A conservative exponential integrators method for fractional conservative differential equations

Yayun Fu; Mengyue Shi; Yayun Fu; Mengyue Shi

doi:10.3934/math.2023973

AIMS Mathematics

2023, Volume 8, Issue 8: 19067-19082. doi: 10.3934/math.2023973

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A conservative exponential integrators method for fractional conservative differential equations

Yayun Fu ¹,
Mengyue Shi ^{2
,
,}

1.
School of Science, Xuchang University, Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, Xuchang 461000, China
2.
China Mobile online services company limited, Zhengzhou 450003, China

Received: 05 March 2023 Revised: 05 May 2023 Accepted: 11 May 2023 Published: 06 June 2023
MSC : 65M06, 65M70

The paper constructs a conservative Fourier pseudo-spectral scheme for some conservative fractional partial differential equations. The scheme is obtained by using the exponential time difference averaged vector field method to approximate the time direction and applying the Fourier pseudo-spectral method to discretize the fractional Laplacian operator so that the FFT technique can be used to reduce the computational complexity in long-time simulations. In addition, the developed scheme can be applied to solve fractional Hamiltonian differential equations because the scheme constructed is built upon the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated by solving the fractional Schrödinger equation.
- fractional conservative equations,
- energy-preserving,
- exponential time difference,
- averaged vector field method,
- fourier pseudo-spectral method
Citation: Yayun Fu, Mengyue Shi. A conservative exponential integrators method for fractional conservative differential equations[J]. AIMS Mathematics, 2023, 8(8): 19067-19082. doi: 10.3934/math.2023973

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Abstract

The paper constructs a conservative Fourier pseudo-spectral scheme for some conservative fractional partial differential equations. The scheme is obtained by using the exponential time difference averaged vector field method to approximate the time direction and applying the Fourier pseudo-spectral method to discretize the fractional Laplacian operator so that the FFT technique can be used to reduce the computational complexity in long-time simulations. In addition, the developed scheme can be applied to solve fractional Hamiltonian differential equations because the scheme constructed is built upon the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated by solving the fractional Schrödinger equation.

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