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December  2020, 28(4): 1503-1528. doi: 10.3934/era.2020079

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

Xuefei He 1, , Kun Wang 1, and  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  April 2020 Revised  June 2020 Published  July 2020

Fund Project: This work is supported by the Natural Science Foundation of China (No. 91630205)

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.

Keywords: Nonlinear Helmholtz equation, finite difference method, iteration method, discontinuous coefficient, Kerr medium.
Mathematics Subject Classification: Primary: 65J15; Secondary: 65N06.
Citation: Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
References:
[1]

G. Baruch, G. Fibich and S. Tsynkov, High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry, J. Comput. Appl. Math., 204 (2007), 477-492.  doi: 10.1016/j.cam.2006.01.048.  Google Scholar

[2]

G. Baruch, G. Fibich and S. Tsynkov, High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension, J. Comput. Phys., 227 (2007), 820-850.  doi: 10.1016/j.jcp.2007.08.022.  Google Scholar

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W. Chen and D. L. Mills, Optical response of a nonlinear dielectric film, Phys. Rev. B, 35 (1987), 524-532.  doi: 10.1103/PhysRevB.35.524.  Google Scholar

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W. Chen and D. L. Mills, Optical response of nonlinear multilayer structures: Bilayers and superlattices, Phys. Rev. B, 36 (1987), 524-532.  doi: 10.1103/PhysRevB.36.6269.  Google Scholar

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G. Evequoz and T. Weth, Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728.  doi: 10.1016/j.aim.2015.04.017.  Google Scholar

[12]

G. Evéquoz, Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane, Analysis (Berlin), 37 (2017), 55-68.  doi: 10.1515/anly-2016-0023.  Google Scholar

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A. Suryanto, E. van Groesen and M. Hammer, A finite element scheme to study the nonlinear optical response of a finite grating without and with defect, Optical and Quantum Electronics, 35 (2003), 313-332. doi: 10.1023/A:1022901201632.  Google Scholar

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K. Wang and Y. S. Wong, Error correction method for Navier-Stokes equations at high Reynolds numbers, J. Comput. Phys., 255 (2013), 245-265.  doi: 10.1016/j.jcp.2013.07.042.  Google Scholar

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K. Wang and Y. S. Wong, Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model., 11 (2014), 787-815.   Google Scholar

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K. Wang and Y. S. Wong, Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?, Commun. Comput. Phys., 21 (2017), 490-514.  doi: 10.4208/cicp.OA-2016-0057.  Google Scholar

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K. Wang, Y. S. Wong and J. Huang, Analysis of pollution-free approaches for multi-dimensional Helmholtz equations, Int. J. Numer. Anal. Model., 16 (2019), 412-435.   Google Scholar

[29]

H. Wu and J. Zou, Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 56 (2018), 1338-1359.  doi: 10.1137/17M111314X.  Google Scholar

[30]

Z. Xu and G. Bao, A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects, Journal of the Optical Society of America(A), 27 (2010), 2347-2353.  doi: 10.1364/JOSAA.27.002347.  Google Scholar

[31]

L. Yuan and Y. Y. Lu, Robust iterative method for nonlinear Helmholtz equation, J. Comput. Phys., 343 (2017), 1-9.  doi: 10.1016/j.jcp.2017.04.046.  Google Scholar

[32]

S. Zhai, X. Feng and Y. He, A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation, J. Sci. Comput., 54 (2013), 97-120.  doi: 10.1007/s10915-012-9607-6.  Google Scholar

[33]

S. Zhai, X. Feng and Y. He, A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation, Appl. Math. Comput., 230 (2014), 9-26.  doi: 10.1016/j.amc.2013.12.096.  Google Scholar

show all references

References:
[1]

G. Baruch, G. Fibich and S. Tsynkov, High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry, J. Comput. Appl. Math., 204 (2007), 477-492.  doi: 10.1016/j.cam.2006.01.048.  Google Scholar

[2]

G. Baruch, G. Fibich and S. Tsynkov, High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension, J. Comput. Phys., 227 (2007), 820-850.  doi: 10.1016/j.jcp.2007.08.022.  Google Scholar

[3]

G. Baruch, G. Fibich and S. Tsynkov, A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media, J. Comput. Phys., 228 (2009), 3789-3815.  doi: 10.1016/j.jcp.2009.02.014.  Google Scholar

[4]

V. A. Bokil, Y. Cheng, Y. Jiang and F. Li, Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media, J. Comput. Phys., 350 (2017), 420-452.  doi: 10.1016/j.jcp.2017.08.009.  Google Scholar

[5] R. W. Boyd, Nonlinear Optics, Elsevier/Academic Press, Amsterdam, 2008.   Google Scholar
[6]

