American Institute of Mathematical Sciences

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January  2021, 26(1): 645-666. doi: 10.3934/dcdsb.2020262

Efficient and accurate sav schemes for the generalized Zakharov systems

Jie Shen 1, and  Nan Zheng 2,

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
2. 

School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling, and High-Performance Scientific Computing, Xiamen University, Xiamen 361005, China

Received  March 2020 Revised  July 2020 Published  August 2020

Fund Project: This work is partially supported by NSF grant DMS-1720442 and NSFC grant 11971407

We develop in this paper efficient and accurate numerical schemes based on the scalar auxiliary variable (SAV) approach for the generalized Zakharov system and generalized vector Zakharov system. These schemes are second-order in time, linear, unconditionally stable, only require solving linear systems with constant coefficients at each time step, and preserve exactly a modified Hamiltonian. Ample numerical results are presented to demonstrate the accuracy and robustness of the schemes.

Keywords: generalized Zakharov system, scalar auxiliary variable, Hamiltonian system, stability.
Mathematics Subject Classification: 35Q55, 65M12, 65M70.
Citation: Jie Shen, Nan Zheng. Efficient and accurate sav schemes for the generalized Zakharov systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 645-666. doi: 10.3934/dcdsb.2020262
References:
[1]

K. Amaratunga, J. R. Williams, S. Qian and J. Weiss, Wavelet-galerkin solutions for one-dimensional partial differential equations, International Journal for Numerical Methods in Engineering, 37 (1994), 2703-2716.  doi: 10.1002/nme.1620371602.  Google Scholar

[2]

X. Antoine, J. Shen and Q. Tang, An explicit scalar auxiliary variable pseudospectral scheme for the dynamics of nonlinear schrödinger and gross-pitaevskii equations, Preprint. Google Scholar

[3]

W. Bao and C. Su, A uniformly and optimally accurate method for the zakharov system in the subsonic limit regime, SIAM Journal on Scientific Computing, 40 (2018), A929-A953. doi: 10.1137/17M1113333.  Google Scholar

[4]

W. Bao and F. Sun, Efficient and stable numerical methods for the generalized and vector zakharov system, SIAM Journal on Scientific Computing, 26 (2005), 1057-1088.  doi: 10.1137/030600941.  Google Scholar

[5]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Courier Corporation, 2001.  Google Scholar

[6]

W. Cai, C. Jiang, Y. Wang and Y. Song, Structure-preserving algorithms for the two-dimensional sine-gordon equation with neumann boundary conditions, Journal of Computational Physics, 395 (2019), 166-185.  doi: 10.1016/j.jcp.2019.05.048.  Google Scholar

[7]

Q. Chang and H. Jiang, A conservative difference scheme for the zakharov equations, Journal of Computational Physics, 113 (1994), 309-319.  doi: 10.1006/jcph.1994.1138.  Google Scholar

[8]

R. T. Glassey, Convergence of an energy-preserving scheme for the zakharov equations in one space dimension, Mathematics of Computation, 58 (1992), 83-102.  doi: 10.1090/S0025-5718-1992-1106968-6.  Google Scholar

[9]

H. Hadouaj, B. A. Malomed and G. A. Maugin, Dynamics of a soliton in a generalized zakharov system with dissipation, Physical Review A, 44 (1991), 3925-3931.  doi: 10.1103/PhysRevA.44.3925.  Google Scholar

[10]

H. Hadouaj, B. A. Malomed and G. A. Maugin, Soliton-soliton collisions in a generalized zakharov system, Physical Review A, 44 (1991), 3932-3940.  doi: 10.1103/PhysRevA.44.3932.  Google Scholar

[11]

P. K. Newton, Wave interactions in the singular zakharov system, Journal of Mathematical Physics, 32 (1991), 431-440.  doi: 10.1063/1.529430.  Google Scholar

[12]

G. L. Payne, D. R. Nicholson and R. M. Downie, Numerical solution of the zakharov equations, Journal of Computational Physics, 50 (1983), 482-498.  doi: 10.1016/0021-9991(83)90107-9.  Google Scholar

[13]

B. F. Sanders, N. D. Katopodes and J. P. Boyd, Spectral modeling of nonlinear dispersive waves, Journal of Hydraulic Engineering, 124 (1998), 2-12.   Google Scholar

