Structure-Preserving Exponential Schemes with High Accuracy for the 2D/3D Nonlinear Schrödinger Type Equation
Year: 2025
Author: Yayun Fu, Hongliang Liu
Advances in Applied Mathematics and Mechanics, Vol. 17 (2025), Iss. 4 : pp. 1088–1110
Abstract
The paper proposes a family of novel arbitrary high-order structure-preserving exponential schemes for the nonlinear Schrödinger equation. First, we introduce a quadratic auxiliary variable to reformulate the original nonlinear Schrödinger equation into an equivalent equation with modified energy. With that, the Lawson transformation technique is applied to the equation and deduces a conservative exponential system. Then, the symplectic Runge-Kutta method approximates the exponential system in the time direction and leads to a semi-discrete conservative scheme. Subsequently, the Fourier pseudo-spectral method is applied to approximate the space of the semi-discrete to obtain a class of fully-discrete schemes. The constructed schemes are proved to inherit quadratic invariants and are stable. Some numerical examples are given to confirm the accuracy and conservation of the developed schemes at last.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2023-0095
Advances in Applied Mathematics and Mechanics, Vol. 17 (2025), Iss. 4 : pp. 1088–1110
Published online: 2025-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Nonlinear Schrödinger equation structure-preserving exponential integrators quadratic auxiliary variable symplectic Runge-Kutta methods.