August 2018, 11(4): 1037-1062. doi: 10.3934/krm.2018040

Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes

1. 

Beijing Computational Science Research Center, Beijing 100193, China

2. 

Department of Mathematics, National University of Singapore, 119076, Singapore

* Corresponding author: Chunmei Su

Received  August 2017 Revised  February 2018 Published  April 2018

Fund Project: The first author is supported by Singapore Ministry of Education Academic Research Fund Tier 2 R-146-000-223-112. The second author was supported by Natural Science Foundation of China Grant U1530401 and the Postdoctoral Science Foundation of China Grant 2016M600904

We establish a uniform error estimate of a finite difference method for the Klein-Gordon-Schrödinger (KGS) equations with two dimensionless parameters $0<γ≤1$ and $0<\varepsilon≤1$, which are the mass ratio and inversely proportional to the speed of light, respectively. In the simultaneously nonrelativistic and massless limit regimes, i.e., $γ\sim\varepsilon$ and $\varepsilon \to 0^+$, the KGS equations converge singularly to the Schrödinger-Yukawa (SY) equations. When $0<\varepsilon\ll 1$, due to the perturbation of the wave operator and/or the incompatibility of the initial data, which is described by two parameters $α≥0$ and $β≥-1$, the solution of the KGS equations oscillates in time with $O(\varepsilon)$-wavelength, which requires harsh meshing strategy for classical numerical methods. We propose a uniformly accurate method based on two key points: (ⅰ) reformulating KGS system into an asymptotic consistent formulation, and (ⅱ) applying an integral approximation of the oscillatory term. Using the energy method and the limiting equation via the SY equations with an oscillatory potential, we establish two independent error bounds at $O(h^2+τ^2/\varepsilon)$ and $O(h^2+τ^2+τ\varepsilon^{α^*}+\varepsilon^{1+α^*})$ with $h$ mesh size, $τ$ time step and $α^* = \min\{1, α, 1+β\}$. This implies that the method converges uniformly and optimally with quadratic convergence rate in space and uniformly in time at $O(τ^{4/3})$ and $O(τ^{1+\frac{α^*}{2+α^*}})$ for well-prepared ($α^* = 1$) and ill-prepared ($0≤α^*<1$) initial data, respectively. Thus the $\varepsilon$-scalability of the method is $τ = O(1)$ and $h = O(1)$ for $0<\varepsilon≤ 1$, which is significantly better than classical methods. Numerical results are reported to confirm our error bounds. Finally, the method is applied to study the convergence rates of KGS equations to its limiting models in the simultaneously nonrelativistic and massless limit regimes.

Citation: Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040
References:
[1]

W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp., 82 (2013), 99-128. doi: 10.1090/S0025-5718-2012-02617-2.

[2]

W. BaoX. Dong and S. Wang, Singular limits of Klein-Gordon-Schrödinger equations to Schrödinger-Yukawa equations, Multiscale Model. Simul., 8 (2010), 1742-1769. doi: 10.1137/100790586.

[3]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Zakharov system in the subsnic limit regime via an asymptotic consistent formulation, Multiscale Model. Simul., 15 (2017), 977-1002. doi: 10.1137/16M1078112.

[4]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime, Math. Comp, (2017). doi: 10.1090/mcom/3278.

[5]

W. Bao and L. Yang, Efficient and accurate numerical methods for the Klein-Gordon-Schrödinger equations, J. Comput. Phys., 225 (2007), 1863-1893. doi: 10.1016/j.jcp.2007.02.018.

[6]

W. Bao and X. Zhao, A uniformly accurate (UA) multiscale time integrator Fourier pseoduspectral method for the Klein-Gordon-Schrödinger equations in the nonrelativistic limit regime, Numer. Math., 135 (2017), 833-873. doi: 10.1007/s00211-016-0818-x.

[7]

P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065.

[8]

Y. Cai and Y. Yuan, Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime, Math. Comp., 87 (2018), 1191-1225. doi: 10.1090/mcom/3269.

[9]

A. Darwish and E. G. Fan, A series of new explicit exact solutions for the coupled Klein-Gordon-Schrödinger equations, Chaos Solitons Fractals, 20 (2004), 609-617. doi: 10.1016/S0960-0779(03)00419-3.

[10]

M. Dehghan and A. Taleei, Numerical solution of the Yukawa-coupled Klein-Gordon-Schrödinger equations via a Chebyshev pseudospectral multidomain method, Appl. Math. Model., 36 (2012), 2340-2349. doi: 10.1016/j.apm.2011.08.030.

[11]

J. M. Dixon, J. A. Tuszynski and P. J. Clarkson, From Nonlinearity To Coherence: Universal Features Of Nonlinear Behavior In Many-body Physics, Cambridge University Press, Cambridge, 1997.

[12]

I. Fukuda and M. Tsutsumi, On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions, Proc. Japan Acad., 51 (1975), 402-405. doi: 10.3792/pja/1195518563.

[13]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅱ, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.

[14]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅲ, Math. Japan, 24 (1979), 307-321.

[15]

B. Guo, The global solutions of some problems for a system of equations of Schrödinger-Klein-Gordon field, Sci. China Ser. A, 25 (1982), 897-910.

[16]

B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Differ. Eq., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.

[17]

B. Guo and C. Miao, Global existence and asymptotic behavior of solutions for the coupled Klein-Gordon-Schrödinger equations, Sci. China Ser. A, 38 (1995), 1444-1456.

[18]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, New York, 1995.

[19]

N. Hayashi and W. V. Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.

[20]

S. Herr and K. Schratz, Trigonometric time integrators for the Zakharov system, IMA J. Numer. Anal., 37 (2017), 2042-2066. doi: 10.1093/imanum/drw059.

[21]

F. T. Hioe, Periodic solitary waves for two coupled nonlinear Klein-Gordon and Schrödinger equations, J. Phys. A: Math. Gen., 36 (2003), 7307-7330. doi: 10.1088/0305-4470/36/26/307.

[22]

J. HongS. Jiang and C. Li, Explicit multi-symplectic methods for Klein-Gordon-Schrödinger equations, J. Comput. Phys., 228 (2009), 3517-3532. doi: 10.1016/j.jcp.2009.02.006.

[23]

L. KongR. Liu and Z. Xu, Numerical simulation of interaction between Schrödinger field and Klein-Gordon field by multisymplectic method, Appl. Math. Comput., 181 (2006), 342-350. doi: 10.1016/j.amc.2006.01.044.

[24]

L. KongJ ZhangY. CaoY. Duan and H. Huang, Semi-explicit symplectic partitioned Runge-Kutta Fourier pseudo-spectral scheme for Klein-Gordon-Schrödinger equations, Comput. Phys. Comm., 181 (2010), 1369-1377. doi: 10.1016/j.cpc.2010.04.003.

[25]

Y. Li and B. Guo, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5.

[26]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differ. Eq., 170 (2001), 281-316. doi: 10.1006/jdeq.2000.3827.

[27]

V. G. Makhankov, Dynamics of classical solitons (in nonintegrable systems), Phys. Rep., 35 (1978), 1-128. doi: 10.1016/0370-1573(78)90074-1.

[28]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Proc. Japan Acad. A, 67 (1991), 113-116. doi: 10.3792/pjaa.67.113.

[29]

T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schrödinger equations, Adv. Stud. Pure Math., 23 (1994), 295-305.

[30]

X. Pan and L. Zhang, High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein-Gordon equation, Nonlinear Anal., 92 (2013), 108-118. doi: 10.1016/j.na.2013.07.003.

[31]

V. Petviashvili and O. Pokhotelov, Solitary Waves in Plasmas and in The Atmosphere, Gordon and Breach, Philadelphia, 1992.

[32]

S. H. Schochet and M. I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. doi: 10.1007/BF01463396.

[33]

Y. R. Shen, Principles of Nonlinear Optics, Wiley, New York, 1984.

[34]

C. Su and W. Yi, Error estimates of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime, IMA J. Numer. Anal., (2017). doi: 10.1093/imanum/drx044.