[14]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, volume 41, Springer Science & Business Media, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[15]

J. Shen, J. Xu and J. Yang, The scalar auxiliary variable (sav) approach for gradient flows, Journal of Computational Physics, 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.  Google Scholar

[16]

J. Shen, J. Xu and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Review, 61 (2019), 474-506.  doi: 10.1137/17M1150153.  Google Scholar

[17]

C. Sulem and P. L. Sulem, Regularity properties for the equations of langmuir turbulence, Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences Serie A, 289 (1979), 173-176.   Google Scholar

[18]

G. W. Wei, Discrete singular convolution for the solution of the fokker-planck equation, The Journal of Chemical Physics, 110 (1999), 8930-8942.  doi: 10.1063/1.478812.  Google Scholar

[19]

V. E. Zakharov et al., Collapse of langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914. Google Scholar

show all references

References:
[1]

K. Amaratunga, J. R. Williams, S. Qian and J. Weiss, Wavelet-galerkin solutions for one-dimensional partial differential equations, International Journal for Numerical Methods in Engineering, 37 (1994), 2703-2716.  doi: 10.1002/nme.1620371602.  Google Scholar

[2]

X. Antoine, J. Shen and Q. Tang, An explicit scalar auxiliary variable pseudospectral scheme for the dynamics of nonlinear schrödinger and gross-pitaevskii equations, Preprint. Google Scholar

[3]

W. Bao and C. Su, A uniformly and optimally accurate method for the zakharov system in the subsonic limit regime, SIAM Journal on Scientific Computing, 40 (2018), A929-A953. doi: 10.1137/17M1113333.  Google Scholar

[4]

W. Bao and F. Sun, Efficient and stable numerical methods for the generalized and vector zakharov system, SIAM Journal on Scientific Computing, 26 (2005), 1057-1088.  doi: 10.1137/030600941.  Google Scholar

[5]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Courier Corporation, 2001.  Google Scholar

[6]

W. Cai, C. Jiang, Y. Wang and Y. Song, Structure-preserving algorithms for the two-dimensional sine-gordon equation with neumann boundary conditions, Journal of Computational Physics, 395 (2019), 166-185.  doi: 10.1016/j.jcp.2019.05.048.  Google Scholar

[7]

Q. Chang and H. Jiang, A conservative difference scheme for the zakharov equations, Journal of Computational Physics, 113 (1994), 309-319.  doi: 10.1006/jcph.1994.1138.  Google Scholar

[8]

R. T. Glassey, Convergence of an energy-preserving scheme for the zakharov equations in one space dimension, Mathematics of Computation, 58 (1992), 83-102.  doi: 10.1090/S0025-5718-1992-1106968-6.  Google Scholar

[9]

H. Hadouaj, B. A. Malomed and G. A. Maugin, Dynamics of a soliton in a generalized zakharov system with dissipation, Physical Review A, 44 (1991), 3925-3931.  doi: 10.1103/PhysRevA.44.3925.  Google Scholar

[10]

H. Hadouaj, B. A. Malomed and G. A. Maugin, Soliton-soliton collisions in a generalized zakharov system, Physical Review A, 44 (1991), 3932-3940.  doi: 10.1103/PhysRevA.44.3932.  Google Scholar

[11]

P. K. Newton, Wave interactions in the singular zakharov system, Journal of Mathematical Physics, 32 (1991), 431-440.  doi: 10.1063/1.529430.  Google Scholar

[12]

G. L. Payne, D. R. Nicholson and R. M. Downie, Numerical solution of the zakharov equations, Journal of Computational Physics, 50 (1983), 482-498.  doi: 10.1016/0021-9991(83)90107-9.  Google Scholar

[13]

B. F. Sanders, N. D. Katopodes and J. P. Boyd, Spectral modeling of nonlinear dispersive waves, Journal of Hydraulic Engineering, 124 (1998), 2-12.   Google Scholar

[14]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, volume 41, Springer Science & Business Media, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[15]

J. Shen, J. Xu and J. Yang, The scalar auxiliary variable (sav) approach for gradient flows, Journal of Computational Physics, 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.  Google Scholar

[16]