E. Centeno and D. Felbacq, Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity, Phys. Rev. B, 62 (2000), 7683-7686.  doi: 10.1103/PhysRevB.62.R7683.  Google Scholar

[7]

W. Chen and D. L. Mills, Optical response of a nonlinear dielectric film, Phys. Rev. B, 35 (1987), 524-532.  doi: 10.1103/PhysRevB.35.524.  Google Scholar

[8]

W. Chen and D. L. Mills, Optical response of nonlinear multilayer structures: Bilayers and superlattices, Phys. Rev. B, 36 (1987), 524-532.  doi: 10.1103/PhysRevB.36.6269.  Google Scholar

[9]

W. Dai and R. Nassar, Compact ADI method for solving parabolic differential equations, Numer. Methods Partial Differential Equations, 18 (2002), 129-142.  doi: 10.1002/num.1037.  Google Scholar

[10]

W. Dai and R. Nassar, A new ADI scheme for solving three-dimensional parabolic equations with first-order derivatives and variable coefficients, J. Comput. Anal. Appl., 2 (2000), 293-308.  doi: 10.1023/A:1010108620966.  Google Scholar

[11]

G. Evequoz and T. Weth, Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728.  doi: 10.1016/j.aim.2015.04.017.  Google Scholar

[12]

G. Evéquoz, Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane, Analysis (Berlin), 37 (2017), 55-68.  doi: 10.1515/anly-2016-0023.  Google Scholar

[13]

G. Fibich, The Nonlinear Schrödinger Equation. Singular Solutions and Optical Collapse, Applied Mathematical Sciences, 192, Springer, Cham, 2015. doi: 10.1007/978-3-319-12748-4.  Google Scholar

[14]

G. Fibich and S. Tsynkov, High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001), 632-677.  doi: 10.1006/jcph.2001.6800.  Google Scholar

[15]

G. Fibich and S. Tsynkov, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005), 183-224.  doi: 10.1016/j.jcp.2005.04.015.  Google Scholar

[16]

R. Guo, K. Wang and L. Xu, Efficient finite difference methods for acoustic scattering from circular cylindrical obstacle, Int. J. Numer. Anal. Model., 13 (2016), 986-1002.   Google Scholar

[17]

X. He and K. Wang, Uniformly convergent novel finite difference methods for singularly perturbed reaction-diffusion equations, Numer. Methods Partial Differential Equations, 35 (2019), 2120-2148.  doi: 10.1002/num.22405.  Google Scholar

[18]

T. A. Laine and A. T. Friberg, Self-guided waves and exact solutions of the nonlinear Helmholtz equation, J. Opt. Soc. Amer. B Opt. Phys., 17 (2000), 751-757.  doi: 10.1364/JOSAB.17.000751.  Google Scholar

[19]

R. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions for nonlinear Helmholtz equations, Z. Angew. Math. Phys., 68 (2017), 19pp. doi: 10.1007/s00033-017-0859-8.  Google Scholar

[20]

G. I. Stegeman and M. Segev, Optical spatial solitons and their interactions: Universality and diversity, Science, 286 (1999), 1518-1523.  doi: 10.1126/science.286.5444.1518.  Google Scholar

[21]

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004. doi: 10.1137/1.9780898717938.  Google Scholar

[22]

A. Suryanto, E. van Groesen and M. Hammer, Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with a Defect, J. Nonlinear Optical Phys. Materials, 12 (2003), 187-204.  doi: 10.1142/S0218863503001328.  Google Scholar

[23]

A. Suryanto, E. van Groesen and M. Hammer, A finite element scheme to study the nonlinear optical response of a finite grating without and with defect, Optical and Quantum Electronics, 35 (2003), 313-332. doi: 10.1023/A:1022901201632.  Google Scholar

[24]

K. Wang and Y. S. Wong, Error correction method for Navier-Stokes equations at high Reynolds numbers, J. Comput. Phys., 255 (2013), 245-265.  doi: 10.1016/j.jcp.2013.07.042.  Google Scholar

[25]

K. Wang and Y. S. Wong, Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model., 11 (2014), 787-815.   Google Scholar

[26]

K. Wang and Y. S. Wong, Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?, Commun. Comput. Phys., 21 (2017), 490-514.  doi: 10.4208/cicp.OA-2016-0057.  Google Scholar

[27]

K. Wang, Y. S. Wong and J. Deng, Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates, Commun. Comput. Phys., 17 (2015), 779-807.  doi: 10.4208/cicp.110214.101014a.  Google Scholar

[28]

