Work

Efficient finite element methods for semiclassical nonlinear Schrödinger equations with random potentials

Year: 2025

Type: article

Abstract: In this paper, we propose two time-splitting finite element methods to solve the semiclassical nonlinear Schrödinger equation (NLSE) with random potentials. We then introduce a multiscale method to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze the corresponding time-splitting multiscale reduced methods for the semiclassical NLSE with random potentials. We provide the L 2 error estimate of the proposed methods and show that they achieve second-order accuracy in both spatial and temporal spaces and an almost first-order convergence rate in the random space. Additionally, we introduce the proper orthogonal decomposition method to reduce the computational cost of constructing basis functions for solving random NLSEs. Finally, we carry out several 1D and 2D numerical examples to validate the convergence of our methods and investigate wave propagation behaviors in the NLSE with random potentials.

Source: ESAIM Mathematical Modelling and Numerical Analysis

DOI: 10.1051/m2an/2025072

Authors: Panchi Li, Zhiwen Zhang

Institution: University of Hong Kong

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid:

Funder: National Natural Science Foundation of China