J. Shen, J. Xu and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Review, 61 (2019), 474-506.  doi: 10.1137/17M1150153.  Google Scholar

[17]

C. Sulem and P. L. Sulem, Regularity properties for the equations of langmuir turbulence, Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences Serie A, 289 (1979), 173-176.   Google Scholar

[18]

G. W. Wei, Discrete singular convolution for the solution of the fokker-planck equation, The Journal of Chemical Physics, 110 (1999), 8930-8942.  doi: 10.1063/1.478812.  Google Scholar

[19]

V. E. Zakharov et al., Collapse of langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914. Google Scholar

Figure 4.1.  Numerical solutions of the electric field $ |E(x,t)|^2 $ at T = 1
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Figure 4.2.  Numerical solutions of the electric field $ |E(x,t)|^2 $ at $ [0,1] $
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Figure 4.3.  Numerical and reference solutions for Case Ⅲ at T = 2 (left column) and T = 4 (right column)
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Figure 4.4.  Numerical solutions in (4.2): surface-plots of the electron density $ |E(x,y,t)|^2 $ (left) and ion density fluctuation $ N(x,y,t) $ (right) for Case 1 with $ \gamma = 0.8 $ and $ \lambda = 20 $
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Figure 4.5.  Numerical solutions in (4.2): surface-plots of the electron density $ |E(x,y,t)|^2 $ (left) and ion density fluctuation $ N(x,y,t) $ (left) with $ \gamma = 0.1 $ and $ \lambda = 20 $
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Figure 4.6.  Numerical solutions in (4.2): surface-plots of the electron density $ |E(x,y,t)|^2 $ (left) and ion density fluctuation $ N(x,y,t) $ (left) for Case 1 with $ \gamma = 0.1 $ and $ \lambda = 100 $
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Figure 4.7.  Numerical solutions in (4.3) for Case Ⅰ with $ \lambda = 2 $
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Figure 4.8.  Numerical solutions in (4.3) for Case Ⅱ with $ \lambda = 2 $
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Figure 4.9.  Numerical solutions in (4.3) for Case Ⅲ with $ \lambda = 2 $
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Figure 4.10.  Numerical solutions in (4.3) for Case Ⅲ with $ \lambda = 100 $
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Figure 4.11.  Evolution of the total wave energy $ \|\textbf{E}(t)\|^2 $ and the wave energy of the three components of the electric field $ \|E_1(t)\|^2 $, $ \|E_2(t)\|^2 $ and $ \|E_3(t)\|^2 $ in (4.4) for Case 1 (left) and Case 2 (right)
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Table 1.  Error and convergence rates in time
$ \delta t $ $ |e_E|_{L^{\infty}(0,T;L^{\infty})} $ $ Rate $ $ |e_N|_{L^{\infty}(0,T;L^{\infty})} $ $ Rate $
$ 2\times 10^{-2} $ 1.90E(-3) - 1.60E(-3) -
