A high-accuracy conservative numerical scheme for the generalized nonlinear Schrödinger equation with wave operator

Xintian Pan; Xintian Pan

doi:10.3934/math.20241330

AIMS Mathematics

2024, Volume 9, Issue 10: 27388-27402. doi: 10.3934/math.20241330

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Research article

A high-accuracy conservative numerical scheme for the generalized nonlinear Schrödinger equation with wave operator

Xintian Pan ^,

School of Mathematics and statistics, Weifang University, Weifang 261061, China

Received: 26 June 2024 Revised: 08 August 2024 Accepted: 14 August 2024 Published: 23 September 2024
MSC : 65N06, 65N12

In this article, we establish a novel high-order energy-preserving numerical approximation scheme to study the initial and periodic boundary problem of the generalized nonlinear Schrödinger equation with wave operator, which is proposed by the finite difference method. The scheme is of fourth-order accuracy in space and second-order one in time. The conservation property of energy as well as a priori estimate are described. The convergence of the proposed scheme is discussed in detail by using the energy method. Some comparisons have been made between the proposed method and the others. Numerical examples are presented to illustrate the validity and accuracy of the method.
- the nonlinear Schrödinger equation with wave operator,
- high-accuracy scheme,
- conservative property,
- error estimate,
- convergence
Citation: Xintian Pan. A high-accuracy conservative numerical scheme for the generalized nonlinear Schrödinger equation with wave operator[J]. AIMS Mathematics, 2024, 9(10): 27388-27402. doi: 10.3934/math.20241330

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Abstract

In this article, we establish a novel high-order energy-preserving numerical approximation scheme to study the initial and periodic boundary problem of the generalized nonlinear Schrödinger equation with wave operator, which is proposed by the finite difference method. The scheme is of fourth-order accuracy in space and second-order one in time. The conservation property of energy as well as a priori estimate are described. The convergence of the proposed scheme is discussed in detail by using the energy method. Some comparisons have been made between the proposed method and the others. Numerical examples are presented to illustrate the validity and accuracy of the method.

References

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