An energy-preserving exponential scheme with scalar auxiliary variable approach for the nonlinear Dirac equations

Hongquan Wang; Yancai Liu; Xiujun Cheng; Hongquan Wang; Yancai Liu; Xiujun Cheng

doi:10.3934/era.2025014

Electronic Research Archive

2025, Volume 33, Issue 1: 263-276. doi: 10.3934/era.2025014

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An energy-preserving exponential scheme with scalar auxiliary variable approach for the nonlinear Dirac equations

1.
College of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
2.
School of Mathematics and Statistics, Henan University of Technology, Zhengzhou 450001, China

Received: 17 November 2024 Revised: 02 January 2025 Accepted: 14 January 2025 Published: 22 January 2025

In this work, an energy-preserving scheme is proposed for the nonlinear Dirac equation by combining the exponential time differencing method with the scalar auxiliary variable (SAV) approach. First, the original equations can be transformed into the equivalent systems by utilizing the SAV technique. Then the exponential time integrator method is applied for discretizing the temporal derivative, and the standard central difference scheme is used for approximating the spatial derivative for the equivalent one. Finally, the reformulated systems, the semi-discrete spatial scheme, and the fully-discrete, linearly implicit exponential scheme are proven to be energy conserving. The numerical experiments confirm the theoretical results.
- nonlinear Dirac equation,
- exponential time integrator method,
- scalar auxiliary variable,
- energy conserving
Citation: Hongquan Wang, Yancai Liu, Xiujun Cheng. An energy-preserving exponential scheme with scalar auxiliary variable approach for the nonlinear Dirac equations[J]. Electronic Research Archive, 2025, 33(1): 263-276. doi: 10.3934/era.2025014

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Abstract

In this work, an energy-preserving scheme is proposed for the nonlinear Dirac equation by combining the exponential time differencing method with the scalar auxiliary variable (SAV) approach. First, the original equations can be transformed into the equivalent systems by utilizing the SAV technique. Then the exponential time integrator method is applied for discretizing the temporal derivative, and the standard central difference scheme is used for approximating the spatial derivative for the equivalent one. Finally, the reformulated systems, the semi-discrete spatial scheme, and the fully-discrete, linearly implicit exponential scheme are proven to be energy conserving. The numerical experiments confirm the theoretical results.

References

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