Exponential integrator method for solving the nonlinear Helmholtz equation

Shuqi He; Kun Wang; Shuqi He; Kun Wang

doi:10.3934/math.2022953

AIMS Mathematics

2022, Volume 7, Issue 9: 17313-17326. doi: 10.3934/math.2022953

Previous Article Next Article

Research article

Exponential integrator method for solving the nonlinear Helmholtz equation

Shuqi He ¹,
Kun Wang ^{2
,
,}

1.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2.
State Key Laboratory of Coal Mine Disaster Dynamics and Control, College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received: 16 May 2022 Revised: 14 July 2022 Accepted: 20 July 2022 Published: 25 July 2022
MSC : 65L05, 65N06, 65N15

In this paper, we study the exponential integrator method (EIM) for solving the nonlinear Helmholtz equation (NLHE). As the wave number or the characteristic coefficient in the nonlinear term is large, the NLHE becomes a highly oscillatory and indefinite nonlinear problem, which makes most of numerical methods lose their expected computational effects. Based on the shooting method, the NLHE is firstly transformed into an initial-value-type problem. Then, the EIM is utilized for solving the deduced problem, by which we not only can capture the oscillation very well, but also avoid to search the nonlinear iteration method and to solve indefinite linear equations at each iteration step. Therefore, the high accuracy simulations with relative large physical parameters in the NLHE become possible and lots of computational costs can be saved. Some numerical examples, including the extension to the nonlinear Helmholtz system, are shown to verify the accuracy and efficiency of the proposed method.
- nonlinear Helmholtz equation,
- exponential integrator method,
- shooting method,
- iteration method,
- third-harmonic signal generation
Citation: Shuqi He, Kun Wang. Exponential integrator method for solving the nonlinear Helmholtz equation[J]. AIMS Mathematics, 2022, 7(9): 17313-17326. doi: 10.3934/math.2022953

Related Papers:

Abstract

In this paper, we study the exponential integrator method (EIM) for solving the nonlinear Helmholtz equation (NLHE). As the wave number or the characteristic coefficient in the nonlinear term is large, the NLHE becomes a highly oscillatory and indefinite nonlinear problem, which makes most of numerical methods lose their expected computational effects. Based on the shooting method, the NLHE is firstly transformed into an initial-value-type problem. Then, the EIM is utilized for solving the deduced problem, by which we not only can capture the oscillation very well, but also avoid to search the nonlinear iteration method and to solve indefinite linear equations at each iteration step. Therefore, the high accuracy simulations with relative large physical parameters in the NLHE become possible and lots of computational costs can be saved. Some numerical examples, including the extension to the nonlinear Helmholtz system, are shown to verify the accuracy and efficiency of the proposed method.

