A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation

Tingting Ma; Yayun Fu; Yuehua He; Wenjie Yang; Tingting Ma; Yayun Fu; Yuehua He; Wenjie Yang

doi:10.3934/nhm.2023048

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1105-1117. doi: 10.3934/nhm.2023048

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A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation

1.
Zhoukou Normal University, Zhoukou 466000, China
2.
School of Science, Xuchang University, Xuchang 461000, China

Received: 13 December 2022 Revised: 16 February 2023 Accepted: 16 March 2023 Published: 03 April 2023

In this paper, we present a new method to solve the fractional nonlinear Schrödinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.
Citation: Tingting Ma, Yayun Fu, Yuehua He, Wenjie Yang. A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation[J]. Networks and Heterogeneous Media, 2023, 18(3): 1105-1117. doi: 10.3934/nhm.2023048

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Abstract

In this paper, we present a new method to solve the fractional nonlinear Schrödinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.

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