Special Issues

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

  • 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.

    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

    Related Papers:

  • 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.



    加载中


    [1] High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry. J. Comput. Appl. Math. (2007) 204: 477-492.
    [2] High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension. J. Comput. Phys. (2007) 227: 820-850.
    [3] A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media. J. Comput. Phys. (2009) 228: 3789-3815.
    [4] Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media. J. Comput. Phys. (2017) 350: 420-452.
    [5] (2008) Nonlinear Optics.Elsevier/Academic Press.
    [6] Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity. Phys. Rev. B (2000) 62: 7683-7686.
    [7] Optical response of a nonlinear dielectric film. Phys. Rev. B (1987) 35: 524-532.
    [8] Optical response of nonlinear multilayer structures: Bilayers and superlattices. Phys. Rev. B (1987) 36: 524-532.
    [9] Compact ADI method for solving parabolic differential equations. Numer. Methods Partial Differential Equations (2002) 18: 129-142.
    [10] A new ADI scheme for solving three-dimensional parabolic equations with first-order derivatives and variable coefficients. J. Comput. Anal. Appl. (2000) 2: 293-308.
    [11] Dual variational methods and nonvanishing for the nonlinear Helmholtz equation. Adv. Math. (2015) 280: 690-728.
    [12] Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane. Analysis (Berlin) (2017) 37: 55-68.
    [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
    [14] High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering. J. Comput. Phys. (2001) 171: 632-677.
    [15] Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions. J. Comput. Phys. (2005) 210: 183-224.
    [16] Efficient finite difference methods for acoustic scattering from circular cylindrical obstacle. Int. J. Numer. Anal. Model. (2016) 13: 986-1002.
    [17] Uniformly convergent novel finite difference methods for singularly perturbed reaction-diffusion equations. Numer. Methods Partial Differential Equations (2019) 35: 2120-2148.
    [18] Self-guided waves and exact solutions of the nonlinear Helmholtz equation. J. Opt. Soc. Amer. B Opt. Phys. (2000) 17: 751-757.
    [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
    [20] Optical spatial solitons and their interactions: Universality and diversity. Science (1999) 286: 1518-1523.
    [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
    [22] Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with a Defect. J. Nonlinear Optical Phys. Materials (2003) 12: 187-204.
    [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
    [24] Error correction method for Navier-Stokes equations at high Reynolds numbers. J. Comput. Phys. (2013) 255: 245-265.
    [25] Pollution-free finite difference schemes for non-homogeneous Helmholtz equation. Int. J. Numer. Anal. Model. (2014) 11: 787-815.
    [26] Is pollution effect of finite difference schemes avoidable for multi-dimensional Helmholtz equations with high wave numbers?. Commun. Comput. Phys. (2017) 21: 490-514.
    [27] Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates. Commun. Comput. Phys. (2015) 17: 779-807.
    [28] Analysis of pollution-free approaches for multi-dimensional Helmholtz equations. Int. J. Numer. Anal. Model. (2019) 16: 412-435.
    [29] Finite element method and its analysis for a nonlinear Helmholtz equation with high wave numbers. SIAM J. Numer. Anal. (2018) 56: 1338-1359.
    [30] A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects. Journal of the Optical Society of America(A) (2010) 27: 2347-2353.
    [31] Robust iterative method for nonlinear Helmholtz equation. J. Comput. Phys. (2017) 343: 1-9.
    [32] A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation. J. Sci. Comput. (2013) 54: 97-120.
    [33] A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation. Appl. Math. Comput. (2014) 230: 9-26.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1688) PDF downloads(246) Cited by(1)

Article outline

Figures and Tables

Figures(9)  /  Tables(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog