An efficient linearly-implicit energy-preserving scheme with fast solver for the fractional nonlinear wave equation

Tingting Ma; Yuehua He; Tingting Ma; Yuehua He

doi:10.3934/math.20231358

AIMS Mathematics

2023, Volume 8, Issue 11: 26574-26589. doi: 10.3934/math.20231358

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An efficient linearly-implicit energy-preserving scheme with fast solver for the fractional nonlinear wave equation

Tingting Ma ¹,
Yuehua He ^{2
,
,}

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

Received: 21 June 2023 Revised: 07 August 2023 Accepted: 23 August 2023 Published: 18 September 2023
MSC : 65M06, 65M70

The paper considers the Hamiltonian structure and develops efficient energy-preserving schemes for the nonlinear wave equation with a fractional Laplacian operator. To this end, we first derive the Hamiltonian form of the equation by using the fractional variational derivative and then applying the finite difference method to the original equation to obtain a semi-discrete Hamiltonian system. Furthermore, the scalar auxiliary variable method and extrapolation technique is used to approximate a semi-discrete system to construct an efficient linearly-implicit energy-preserving scheme. A fast solver for the proposed scheme is presented to reduce CPU consumption. Ample numerical results are given to finally confirm the efficiency and conservation of the developed scheme.
- fractional nonlinear wave equation,
- Hamiltonian system,
- scalar auxiliary variable,
- conservative scheme
Citation: Tingting Ma, Yuehua He. An efficient linearly-implicit energy-preserving scheme with fast solver for the fractional nonlinear wave equation[J]. AIMS Mathematics, 2023, 8(11): 26574-26589. doi: 10.3934/math.20231358

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Abstract

The paper considers the Hamiltonian structure and develops efficient energy-preserving schemes for the nonlinear wave equation with a fractional Laplacian operator. To this end, we first derive the Hamiltonian form of the equation by using the fractional variational derivative and then applying the finite difference method to the original equation to obtain a semi-discrete Hamiltonian system. Furthermore, the scalar auxiliary variable method and extrapolation technique is used to approximate a semi-discrete system to construct an efficient linearly-implicit energy-preserving scheme. A fast solver for the proposed scheme is presented to reduce CPU consumption. Ample numerical results are given to finally confirm the efficiency and conservation of the developed scheme.

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