the time integration of the Schr\"odinger equation

TL;DR

This paper introduces a new set of symplectic splitting algorithms for efficiently solving the Schrödinger equation in time, outperforming Chebyshev-based methods across various tolerances and intervals.

Contribution

The paper develops highly efficient symplectic integrators tailored for the Schrödinger equation, with optimized error bounds and automatic scheme selection based on problem parameters.

Findings

New splitting methods outperform Chebyshev schemes in efficiency.

Algorithms adaptively select optimal schemes for given tolerances.

Numerical experiments confirm theoretical efficiency gains.

Abstract

We present a practical algorithm based on symplectic splitting methods to integrate numerically in time the Schr\"odinger equation. When discretized in space, the Schr\"odinger equation can be recast as a classical Hamiltonian system corresponding to a generalized high-dimensional separable harmonic oscillator. The particular structure of this system combined with previously obtained stability and error analyses allows us to construct a set of highly efficient symplectic integrators with sharp error bounds and optimized for different tolerances and time integration intervals. They can be considered, in this setting, as polynomial approximations to the matrix exponential in a similar way as methods based on Chebyshev and Taylor polynomials. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on Chebyshev polynomials for all tolerances and time intervals. The algorithm we present incorporates the new splitting methods and automatically selects the most efficient scheme given a tolerance, a time integration interval and an estimate on the spectral radius of the Hamiltonian.