$ 1\times 10^{-2} $ 4.82E(-4) 1.98 3.94E(-4) 1.98
$ 5\times 10^{-3} $ 1.21E(-4) 1.99 9.93E(-5) 1.99
$ 2.5\times 10^{-3} $ 3.04E(-5) 2.00 2.49E(-5) 2.00
$ 1.25\times 10^{-3} $ 7.61E(-6) 2.00 6.23E(-6) 2.00
$ 6.25\times 10^{-4} $ 1.90E(-6) 2.00 1.56E(-6) 2.00
$ 3.125\times 10^{-4} $ 4.76E(-7) 2.00 3.93E(-7) 1.99
$ 1.5625\times 10^{-4} $ 1.21E(-7) 1.97 1.08E(-7) 1.87
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$ \delta t $ $ |e_E|_{L^{\infty}(0,T;L^{\infty})} $ $ Rate $ $ |e_N|_{L^{\infty}(0,T;L^{\infty})} $ $ Rate $
$ 2\times 10^{-2} $ 1.90E(-3) - 1.60E(-3) -
$ 1\times 10^{-2} $ 4.82E(-4) 1.98 3.94E(-4) 1.98
$ 5\times 10^{-3} $ 1.21E(-4) 1.99 9.93E(-5) 1.99
$ 2.5\times 10^{-3} $ 3.04E(-5) 2.00 2.49E(-5) 2.00
$ 1.25\times 10^{-3} $ 7.61E(-6) 2.00 6.23E(-6) 2.00
$ 6.25\times 10^{-4} $ 1.90E(-6) 2.00 1.56E(-6) 2.00
$ 3.125\times 10^{-4} $ 4.76E(-7) 2.00 3.93E(-7) 1.99
$ 1.5625\times 10^{-4} $ 1.21E(-7) 1.97 1.08E(-7) 1.87
Table 2.  Discretization Error in space
$ N $ 32 64 128 256
$ |e_E|_{L^{\infty}(0,T;L^{\infty})} $ 5.40E(-1) 7.84E(-2) 1.91E(-4) 8.88E(-7)
$ |e_N|_{L^{\infty}(0,T;L^{\infty})} $ 2.64E(-1) 1.23E(-1) 1.10E(-3) 5.19E(-6)
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$ N $ 32 64 128 256
$ |e_E|_{L^{\infty}(0,T;L^{\infty})} $ 5.40E(-1) 7.84E(-2) 1.91E(-4) 8.88E(-7)
$ |e_N|_{L^{\infty}(0,T;L^{\infty})} $ 2.64E(-1) 1.23E(-1) 1.10E(-3) 5.19E(-6)
Table 3.  Error and convergence rates for the conserved quantities
$ \delta t $ $ |e_{D^{GZS}}|_{L^{\infty}(0,T)} $ $ Rate $ $ |e_P^{GZS}|_{L^{\infty}(0,T)} $ $ Rate $ $ |e_H^{GZS}|_{L^{\infty}(0,T)} $ $ Rate $
$ 8\times 10^{-2} $ 1.50E(-3) - 2.10E(-3) - 2.00E(-3) -
$ 4\times 10^{-2} $ 1.69E(-4) 3.14 4.68E(-4) 2.16 2.26E(-4) 3.10
$ 2\times 10^{-2} $ 2.05E(-5) 3.04 1.12E(-4) 2.07 2.90E(-5) 3.02
$ 1\times 10^{-2} $ 2.53E(-6) 3.02 2.73E(-5) 2.03 3.60E(-6) 3.01
$ 5\times 10^{-3} $ 3.14E(-7) 3.01 6.74E(-6) 2.02 4.49E(-7) 3.00
$ 2.5\times 10^{-3} $ 3.91E(-8) 3.01 1.68E(-6) 2.01 5.61E(-8) 3.00
$ 1.25\times 10^{-3} $ 4.88E(-9) 3.00 4.18E(-7) 2.00 7.09E(-9) 2.98
$ 6.25\times 10^{-4} $ 6.09E(-10) 3.00 1.05E(-7) 1.99 1.18E(-9) 2.58
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$ \delta t $ $ |e_{D^{GZS}}|_{L^{\infty}(0,T)} $ $ Rate $ $ |e_P^{GZS}|_{L^{\infty}(0,T)} $ $ Rate $ $ |e_H^{GZS}|_{L^{\infty}(0,T)} $ $ Rate $
$ 8\times 10^{-2} $ 1.50E(-3) - 2.10E(-3) - 2.00E(-3) -
$ 4\times 10^{-2} $ 1.69E(-4) 3.14 4.68E(-4) 2.16 2.26E(-4) 3.10
$ 2\times 10^{-2} $ 2.05E(-5) 3.04 1.12E(-4) 2.07 2.90E(-5) 3.02
$ 1\times 10^{-2} $ 2.53E(-6) 3.02 2.73E(-5) 2.03 3.60E(-6) 3.01
$ 5\times 10^{-3} $ 3.14E(-7) 3.01 6.74E(-6) 2.02 4.49E(-7) 3.00
$ 2.5\times 10^{-3} $ 3.91E(-8) 3.01 1.68E(-6) 2.01 5.61E(-8) 3.00
$ 1.25\times 10^{-3} $ 4.88E(-9) 3.00 4.18E(-7) 2.00 7.09E(-9) 2.98
$ 6.25\times 10^{-4} $ 6.09E(-10) 3.00 1.05E(-7) 1.99 1.18E(-9) 2.58
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