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Efficient Structure Preserving Schemes for the Klein–Gordon–Schrödinger Equations

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Abstract

We construct three efficient and accurate numerical methods for solving the Klein–Gordon–Schrödinger (KGS) equations with/without damping terms. The first one is based on the original SAV approach, it preserves a modified Hamiltonian but does not preserve the wave energy. The second one is based on the Lagrange multiplier SAV approach, it preserves both the original Hamiltonian and wave energy, but requires solving a nonlinear algebraic system which may require smaller time steps to have real solutions. The third one is also based on the Lagrange multiplier approach and preserves the Hamiltonian and wave energy in a slightly different form, but it leads to a nonlinear quadratic system for the Lagrange multiplier which can always be explicitly solved. We present ample numerical tests to validate the three schemes, and provide a comparison on the efficiency and accuracy of the three schemes for the KGS equations.

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Acknowledgements

This work is supported in part by NSFC 11971407, NSF DMS-2012585 and AFOSR FA9550-20-1-0309.

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Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, China

    Yanrong Zhang

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Jie Shen

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Correspondence to Jie Shen.

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Appendix A: \(F_1\) and \(F_2\) in (3.23)

Appendix A: \(F_1\) and \(F_2\) in (3.23)

The exact forms of nonlinear functions \(F_1\) and \(F_2\) in (3.23) are

$$\begin{aligned} F_{1}\left( \eta ^{n+1}, \lambda ^{n+1}\right)= & {} \left( \eta ^{n+1}\right) ^{3}\left( \left( p_{2}^{n+1}\right) ^{2}+\left( q_{2}^{n+1}\right) ^{2}, \phi _{2}^{n+1}\right) + \left( \lambda ^{n+1}\right) ^{3}\left( \left( p_{3}^{n+1}\right) ^{2}+\left( q_{3}^{n+1}\right) ^{2}, \phi _{3}^{n+1}\right) \\&\quad +\lambda ^{n+1}\left( \eta ^{n+1}\right) ^{2}\left\{ 2\left( p_{2}^{n+1}p_{3}^{n+1}+q_{2}^{n+1}q_{3}^{n+1}, \phi _{2}^{n+1}\right) + \left( \left( p_{2}^{n+1}\right) ^{2}+\left( q_{2}^{n+1}\right) ^{2}, \phi _{3}^{n+1}\right) \right\} \\&\quad + \left( \lambda ^{n+1}\right) ^{2}\eta ^{n+1}\left\{ 2\left( p_{2}^{n+1}p_{3}^{n+1}+q_{2}^{n+1}q_{3}^{n+1}, \phi _{3}^{n+1}\right) + \left( \left( p_{3}^{n+1}\right) ^{2}+\left( q_{3}^{n+1}\right) ^{2}, \phi _{2}^{n+1}\right) \right\} \\&\quad + \left( \eta ^{n+1}\right) ^{2}\left\{ 2\left( p_{1}^{n+1}p_{2}^{n+1}+q_{1}^{n+1}q_{2}^{n+1}, \phi _{2}^{n+1}\right) + \left( \left( p_{2}^{n+1}\right) ^{2}+\left( q_{2}^{n+1}\right) ^{2}, \phi _{1}^{n+1}\right) \right. \\&\quad \left. - \frac{1}{2}\left[ 2\left( p^{n}p_{2}^{n+1}+q^{n}q_2^{n+1}, \phi ^{n}\right) + \left( \left( p^{n}\right) ^{2}+\left( q^{n}\right) ^{2},\phi _{2}^{n+1}\right) \right] \right\} \\&\quad + \left( \lambda ^{n+1}\right) ^{2}\left\{ 2\left( p_{1}^{n+1}p_{3}^{n+1}+q_{1}^{n+1}q_{3}^{n+1}, \phi _{3}^{n+1}\right) + \left( \left( p_{3}^{n+1}\right) ^{2}+\left( q_{3}^{n+1}\right) ^{2}, \phi _{1}^{n+1}\right) \right. \\&\quad \left. -\left[ \left( p^{n}, p_{3}^{n+1}\right) +\left( q^{n}, q_{3}^{n+1}\right) \right] \right\} \\&\quad + \lambda ^{n+1}\eta ^{n+1}\left\{ 2\left( p_{2}^{n+1}p_{3}^{n+1}+q_{2}^{n+1}q_{3}^{n+1}, \phi _{1}^{n+1}\right) + 2\left( p_{1}^{n+1}p_{3}^{n+1}+q_{1}^{n+1}q_{3}^{n+1}, \phi _{2}^{n+1}\right) \right. \\&\quad \left. + 