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Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime


Authors: Yongyong Cai and Yongjun Yuan
Journal: Math. Comp. 87 (2018), 1191-1225
MSC (2010): Primary 35Q55, 65M06, 65M12, 65M15
DOI: https://doi.org/10.1090/mcom/3269
Published electronically: October 12, 2017
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Abstract: We rigorously analyze the error estimates of the conservative finite difference method (CNFD) for the Zakharov system (ZS) with a dimensionless parameter $ \varepsilon \in (0,1]$, which is inversely proportional to the ion acoustic speed. When $ \varepsilon \to 0^+$, ZS collapses to the standard nonlinear Schrödinger equation (NLS). In the subsonic limit regime, i.e., $ \varepsilon \to 0^+$, there exist highly oscillatory initial layers in the solution. The initial layers propagate with $ O(\varepsilon )$ wavelength in time, $ O(1)$ and $ O(\varepsilon ^2)$ amplitudes, for the ill-prepared initial data and well-prepared initial data, respectively. This oscillatory behavior brings significant difficulties in analyzing the errors of numerical methods for solving the Zakharov system. In this work, we show the CNFD possesses the error bounds $ h^2/\varepsilon +\tau ^2/\varepsilon ^3$ in the energy norm for the ill-prepared initial data, where $ h$ is mesh size and $ \tau $ is time step. For the well-prepared initial data, CNFD is uniformly convergent for $ \varepsilon \in (0,1]$, with second-order accuracy in space and $ O(\tau ^{4/3})$ accuracy in time. The main tools involved in the analysis include cut-off technique, energy methods, $ \varepsilon $-dependent error estimates of the ZS, and $ \varepsilon $-dependent error bounds between the numerical approximate solution of the ZS and the solution of the limit NLS. Our approach works in one, two and three dimensions, and can be easily extended to the generalized Zakharov system and nonconservative schemes. Numerical results suggest that the error bounds are sharp for the plasma densities and the error bounds of the CNFD for the electric fields are the same as those of the splitting methods.


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  • [1] Hélène Added and Stéphane Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal. 79 (1988), no. 1, 183-210. MR 950090, https://doi.org/10.1016/0022-1236(88)90036-5
  • [2] Georgios D. Akrivis, Vassilios A. Dougalis, and Ohannes A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math. 59 (1991), no. 1, 31-53. MR 1103752, https://doi.org/10.1007/BF01385769
  • [3] Weizhu Bao and Yongyong Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp. 82 (2013), no. 281, 99-128. MR 2983017, https://doi.org/10.1090/S0025-5718-2012-02617-2
  • [4] Weizhu Bao and Yongyong Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 50 (2012), no. 2, 492-521. MR 2914273, https://doi.org/10.1137/110830800
  • [5] Weizhu Bao and Fangfang Sun, Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput. 26 (2005), no. 3, 1057-1088. MR 2126126, https://doi.org/10.1137/030600941
  • [6] Weizhu Bao, Fangfang Sun, and G. W. Wei, Numerical methods for the generalized Zakharov system, J. Comput. Phys. 190 (2003), no. 1, 201-228. MR 2046763, https://doi.org/10.1016/S0021-9991(03)00271-7
  • [7] A. H. Bhrawy, An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput. 247 (2014), 30-46. MR 3270819, https://doi.org/10.1016/j.amc.2014.08.062
  • [8] Qian Shun Chang, Bo Ling Guo, and Hong Jiang, Finite difference method for generalized Zakharov equations, Math. Comp. 64 (1995), no. 210, 537-553, S7-S11. MR 1284664, https://doi.org/10.2307/2153438
  • [9] J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), no. 2, 384-436. MR 1491547, https://doi.org/10.1006/jfan.1997.3148
  • [10] R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp. 58 (1992), no. 197, 83-102. MR 1106968, https://doi.org/10.2307/2153022
  • [11] Yuanyuan Ji and Heping Ma, Uniform convergence of the Legendre spectral method for the Zakharov equations, Numer. Methods Partial Differential Equations 29 (2013), no. 2, 475-495. MR 3022895, https://doi.org/10.1002/num.21716
  • [12] Shi Jin, Peter A. Markowich, and Chunxiong Zheng, Numerical simulation of a generalized Zakharov system, J. Comput. Phys. 201 (2004), no. 1, 376-395. MR 2098862, https://doi.org/10.1016/j.jcp.2004.06.001
  • [13] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal. 127 (1995), no. 1, 204-234. MR 1308623, https://doi.org/10.1006/jfan.1995.1009
  • [14] Nader Masmoudi and Kenji Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math. 172 (2008), no. 3, 535-583. MR 2393080, https://doi.org/10.1007/s00222-008-0110-5
  • [15] Frank Merle, Blow-up results of virial type for Zakharov equations, Comm. Math. Phys. 175 (1996), no. 2, 433-455. MR 1370102
  • [16] Tohru Ozawa and Yoshio Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Differential Integral Equations 5 (1992), no. 4, 721-745. MR 1167491
  • [17] G. L. Payne, D. R. Nicholson, R. M. Downie, , and Numerical solution of the Zakharov equations, J. Comput. Phys. 50 (1983), no. 3, 482-498. MR 710406, https://doi.org/10.1016/0021-9991(83)90107-9
  • [18] Steven H. Schochet and Michael I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys. 106 (1986), no. 4, 569-580. MR 860310
  • [19] Benjamin Texier, Derivation of the Zakharov equations, Arch. Ration. Mech. Anal. 184 (2007), no. 1, 121-183. MR 2289864, https://doi.org/10.1007/s00205-006-0034-4
  • [20] Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170
  • [21] Jian Wang, Multisymplectic numerical method for the Zakharov system, Comput. Phys. Comm. 180 (2009), no. 7, 1063-1071. MR 2678347, https://doi.org/10.1016/j.cpc.2008.12.028
  • [22] Yinhua Xia, Yan Xu, and Chi-Wang Shu, Local discontinuous Galerkin methods for the generalized Zakharov system, J. Comput. Phys. 229 (2010), no. 4, 1238-1259. MR 2576247, https://doi.org/10.1016/j.jcp.2009.10.029
  • [23] V. E. ZAKHAROV, Collapse of Langmuir waves, JETP, 35 (1972), pp. 908-914.

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Additional Information

Yongyong Cai
Affiliation: Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China – and – Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: yongyong.cai@csrc.ac.cn

Yongjun Yuan
Affiliation: Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha 410006, People’s Republic of China – and – Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
Email: yuanyongjun0301@163.com

DOI: https://doi.org/10.1090/mcom/3269
Keywords: Zakharov system, error estimates, subsonic limit, finite difference method, conservative scheme
Received by editor(s): September 28, 2015
Received by editor(s) in revised form: June 19, 2016, and December 23, 2016
Published electronically: October 12, 2017
Additional Notes: The first author was partially supported by NSF grants DMS-1217066 and DMS-1419053, and by the NSAF grant U1530401; the second and corresponding author was partially supported by the National Natural Science Foundation of China under grant 11601148, and by the Construct Program of the Key Discipline in Hunan Province.
Article copyright: © Copyright 2017 American Mathematical Society

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