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A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation.

Siriguleng He1, Yang Liu2, Hong Li2

  • 1School of Mathematics and Big Data, Hohhot Minzu College, Hohhot 010051, China.

Entropy (Basel, Switzerland)
|June 24, 2022
PubMed
Summary
This summary is machine-generated.

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A new fast algorithm improves computing efficiency for the nonlinear Schrödinger equation using a time two-mesh compact difference scheme. This method enhances accuracy and reduces computation time for quantum state studies.

Area of Science:

  • Computational Physics
  • Quantum Mechanics
  • Numerical Analysis

Background:

  • The nonlinear Schrödinger equation is crucial for modeling quantum states in physical systems.
  • Existing numerical methods for this equation face challenges in computational efficiency and accuracy.

Purpose of the Study:

  • To develop a fast and accurate algorithm for solving the nonlinear Schrödinger equation.
  • To improve computational efficiency and calculation accuracy compared to existing methods.

Main Methods:

  • A fourth-order compact difference scheme approximates spatial derivatives.
  • A time two-mesh method is employed for efficient nonlinear system solving.
  • The fine mesh solution is utilized as an initial guess for the linear system to enhance accuracy.
Keywords:
conservation lawerror estimatehigh-order compact difference schemesolitontime two-mesh algorithm

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Main Results:

  • The algorithm achieves an accuracy of O(τC4+τF2+h4) in the discrete L2-norm.
  • Numerical experiments demonstrate high accuracy and preservation of conservation laws (charge and energy).
  • The method offers reduced CPU time compared to standard nonlinear implicit compact difference schemes without sacrificing accuracy.

Conclusions:

  • The proposed time two-mesh high-order compact difference algorithm is efficient and accurate for the nonlinear Schrödinger equation.
  • The novel use of the fine mesh solution as an initial guess improves calculation accuracy.
  • This algorithm provides a simpler numerical calculation and faster computation for quantum state modeling.