[35]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1997. doi: 10.1007/3-540-33122-0.

[36]

M. Wang and Y. Zhou, The periodic wave solutions for the Klein-Gordon-Schrödinger equations, Phys. Lett. A, 318 (2003), 84-92. doi: 10.1016/j.physleta.2003.07.026.

[37]

S. Wang and L. Zhang, A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schrödinger equations, Appl. Math. Comput., 203 (2008), 799-812. doi: 10.1016/j.amc.2008.05.089.

[38]

T. Wang, Optimal point-wise error estimate of a compact difference scheme for the Klein-Gordon-Schrödinger equation, J. Math. Anal. Appl., 412 (2014), 155-167. doi: 10.1016/j.jmaa.2013.10.038.

[39]

X. Xiang, Spectral method for solving the system of equations of Schrödinger-Klein-Gordon field, J. Comput. Appl. Math., 21 (1988), 161-171. doi: 10.1016/0377-0427(88)90265-8.

[40]

H. Yukawa, On the interaction of elementary particles, Ⅰ, Proc. Phys. Math. Soc. Japan, 17 (1935), 48-57.

[41]

L. Zhang, Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space, Appl. Math. Comput., 163 (2005), 343-355. doi: 10.1016/j.amc.2004.02.010.

show all references

References:
[1]

W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp., 82 (2013), 99-128. doi: 10.1090/S0025-5718-2012-02617-2.

[2]

W. BaoX. Dong and S. Wang, Singular limits of Klein-Gordon-Schrödinger equations to Schrödinger-Yukawa equations, Multiscale Model. Simul., 8 (2010), 1742-1769. doi: 10.1137/100790586.

[3]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Zakharov system in the subsnic limit regime via an asymptotic consistent formulation, Multiscale Model. Simul., 15 (2017), 977-1002. doi: 10.1137/16M1078112.

[4]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime, Math. Comp, (2017). doi: 10.1090/mcom/3278.

[5]

W. Bao and L. Yang, Efficient and accurate numerical methods for the Klein-Gordon-Schrödinger equations, J. Comput. Phys., 225 (2007), 1863-1893. doi: 10.1016/j.jcp.2007.02.018.

[6]

W. Bao and X. Zhao, A uniformly accurate (UA) multiscale time integrator Fourier pseoduspectral method for the Klein-Gordon-Schrödinger equations in the nonrelativistic limit regime, Numer. Math., 135 (2017), 833-873. doi: 10.1007/s00211-016-0818-x.

[7]

P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065.

[8]

Y. Cai and Y. Yuan, Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime, Math. Comp., 87 (2018), 1191-1225. doi: 10.1090/mcom/3269.

[9]

A. Darwish and E. G. Fan, A series of new explicit exact solutions for the coupled Klein-Gordon-Schrödinger equations, Chaos Solitons Fractals, 20 (2004), 609-617. doi: 10.1016/S0960-0779(03)00419-3.

[10]

M. Dehghan and A. Taleei, Numerical solution of the Yukawa-coupled Klein-Gordon-Schrödinger equations via a Chebyshev pseudospectral multidomain method, Appl. Math. Model., 36 (2012), 2340-2349. doi: 10.1016/j.apm.2011.08.030.

[11]

J. M. Dixon, J. A. Tuszynski and P. J. Clarkson, From Nonlinearity To Coherence: Universal Features Of Nonlinear Behavior In Many-body Physics, Cambridge University Press, Cambridge, 1997.

[12]

I. Fukuda and M. Tsutsumi, On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions, Proc. Japan Acad., 51 (1975), 402-405. doi: 10.3792/pja/1195518563.

[13]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅱ, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.

[14]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅲ, Math. Japan, 24 (1979), 307-321.

[15]

B. Guo, The global solutions of some problems for a system of equations of Schrödinger-Klein-Gordon field, Sci. China Ser. A, 25 (1982), 897-910.

[16]

B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Differ. Eq., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.

