Structure-preserving scheme for one dimension and two dimension fractional KGS equations

Junjie Wang; Yaping Zhang; Liangliang Zhai; Junjie Wang; Yaping Zhang; Liangliang Zhai

doi:10.3934/nhm.2023019

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 463-493. doi: 10.3934/nhm.2023019

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Structure-preserving scheme for one dimension and two dimension fractional KGS equations

1.
School of Mathematics and Statistical, Pu'er University, Yunnan, 665000, China
2.
School of Science, Shaoyang University, Hunan, 422000, China
3.
School of Science, Xi'an Shiyou University, Shaanxi, 710065, China

Received: 30 October 2022 Revised: 07 December 2022 Accepted: 21 December 2022 Published: 13 January 2023

In the paper, we study structure-preserving scheme to solve general fractional Klein-Gordon-Schrödinger equations, including one dimension case and two dimension case. First, the high central difference scheme and Crank-Nicolson scheme are used to one dimension fractional Klein-Gordon-Schrödinger equations. We show that the arising scheme is uniquely solvable, and approximate solutions converge to the exact solution at the rate $ O(\tau^2+h^4) $. Moreover, we prove that the resulting scheme can preserve the mass and energy conservation laws. Second, we show Crank-Nicolson scheme for two dimension fractional Klein-Gordon-Schrödinger equations, and the proposed scheme preserves the mass and energy conservation laws in discrete formulations. However, the obtained discrete system is nonlinear system. Then, we show a equivalent form of fractional Klein-Gordon-Schrödinger equations by introducing some new auxiliary variables. The new system is discretized by the high central difference scheme and scalar auxiliary variable scheme, and a linear discrete system is obtained, which can preserve the energy conservation law. Finally, the numerical experiments including one dimension and two dimension fractional Klein-Gordon-Schrödinger systems are given to verify the correctness of theoretical results.
- Fractional Klein-Gordon-Schrödinger equations,
- conservation law,
- convergence,
- high central difference scheme,
- scalar auxiliary variable scheme
Citation: Junjie Wang, Yaping Zhang, Liangliang Zhai. Structure-preserving scheme for one dimension and two dimension fractional KGS equations[J]. Networks and Heterogeneous Media, 2023, 18(1): 463-493. doi: 10.3934/nhm.2023019

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Abstract

In the paper, we study structure-preserving scheme to solve general fractional Klein-Gordon-Schrödinger equations, including one dimension case and two dimension case. First, the high central difference scheme and Crank-Nicolson scheme are used to one dimension fractional Klein-Gordon-Schrödinger equations. We show that the arising scheme is uniquely solvable, and approximate solutions converge to the exact solution at the rate $ O(\tau^2+h^4) $. Moreover, we prove that the resulting scheme can preserve the mass and energy conservation laws. Second, we show Crank-Nicolson scheme for two dimension fractional Klein-Gordon-Schrödinger equations, and the proposed scheme preserves the mass and energy conservation laws in discrete formulations. However, the obtained discrete system is nonlinear system. Then, we show a equivalent form of fractional Klein-Gordon-Schrödinger equations by introducing some new auxiliary variables. The new system is discretized by the high central difference scheme and scalar auxiliary variable scheme, and a linear discrete system is obtained, which can preserve the energy conservation law. Finally, the numerical experiments including one dimension and two dimension fractional Klein-Gordon-Schrödinger systems are given to verify the correctness of theoretical results.

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