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Mass-, and Energy Preserving Schemes with Arbitrarily High Order for the Klein–Gordon–Schrödinger Equations

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Abstract

We present a class of arbitrarily high-order conservative schemes for the Klein–Gordon Schrödinger equations. These schemes combine the symplectic Runge–Kutta method with the quadratic auxiliary variable approach. We first introduce an auxiliary variable that satisfies a quadratic equation to reformulate the original system into an equivalent one. This reformulated system possesses two strong quadratic invariants: energy and mass. Next, we discretize the reformulated system using symplectic Runge–Kutta methods, yielding a class of semi-discrete systems with arbitrarily high-order accuracy in time. The semi-discrete systems naturally preserve the discrete contour part of the strong invariants and the relationship of the quadratic equation. By eliminating the intermediate variable, we obtain the original energy conservation law. Then, the Fourier pseudo-spectral method is employed to obtain the fully discrete scheme that preserves the original energy and mass. We provide a fast solver to implement the proposed methods effectively. Numerical experiments demonstrate the expected accuracy and conservation of proposed schemes.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 12171245, 11971416, 11971242), the Natural Science Foundation of Henan Province (No. 222300420280), the China Postdoctoral Science Foundation (No. 2023T160589), the Natural Science Foundation of Hunan Province (No. 2023JJ40656), and the scientific research Fund of Xuchang University (2024ZD010).

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Authors and Affiliations

  1. Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, School of Science, Xuchang University, Xuchang, 461000, China

    Yayun Fu

  2. Key Laboratory of NSLSCS, Ministry of Education, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Xuelong Gu, Yushun Wang & Wenjun Cai

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  1. Yayun Fu
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  2. Xuelong Gu
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Correspondence to Wenjun Cai.

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Fu, Y., Gu, X., Wang, Y. et al. Mass-, and Energy Preserving Schemes with Arbitrarily High Order for the Klein–Gordon–Schrödinger Equations. J Sci Comput 97, 75 (2023). https://doi.org/10.1007/s10915-023-02388-y

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  • DOI: https://doi.org/10.1007/s10915-023-02388-y

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