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A Class of Meshless Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation

  • Jialing Wang EMAIL logo , Zhengting Zhou and Zhoujin Lin
Published/Copyright: November 14, 2024

Abstract

This paper aims to give a unified construction framework of meshless structure-preserving algorithms to solve the d-dimensional ( d = 1 or 2) nonlinear Schrödinger equation. Based on the method of lines, we first derive a finite-dimensional Hamiltonian system by using the radial basis function method of the quasi-interpolation and the technique of left-multiplying a diagonal matrix to discretize the space direction. Then suitable geometric numerical integrations can be used to discretize the time direction, which yields a class of meshless structure-preserving algorithms. In addition to the construction, the structure-preserving properties and their proofs are also provided in detail. Besides the uniform and nonuniform grids, the numerical experiments on the random grids are also emphasized to verify the theoretical research well, which is of great significance for scattering points based on the characteristics of actual problems.

Award Identifier / Grant number: 11801277

Funding statement: Jialing Wang was supported by the National Natural Science Foundation of China (under Grant No. 11801277).

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Received: 2023-09-28
Revised: 2024-09-01
Accepted: 2024-09-23
Published Online: 2024-11-14

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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