[17]

B. Guo and C. Miao, Global existence and asymptotic behavior of solutions for the coupled Klein-Gordon-Schrödinger equations, Sci. China Ser. A, 38 (1995), 1444-1456.

[18]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, New York, 1995.

[19]

N. Hayashi and W. V. Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japan, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.

[20]

S. Herr and K. Schratz, Trigonometric time integrators for the Zakharov system, IMA J. Numer. Anal., 37 (2017), 2042-2066. doi: 10.1093/imanum/drw059.

[21]

F. T. Hioe, Periodic solitary waves for two coupled nonlinear Klein-Gordon and Schrödinger equations, J. Phys. A: Math. Gen., 36 (2003), 7307-7330. doi: 10.1088/0305-4470/36/26/307.

[22]

J. HongS. Jiang and C. Li, Explicit multi-symplectic methods for Klein-Gordon-Schrödinger equations, J. Comput. Phys., 228 (2009), 3517-3532. doi: 10.1016/j.jcp.2009.02.006.

[23]

L. KongR. Liu and Z. Xu, Numerical simulation of interaction between Schrödinger field and Klein-Gordon field by multisymplectic method, Appl. Math. Comput., 181 (2006), 342-350. doi: 10.1016/j.amc.2006.01.044.

[24]

L. KongJ ZhangY. CaoY. Duan and H. Huang, Semi-explicit symplectic partitioned Runge-Kutta Fourier pseudo-spectral scheme for Klein-Gordon-Schrödinger equations, Comput. Phys. Comm., 181 (2010), 1369-1377. doi: 10.1016/j.cpc.2010.04.003.

[25]

Y. Li and B. Guo, Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 282 (2003), 256-265. doi: 10.1016/S0022-247X(03)00152-5.

[26]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differ. Eq., 170 (2001), 281-316. doi: 10.1006/jdeq.2000.3827.

[27]

V. G. Makhankov, Dynamics of classical solitons (in nonintegrable systems), Phys. Rep., 35 (1978), 1-128. doi: 10.1016/0370-1573(78)90074-1.

[28]

T. Ozawa and Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Proc. Japan Acad. A, 67 (1991), 113-116. doi: 10.3792/pjaa.67.113.

[29]

T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schrödinger equations, Adv. Stud. Pure Math., 23 (1994), 295-305.

[30]

X. Pan and L. Zhang, High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein-Gordon equation, Nonlinear Anal., 92 (2013), 108-118. doi: 10.1016/j.na.2013.07.003.

[31]

V. Petviashvili and O. Pokhotelov, Solitary Waves in Plasmas and in The Atmosphere, Gordon and Breach, Philadelphia, 1992.

[32]

S. H. Schochet and M. I. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Comm. Math. Phys., 106 (1986), 569-580. doi: 10.1007/BF01463396.

[33]

Y. R. Shen, Principles of Nonlinear Optics, Wiley, New York, 1984.

[34]

C. Su and W. Yi, Error estimates of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime, IMA J. Numer. Anal., (2017). doi: 10.1093/imanum/drx044.

[35]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1997. doi: 10.1007/3-540-33122-0.

[36]

M. Wang and Y. Zhou, The periodic wave solutions for the Klein-Gordon-Schrödinger equations, Phys. Lett. A, 318 (2003), 84-92. doi: 10.1016/j.physleta.2003.07.026.

[37]

S. Wang and L. Zhang, A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schrödinger equations, Appl. Math. Comput., 203 (2008), 799-812. doi: 10.1016/j.amc.2008.05.089.

[38]

T. Wang, Optimal point-wise error estimate of a compact difference scheme for the Klein-Gordon-Schrödinger equation, J. Math. Anal. Appl., 412 (2014), 155-167. doi: 10.1016/j.jmaa.2013.10.038.

[39]

X. Xiang, Spectral method for solving the system of equations of Schrödinger-Klein-Gordon field, J. Comput. Appl. Math., 21 (1988), 161-171. doi: 10.1016/0377-0427(88)90265-8.

