An adaptive-size multi-domain pseudospectral approach for solving the time-dependent Schr\"odinger equation

TL;DR

This paper introduces an adaptive multi-domain pseudospectral method for solving the time-dependent Schrödinger equation, combining efficiency and high accuracy by using variable-sized spatial domains and spectral techniques.

Contribution

The paper presents a novel multi-domain pseudospectral approach with variable domain sizes, improving efficiency and accuracy in solving the Schrödinger equation compared to existing methods.

Findings

Accurate simulation of high-harmonic generation near the cutoff.

Efficient diagonalization due to sparse Hamiltonian matrices.

Stable and convergent polynomial propagation using Chebychev propagator.

Abstract

We show that a pseudospectral representation of the wavefunction using multiple spatial domains of variable size yields a highly accurate, yet efficient method to solve the time-dependent Schr\"odinger equation. The overall spatial domain is split into non-overlapping intervals whose size is chosen according to the local de Broglie wavelength. A multi-domain weak formulation of the Schr\"odinger equation is obtained by representing the wavefunction by Lagrange polynomials with compact support in each domain, discretized at the Legendre-Gauss-Lobatto points. The resulting Hamiltonian is sparse, allowing for efficient diagonalization and storage. Accurate time evolution is carried out by the Chebychev propagator, involving only sparse matrix-vector multiplications. Our approach combines the efficiency of mapped grid methods with the accuracy of spectral representations based on Gaussian quadrature rules and the stability and convergence properties of polynomial propagators. We apply this method to high-harmonic generation and examine the role of the initial state for the harmonic yield near the cutoff.