K. Wang, Y. S. Wong and J. Huang, Analysis of pollution-free approaches for multi-dimensional Helmholtz equations, Int. J. Numer. Anal. Model., 16 (2019), 412-435.   Google Scholar

[29]

H. Wu and J. Zou, Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 56 (2018), 1338-1359.  doi: 10.1137/17M111314X.  Google Scholar

[30]

Z. Xu and G. Bao, A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects, Journal of the Optical Society of America(A), 27 (2010), 2347-2353.  doi: 10.1364/JOSAA.27.002347.  Google Scholar

[31]

L. Yuan and Y. Y. Lu, Robust iterative method for nonlinear Helmholtz equation, J. Comput. Phys., 343 (2017), 1-9.  doi: 10.1016/j.jcp.2017.04.046.  Google Scholar

[32]

S. Zhai, X. Feng and Y. He, A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation, J. Sci. Comput., 54 (2013), 97-120.  doi: 10.1007/s10915-012-9607-6.  Google Scholar

[33]

S. Zhai, X. Feng and Y. He, A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation, Appl. Math. Comput., 230 (2014), 9-26.  doi: 10.1016/j.amc.2013.12.096.  Google Scholar

Figure 1.  Kerr medium
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Figure 2.  Computational domain in 2D
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Figure 3.  Numerical solutions with $ \varepsilon = 0.01 $ in 1D problem (Red: new scheme with $ k_0h = 1 $; Blue: reference solution)
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Figure 4.  Numerical solutions with $ \varepsilon = 0.1 $ for the 1D problem (Red: new scheme with $ k_0h = 1 $; Blue: reference solution)
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Figure 5.  $ |T|^2 $ with respect to $ \varepsilon $ for the 1D problem
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Figure 6.  Switchback-type non-uniqueness of $ |T|^2 $ near $ \varepsilon = 0.724 $ for the 1D problem
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Figure 7.  Solutions for the 2D problem with $ k_0 = 100,h = 1/400 $ (Left: numerical solution; Right: exact solution)
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Figure 8.  Transmission of a single soliton
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Figure 9.  Collision of two solitons
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Table 1.  Errors in $ l^\infty $-norm for the 1D problem with $ \varepsilon = 0.01 $
$ N $ 100 200 400 800 1600
$ k_0=10 $
SFD 2.14 1.05 2.69e-1 6.71e-2 1.67e-3
FV[2] 1.59 5.03e-1 1.29e-1 3.23e-2 8.09e-3
CFD 5.38e-1 3.70e-2 3.27e-3 6.67e-4 2.72e-4
Scheme (29) 1.26e-3 2.99e-4 7.43e-5 1.89e-5 5.16e-6
Scheme (30) 2.16e-4 5.68e-5 1.43e-5 3.68e-6 1.16e-6
$ k_0=20 $
SFD 2.31 2.13 1.80 5.33e-1 1.34e-1
FV[2] 2.00 2.00 9.76e-1 2.60e-1 6.52e-2
CFD 2.17 1.02 7.16e-2 5.46e-3 8.00e-4
Scheme (29) 8.56e-3 1.57e-3 3.75e-4 9.57e-5 2.70e-5
Scheme (30) 1.29e-3 3.16e-4 7.56e-5 1.87e-5 4.95e-6
$ k_0=40 $
SFD 1.24 2.35 2.13 2.03 1.03
FV[2] - 2.00 1.99 1.70 5.16e-1
CFD 1.22 2.36 1.76 1.40e-1 9.86e-3
Scheme (29) 1.12e-2 9.70e-3 1.80e-3 4.49e-4 1.31e-4
Scheme (30) 5.64e-3 2.68e-3 6.86e-4 1.05e-4 2.61e-5
$ k_0=80 $
SFD 1.07 1.05 2.32 2.29 2.02
FV[2] - - 2.00 1.98 1.97
CFD 1.04 1.21 2.31 1.99 0.29
Scheme (29) 7.38e-3 8.68e-3 4.92e-3 1.99e-3 1.52e-3