[40]

H. Yukawa, On the interaction of elementary particles, Ⅰ, Proc. Phys. Math. Soc. Japan, 17 (1935), 48-57.

[41]

L. Zhang, Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space, Appl. Math. Comput., 163 (2005), 343-355. doi: 10.1016/j.amc.2004.02.010.

Figure 1.1.  Diagram of different limits of the KGS system (1.3).
Figure 2.2.  The temporal oscillation (a) and rapid outspreading wave in space (b) of the KGS system (1.10).
Figure 2.3.  Time evolution of $ e_\chi ^\varepsilon \left( t \right), e_\psi ^\varepsilon \left( t \right) $ and $ e_\infty ^\varepsilon \left( t \right) $ .
Figure 5.4.  Spatial errors for Case Ⅱ (a) and temporal errors of $\psi^\varepsilon$ for Case Ⅰ (b).
Figure 5.5.  Convergence behavior between the KGS equations (1.10) and the SY equations (2.12) for different initial data.
Figure 5.6.  Convergence behavior between the KGS equations (1.10) and the SY-OP (2.24) with ill-prepared initial data, i.e., $\alpha = 0$, $\beta = -1$.
Table 5.1.  Temporal errors of $\phi^\varepsilon$ for Case Ⅰ initial data.
$e_\phi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$
$\varepsilon=1/2$2.15E-25.48E-31.39E-33.49E-48.75E-52.19E-55.48E-61.38E-6
rate-1.971.981.992.002.002.001.99
$\varepsilon=1/2^{2}$4.72E-21.57E-24.19E-31.07E-32.68E-46.72E-51.68E-54.21E-6
rate-1.591.911.971.992.002.002.00
$\varepsilon=1/2^{3}$2.38E-21.36E-24.60E-31.24E-33.15E-47.92E-51.98E-54.96E-6
rate-0.811.561.891.971.992.002.00
$\varepsilon=1/2^{4}$2.19E-28.12E-34.79E-32.16E-36.21E-41.59E-43.99E-51.00E-5
rate-1.430.761.151.801.971.992.00
$\varepsilon=1/2^{5}$2.45E-25.22E-31.83E-31.37E-39.03E-43.11E-48.20E-52.07E-5
rate-2.231.510.420.601.541.921.99
$\varepsilon=1/2^{6}$2.57E-27.93E-31.70E-34.97E-43.25E-43.06E-41.44E-44.16E-5
rate-1.702.221.770.610.091.091.79
$\varepsilon=1/2^{7}$2.61E-26.58E-31.90E-34.20E-41.25E-47.70E-58.50E-55.75E-5
rate-1.991.792.181.750.69-0.140.57
$\varepsilon=1/2^8$2.62E-26.26E-31.75E-33.85E-41.05E-43.12E-51.97E-51.94E-5
rate-2.071.842.191.871.750.670.02
$\varepsilon=1/2^9$2.62E-26.21E-31.54E-35.00E-41.06E-42.63E-57.80E-64.92E-6
rate-2.082.011.622.242.011.750.67
$\varepsilon=1/2^{10}$2.62E-26.19E-31.50E-33.97E-41.17E-42.62E-56.58E-61.95E-6
rate-2.082.041.921.762.162.001.75
$e_\phi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$
$\varepsilon=1/2$2.15E-25.48E-31.39E-33.49E-48.75E-52.19E-55.48E-61.38E-6
rate-1.971.981.992.002.002.001.99
$\varepsilon=1/2^{2}$4.72E-21.57E-24.19E-31.07E-32.68E-46.72E-51.68E-54.21E-6
rate-1.591.911.971.992.002.002.00
$\varepsilon=1/2^{3}$2.38E-21.36E-24.60E-31.24E-33.15E-47.92E-51.98E-54.96E-6