References

[1]	G. Baruch, G. Fibich, S. Tsynkov, High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry, J. Comput. Appl. Math., 204 (2007), 477–492. http://dx.doi.org/10.1016/j.cam.2006.01.048 doi: 10.1016/j.cam.2006.01.048
[2]	G. Baruch, G. Fibich, 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. http://dx.doi.org/10.1016/j.jcp.2007.08.022 doi: 10.1016/j.jcp.2007.08.022
[3]	H. Barucq, T. Chaumont-Frelet, C. Gout, Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation, Math. Comput., 86 (2017), 2129–2157. http://dx.doi.org/10.1090/mcom/3165 doi: 10.1090/mcom/3165
[4]	R. Boyd, Nonlinear optics, New York: Academic Press, 2008.
[5]	S. Deng, J. Li, A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with energy-preservation for long-time dynamics of the nonlinear Klein-Gordon equation, Appl. Numer. Math., 178 (2022), 166–191. http://dx.doi.org/10.1016/j.apnum.2022.03.019 doi: 10.1016/j.apnum.2022.03.019
[6]	P. Deuflhard, A study of extrapolation methods based on multistep schemes without parasitic solutions, ZAMP, 30 (1979), 177–189. http://dx.doi.org/10.1007/BF01601932 doi: 10.1007/BF01601932
[7]	W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3 (1961), 381–397. http://dx.doi.org/10.1007/BF01386037 doi: 10.1007/BF01386037
[8]	B. Garcia-Archilla, J. Sanz-Serna, R. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput., 20 (1998), 930–963. http://dx.doi.org/10.1137/S1064827596313851 doi: 10.1137/S1064827596313851
[9]	V. Grimm, M. Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations, J. Phys. A: Math. Gen., 39 (2006), 5495.
[10]	X. He, K. Wang, L. Xu, Efficient finite difference methods for the nonlinear Helmholtz equation in kerr medium, Electron. Res. Arch., 28 (2020), 1503–1528. http://dx.doi.org/10.3934/era.2020079 doi: 10.3934/era.2020079
[11]	M. Heath, Scientific computing: an introductory survey, New York: McGraw-Hill Companies, 1997.
[12]	M. Hochbruck, A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209–286. http://dx.doi.org/10.1017/S0962492910000048 doi: 10.1017/S0962492910000048
[13]	M. Hochbruch, A. Ostermann, J. Schweitzer, Exponential Rosenbrock-type methods, SIAM J. Numer. Anal., 47 (2009), 786–803. http://dx.doi.org/10.1137/080717717 doi: 10.1137/080717717
[14]	J. Li, Convergence analysis of a symmetric exponential integrator Fourier pseudo-spectral scheme for the Klein-Gordon-Dirac equation, Math. Comput. Simulat., 190 (2021), 691–713. http://dx.doi.org/10.1016/j.matcom.2021.06.007 doi: 10.1016/j.matcom.2021.06.007
[15]	J. Li, Energy-preserving exponential integrator Fourier pseudo-spectral schemes for the nonlinear Dirac equation, Appl. Numer. Math., 172 (2022), 1–26. http://dx.doi.org/10.1016/j.apnum.2021.09.006 doi: 10.1016/j.apnum.2021.09.006
[16]	J. Li, Error analysis of a time fourth-order exponential wave integrator Fourier pseudo-spectral method for the nonlinear Dirac equation, Int. J. Comput. Math., 99 (2022), 791–807. http://dx.doi.org/10.1080/00207160.2021.1934459 doi: 10.1080/00207160.2021.1934459
[17]	V. Luan, A. Ostermann, Exponential Rosenbrock methods of order five-construction, analysis and numerical comparisons, J. Comput. Appl. Math., 255 (2014), 417–431. http://dx.doi.org/10.1016/j.cam.2013.04.041 doi: 10.1016/j.cam.2013.04.041
[18]	A. Suryanto, E. Groesen, M. Hammer, Finite element analysis of optical bistability in one-dimensional nonlinear photonic band gap structures with a defect, J. Nonlinear Opt. Phys., 12 (2003), 187–204. http://dx.doi.org/10.1142/S0218863503001328 doi: 10.1142/S0218863503001328
[19]	H. Wu, 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. http://dx.doi.org/10.1137/17M111314X doi: 10.1137/17M111314X
[20]	X. Wu, X. You, W. Shi, B. Wang, ERKN integrators for systems of oscillatory second-order differential equations, Comput. Phys. Commun., 181 (2010), 1873–1887. http://dx.doi.org/10.1016/j.cpc.2010.07.046 doi: 10.1016/j.cpc.2010.07.046
[21]	Z. Xu, G. Bao, A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects, J. Opt. Soc. Am. A, 27 (2010), 2347–2353. http://dx.doi.org/10.1364/JOSAA.27.002347 doi: 10.1364/JOSAA.27.002347
[22]	J. Yuan, J. Yang, W. Ai, J. Xiao, T. Shuai, Third-harmonic signal generation and enhancement in nonlinear photonic crystals with an efficient continuation finite-element method, J. Nanophotonics, 10 (2016), 036017. http://dx.doi.org/10.1117/1.JNP.10.036017 doi: 10.1117/1.JNP.10.036017
[23]	L. Yuan, Y. Lu, Robust iterative method for nonlinear Helmholtz equation, J. Comput. Phys., 343 (2017), 1–9. http://dx.doi.org/10.1016/j.jcp.2017.04.046 doi: 10.1016/j.jcp.2017.04.046

Reader Comments

Your name:*

Email:*
© 2022 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)