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FULLY DECOUPLED ARBITRARILY HIGH-ORDER CONSERVATIVE SCHEMES FOR THE SCHRÖDINGER–BOUSSINESQ EQUATION

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  25 July 2025

JIN-LIANG YAN Open the ORCID record for JIN-LIANG YAN [Opens in a new window] ,
LIANGHONG ZHENG Open the ORCID record for LIANGHONG ZHENG [Opens in a new window]  and
LING ZHU Open the ORCID record for LING ZHU [Opens in a new window]
JIN-LIANG YAN*
Affiliation:
Fujian Key Laboratory of Big Data Application and Intellectualization for Tea Industry, College of Mathematics and Computer, https://ror.org/059djzq42 Wuyi University , Fujian, PR China
LIANGHONG ZHENG
Affiliation:
Department of Information and Computer Technology, No. 1 Middle School of Nanping, Fujian, PR China; e-mail: 413845939@qq.com
LING ZHU
Affiliation:
Department of Mathematics and Physics, https://ror.org/03jc41j30 Jiangsu University of Science and Technology , Zhenjiang, PR China; e-mail: 38196700@qq.com
*
e-mail: yanjinliang3333@163.com
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Abstract

A family of arbitrarily high-order energy-preserving methods are developed to solve the coupled Schrödinger–Boussinesq (S-B) system. The system is a nonlinear coupled system and satisfies a series of conservation laws. It is often difficult to construct a high-order decoupling numerical algorithm to solve the nonlinear system. In this paper, the original system is first reformulated into an equivalent Hamiltonian system by introducing multiple auxiliary variables. Next, the reformulated system is discretized by the Fourier pseudo-spectral method and the implicit midpoint scheme in the spatial and temporal directions, respectively, and a second-order conservative scheme is obtained. Finally, the scheme is extended to arbitrarily high-order accuracy by means of diagonally implicit symplectic Runge–Kutta methods or composition methods. Rigorous analyses show that the proposed methods are fully decoupled and can precisely conserve the discrete invariants. Numerical results show that the proposed schemes are effective and can be easily extended to other nonlinear partial differential equations.

Keywords

Schrödinger–Boussinesq systemenergy-preservingmultiple quadratic auxiliary variables methoddiagonally symplectic Runge–Kutta methodscomposition methods

MSC classification

Primary: 65M22: Solution of discretized equations
Secondary: 65L05: Initial value problems 65M70: Spectral, collocation and related methods 65M20: Method of lines

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e27
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Australian Mathematical Society

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