Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Panchi Li, Zhiwen Zhang

TL;DR

This work addresses efficient numerical solution of the semiclassical nonlinear Schrödinger equation with random potentials by marrying time-splitting finite element methods (TS-FEM) with multiscale finite element methods (MsFEM). It develops two Strang-splitting schemes, analyzes their convergence in both deterministic and random settings, and employs KL expansion with qMC sampling to handle randomness. A key contribution is the TS-MsFEM framework, including a convergence theory that accounts for spatial multiscale features and stochastic truncation, plus a reduced-basis variant to lower offline costs. Numerical experiments in 1D and 2D validate second-order temporal and spatial accuracy, demonstrate the efficacy of MSFEM over standard FEM, and reveal localization in linear versus delocalization in nonlinear regimes under random potentials. The results advance scalable, accurate simulations of dispersive wave dynamics in random media, with implications for quantum and optical systems under uncertainty.

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 the multiscale finite element method (MsFEM) to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze two time-splitting MsFEMs 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 present a multiscale reduced basis 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.

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

TL;DR

This work addresses efficient numerical solution of the semiclassical nonlinear Schrödinger equation with random potentials by marrying time-splitting finite element methods (TS-FEM) with multiscale finite element methods (MsFEM). It develops two Strang-splitting schemes, analyzes their convergence in both deterministic and random settings, and employs KL expansion with qMC sampling to handle randomness. A key contribution is the TS-MsFEM framework, including a convergence theory that accounts for spatial multiscale features and stochastic truncation, plus a reduced-basis variant to lower offline costs. Numerical experiments in 1D and 2D validate second-order temporal and spatial accuracy, demonstrate the efficacy of MSFEM over standard FEM, and reveal localization in linear versus delocalization in nonlinear regimes under random potentials. The results advance scalable, accurate simulations of dispersive wave dynamics in random media, with implications for quantum and optical systems under uncertainty.

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 the multiscale finite element method (MsFEM) to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze two time-splitting MsFEMs for the semiclassical NLSE with random potentials. We provide the 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 present a multiscale reduced basis 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.

Paper Structure

This paper contains 21 sections, 13 theorems, 156 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The semiclassical NLSE with random potentials
  3. Numerical methods
  4. TS-FEM for the NLSE
  5. MsFEM for the deterministic NLSE
  6. Convergence analysis
  7. Convergence analysis of the TS-FEM
  8. Convergence analysis of the TS-MsFEM for NLSE with random potentials
  9. TS-MsFEM for the deterministic NLSE
  10. MsFEM for the NLSE with random potentials
  11. Numerical experiments
  12. Numerical accuracy of TS-FEMs
  13. Numerical experiments of TS-MsFEMs
  14. Numerical simulations of NLSE with random potentials
  15. Comparison of FEM and MsFEM
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $\psi^{\epsilon}$ be the solution of equ:NLS_equ, and assume $\psi^{\epsilon}\in L^{\infty}([0, T]; H^{4}) \cap L^{1}([0, T]; H^2)$. If $\partial_t\psi^{\epsilon}(t)\in H^s$ with $s = 0,1,2$ for all $t\in [0, T]$, there exists a constant $C_{\lambda,\epsilon}$ such that where $C_{\lambda,\epsilon}$ mainly depends on $\epsilon$ and $\lambda$. In particular, for $d = 3$ and $s = 1, 2$, we have

Figures (12)

  • Figure 1: Numerical solution computed by the two TS-FEMs with different $\Delta t$.
  • Figure 2: Numerical convergence rate of SI and SII for the discontinuous potential. In the plots, the $L^2$ error and $H^1$ error on the coarse mesh are depicted.
  • Figure 3: The convergence rates of FEM and MsFEM for the NLSE with the discontinuous potential and semiclassical constant $\epsilon = \frac{1}{128}$.
  • Figure 4: Reference solution (FEM) and the spatial error distribution computed by SI, in which the MsFEM is used with $H = 8h$ and $H = 4h$.
  • Figure 5: Reference solution (FEM) and the spatial error distribution computed by SII, in which the MsFEM is used with $H = 8h$ and $H = 4h$.
  • ...and 7 more figures

Theorems & Definitions (28)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1: Wu2022, Lemma 3.2
  • proof
  • Proposition 3.2: Wu2022, Theorem 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 18 more