Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium

Xuefei He; Kun Wang; Liwei Xu; Xuefei He; Kun Wang; Liwei Xu

doi:10.3934/era.2020079

Electronic Research Archive

2020, Volume 28, Issue 4: 1503-1528. doi: 10.3934/era.2020079

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Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium

Xuefei He ^{1
,},
Kun Wang ^{1
,},
Liwei Xu ^{2
,}

1.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan 611731, China

Received: 01 April 2020 Revised: 01 June 2020 Published: 31 July 2020
Primary: 65J15; Secondary: 65N06

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.
- Nonlinear Helmholtz equation,
- finite difference method,
- iteration method,
- discontinuous coefficient,
- Kerr medium
Citation: Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium[J]. Electronic Research Archive, 2020, 28(4): 1503-1528. doi: 10.3934/era.2020079

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Abstract

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

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