Scheme (30) 1.47e-3 1.05e-3 2.48e-4 2.56e-4 2.04e-4
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$ N $ 100 200 400 800 1600
$ k_0=10 $
SFD 2.14 1.05 2.69e-1 6.71e-2 1.67e-3
FV[2] 1.59 5.03e-1 1.29e-1 3.23e-2 8.09e-3
CFD 5.38e-1 3.70e-2 3.27e-3 6.67e-4 2.72e-4
Scheme (29) 1.26e-3 2.99e-4 7.43e-5 1.89e-5 5.16e-6
Scheme (30) 2.16e-4 5.68e-5 1.43e-5 3.68e-6 1.16e-6
$ k_0=20 $
SFD 2.31 2.13 1.80 5.33e-1 1.34e-1
FV[2] 2.00 2.00 9.76e-1 2.60e-1 6.52e-2
CFD 2.17 1.02 7.16e-2 5.46e-3 8.00e-4
Scheme (29) 8.56e-3 1.57e-3 3.75e-4 9.57e-5 2.70e-5
Scheme (30) 1.29e-3 3.16e-4 7.56e-5 1.87e-5 4.95e-6
$ k_0=40 $
SFD 1.24 2.35 2.13 2.03 1.03
FV[2] - 2.00 1.99 1.70 5.16e-1
CFD 1.22 2.36 1.76 1.40e-1 9.86e-3
Scheme (29) 1.12e-2 9.70e-3 1.80e-3 4.49e-4 1.31e-4
Scheme (30) 5.64e-3 2.68e-3 6.86e-4 1.05e-4 2.61e-5
$ k_0=80 $
SFD 1.07 1.05 2.32 2.29 2.02
FV[2] - - 2.00 1.98 1.97
CFD 1.04 1.21 2.31 1.99 0.29
Scheme (29) 7.38e-3 8.68e-3 4.92e-3 1.99e-3 1.52e-3
Scheme (30) 1.47e-3 1.05e-3 2.48e-4 2.56e-4 2.04e-4
Table 2.  Iteration numbers of different iteration methods for the 1D problem
$ k_0 $ 10 20 40 80 160 320 640 1280
$ \varepsilon=0.01 $
Frozen-nonlinearity 5 5 6 7 9 12 17 38
Error Correction 3 4 4 4 4 4 5 6
Modified Newton 5 6 7 8 10 14 22 -
Newton's method 4 4 5 5 6 8 11 -
$ \varepsilon=0.02 $
Frozen-nonlinearity 5 6 7 9 12 19 45 -
Error Correction 4 4 4 4 5 5 7 9
Modified Newton 6 7 8 10 14 23 - -
Newton's method 4 5 5 6 8 11 - -
$ \varepsilon=0.04 $
Frozen-nonlinearity 6 8 9 13 22 55 - -
Error Correction 4 4 5 5 6 8 13 -
Modified Newton 7 9 10 16 25 - - -
Newton's method 5 5 6 8 10 - - -
$ \varepsilon=0.06 $
Frozen-nonlinearity 7 9 12 18 35 - - -
Error Correction 4 5 5 6 7 10 - -
Modified Newton 8 9 14 20 39 - - -
Newton's method 5 6 7 10 - - - -
$ \varepsilon=0.08 $
Frozen-nonlinearity 8 10 14 20 89 - - -
Error Correction 5 5 6 7 9 - - -
Modified Newton 9 11 17 25 - - - -
Newton's method 5 6 8 11 - - - -
$ \varepsilon=0.1 $
Frozen-nonlinearity 9 10 16 35 - - - -
Error Correction 5 6 6 8 12 - - -
Modified Newton 10 13 18 36 - - - -
Newton's method 6 7 9 - - - - -
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$ k_0 $ 10 20 40 80 160 320 640 1280
$ \varepsilon=0.01 $
Frozen-nonlinearity 5 5 6 7 9 12 17 38
Error Correction 3 4 4 4 4 4 5 6
Modified Newton 5 6 7 8 10 14 22 -
Newton's method 4 4 5 5 6 8 11 -
$ \varepsilon=0.02 $
Frozen-nonlinearity 5 6 7 9 12 19 45 -
Error Correction 4 4 4 4 5 5 7 9
Modified Newton 6 7 8 10 14 23 - -
Newton's method 4 5 5 6 8 11 - -
$ \varepsilon=0.04 $
Frozen-nonlinearity 6 8 9 13 22 55 - -
Error Correction 4 4 5 5 6 8 13 -
Modified Newton 7 9 10 16 25 - - -
Newton's method 5 5 6 8 10 - - -
$ \varepsilon=0.06 $
Frozen-nonlinearity 7 9 12 18 35 - - -
Error Correction 4 5 5 6 7 10 - -
Modified Newton 8 9 14 20 39 - - -
Newton's method 5 6 7 10 - - - -
$ \varepsilon=0.08 $
Frozen-nonlinearity 8 10 14 20 89 - - -
Error Correction 5 5 6 7 9 - - -
Modified Newton 9 11 17 25 - - - -
Newton's method 5 6 8 11 - - - -
$ \varepsilon=0.1 $
Frozen-nonlinearity 9 10 16 35 - - - -
Error Correction 5 6 6 8 12 - - -
Modified Newton 10 13 18 36 - - - -
Newton's method 6 7 9 - - - - -
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