2\left( p_{1}^{n+1}p_{2}^{n+1}+q_{1}^{n+1}q_{2}^{n+1}, \phi _{3}^{n+1}\right) - \frac{1}{2}\left[ 2\left( p^{n}p_{3}^{n+1}+q^{n}q_{3}^{n+1}, \phi ^{n}\right) + \left( \left( p^{n}\right) ^{2}+\left( q^{n}\right) ^{2}, \phi _{3}^{n+1}\right) \right. \right. \\&\quad \left. \left. + 2\left( p^{n}, p_{2}^{n+1}\right) +2\left( q^{n}, q_{2}^{n+1}\right) \right] \right\} \\&\quad + \eta ^{n+1}\left\{ 2\left( p_{1}^{n+1}p_{2}^{n+1}+q_{1}^{n+1}q_{2}^{n+1}, \phi _{1}^{n+1}\right) + \left( \left( p_{1}^{n+1}\right) ^{2}+\left( q_{1}^{n+1}\right) ^{2}, \phi _{2}^{n+1}\right) \right. \\&\quad \left. -\frac{1}{2}\left[ 2\left( p^{n}\phi ^{n}, p_{1}^{n+1}-p^{n-1}\right) \right. \right. \\&\quad \left. \left. + 2 \left( q^{n}\phi ^{n}, q_{1}^{n+1}-q^{n-1}\right) + \left( \left( p^{n}\right) ^{2}+\left( q^{n}\right) ^{2}, \phi _{1}^{n+1}-\phi ^{n-1}\right) \right. \right. \\&\quad \left. \left. + 2\eta ^{n-1}\left( p^{n}p_{2}^{n+1}+q^{n}q_{2}^{n+1}, \phi ^{n}\right) + \eta ^{n-1}\left( \left( p^{n}\right) ^{2}+ \left( q^{n}\right) ^{2}, \phi _{2}^{n+1}\right) \right. \right. \\&\quad \left. \left. + 2\lambda ^{n-1}\left( \left( p^{n}, p_{2}^{n+1}\right) + \left( q^{n}, q_{2}^{n+1}\right) \right) \right] \right\} \\&\quad + \lambda ^{n+1}\left\{ 2\left( p_{1}^{n+1}p_{3}^{n+1}+q_{1}^{n+1}q_{3}^{n+1}, \phi _{1}^{n+1}\right) + \left( \left( p_{1}^{n+1}\right) ^{2}+\left( q_{1}^{n+1}\right) ^{2}, \phi _{3}^{n+1}\right) \right. \\&\quad \left. - \frac{1}{2}\left[ 2\left( p^{n}, p_{1}^{n+1}-p^{n-1}\right) + 2 \left( q^{n}, q_{1}^{n+1}-q^{n-1}\right) + 2\eta ^{n-1}\left( p^{n}p_{3}^{n+1}+q^{n}q_{3}^{n+1}, \phi ^{n}\right) \right. \right. \\&\quad \left. \left. + \eta ^{n-1}\left( \left( p^{n}\right) ^{2}+ \left( q^{n}\right) ^{2}, \phi _{3}^{n+1}\right) + 2\lambda ^{n-1}\left( \left( p^{n}, p_{3}^{n+1}\right) + \left( q^{n}, q_{3}^{n+1}\right) \right) \right] \right\} \\&\quad + \left( \left( p_{1}^{n+1}\right) ^{2}+\left( q_{1}^{n+1}\right) ^{2}, \phi _{1}^{n+1}\right) - \left( \left( p^{n-1}\right) ^{2}+\left( q^{n-1}\right) ^{2}, \phi ^{n-1}\right) - \left[ \eta ^{n-1}\left( p^{n}\phi ^{n}, p_{1}^{n+1}-p^{n-1}\right) \right. \\&\quad \left. + \left( q^{n}\phi ^{n}, q_{1}^{n+1}-q^{n-1}\right) +\frac{\eta ^{n-1}}{2}\left( \left( p^{n}\right) ^{2}+\left( q^{n}\right) ^{2}, \phi _{1}^{n+1}-\phi ^{n-1}\right) \right. \\&\quad \left. +\lambda ^{n-1}\left( \left( p^{n}, p_{1}^{n+1}-p^{n-1}\right) + \left( q^{n}, q_{1}^{n+1}-q^{n-1}\right) \right) \right] , \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} F_{2}\left( \eta ^{n+1}, \lambda ^{n+1}\right)&= \left( \eta ^{n+1}\right) ^{2}\left( \left( p_{2}^{n+1}\right) ^{2}+\left( q_{2}^{n+1}\right) ^{2}, 1\right) + \left( \lambda ^{n+1}\right) ^{2}\left( \left( p_{3}^{n+1}\right) ^{2}+\left( q_{3}^{n+1}\right) ^{2}, 1\right) \\&\quad \!+\! 2\eta ^{n+1}\left[ (p_{1}^{n+1}, p_{2}^{n+1}) \!+\! (q_{1}^{n+1}, q_{2}^{n+1})\right] \!+\! 2\lambda ^{n+1} \left[ (p_{1}^{n+1}, p_{3}^{n+1}) \!+\! (q_{1}^{n+1}, q_{3}^{n+1})\right] \!+\! 2\lambda ^{n+1}\eta ^{n+1}\\&\quad \left[ \left( p_{2}^{n+1}, p_{3}^{n+1}\right) + \left( q_{2}^{n+1}, q_{3}^{n+1}\right) \right] + \left( \left( p_{1}^{n+1}\right) ^{2}+ \left( q_{1}^{n+1}\right) ^{2}, 1\right) - \left( \left( p^{0}\right) ^{2}+ \left( q^{0}\right) ^{2}, 1\right) . \end{aligned} \end{aligned}$$

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Zhang, Y., Shen, J. Efficient Structure Preserving Schemes for the Klein–Gordon–Schrödinger Equations. J Sci Comput 89, 47 (2021). https://doi.org/10.1007/s10915-021-01649-y

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