rate-0.811.561.891.971.992.002.00
$\varepsilon=1/2^{4}$2.19E-28.12E-34.79E-32.16E-36.21E-41.59E-43.99E-51.00E-5
rate-1.430.761.151.801.971.992.00
$\varepsilon=1/2^{5}$2.45E-25.22E-31.83E-31.37E-39.03E-43.11E-48.20E-52.07E-5
rate-2.231.510.420.601.541.921.99
$\varepsilon=1/2^{6}$2.57E-27.93E-31.70E-34.97E-43.25E-43.06E-41.44E-44.16E-5
rate-1.702.221.770.610.091.091.79
$\varepsilon=1/2^{7}$2.61E-26.58E-31.90E-34.20E-41.25E-47.70E-58.50E-55.75E-5
rate-1.991.792.181.750.69-0.140.57
$\varepsilon=1/2^8$2.62E-26.26E-31.75E-33.85E-41.05E-43.12E-51.97E-51.94E-5
rate-2.071.842.191.871.750.670.02
$\varepsilon=1/2^9$2.62E-26.21E-31.54E-35.00E-41.06E-42.63E-57.80E-64.92E-6
rate-2.082.011.622.242.011.750.67
$\varepsilon=1/2^{10}$2.62E-26.19E-31.50E-33.97E-41.17E-42.62E-56.58E-61.95E-6
rate-2.082.041.921.762.162.001.75
Table 5.2.  Temporal errors for Case Ⅱ initial data.
$e_\psi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$
$\varepsilon=1/2$1.85E-17.00E-22.19E-25.89E-31.50E-33.76E-49.41E-52.36E-5
rate-1.401.671.901.981.992.002.00
$\varepsilon=1/2^{2}$3.64E-11.99E-16.66E-21.75E-24.40E-31.10E-32.76E-46.90E-5
rate-0.871.581.931.992.002.002.00
$\varepsilon=1/2^{3}$1.31E-15.94E-23.36E-21.62E-24.95E-31.28E-33.23E-48.09E-5
rate-1.140.821.051.711.951.992.00
$\varepsilon=1/2^{4}$1.46E-14.21E-21.12E-22.91E-37.34E-41.84E-44.59E-51.15E-5
rate-1.791.911.951.992.002.002.00
$\varepsilon=1/2^{5}$1.05E-14.15E-21.09E-22.62E-36.38E-41.57E-43.90E-59.75E-6
rate-1.351.932.062.042.022.012.00
$\varepsilon=1/2^{6}$1.00E-13.14E-29.05E-33.02E-36.81E-41.60E-43.86E-59.53E-6
rate-1.671.791.582.152.092.052.02
$\varepsilon=1/2^{7}$1.01E-13.30E-28.75E-32.88E-39.29E-41.93E-44.23E-59.88E-6
rate-1.611.921.611.632.272.192.10
$\varepsilon=1/2^8$1.00E-13.30E-29.80E-32.59E-31.16E-33.30E-46.16E-51.21E-5
rate-1.611.751.921.171.812.422.35
$\varepsilon=1/2^9$1.01E-13.31E-29.84E-33.05E-38.71E-45.22E-41.36E-42.31E-5
rate-1.611.751.691.810.741.942.55
$\varepsilon=1/2^{10}$1.01E-13.34E-29.96E-33.11E-31.08E-33.41E-42.50E-46.16E-5
rate-1.591.751.681.521.670.452.02
$e_\phi^\varepsilon(1)$$\tau_0=0.1$$\tau_0/2$$\tau_0/2^2$$\tau_0/2^3$$\tau_0/2^4$$\tau_0/2^5$$\tau_0/2^6$$\tau_0/2^7$
$\varepsilon=1/2$1.71E-24.30E-31.09E-32.74E-46.88E-51.72E-54.31E-61.08E-6
rate-1.991.981.992.002.002.001.99
$\varepsilon=1/2^{2}$2.76E-29.96E-32.63E-36.69E-41.68E-44.21E-51.05E-52.64E-6
rate-1.471.921.971.992.002.002.00
$\varepsilon=1/2^{3}$9.75E-38.65E-33.62E-31.05E-32.71E-46.83E-51.71E-54.28E-6
rate-0.171.261.791.951.992.002.00
$\varepsilon=1/2^{4}$6.62E-32.61E-32.72E-31.58E-34.52E-41.15E-42.90E-57.25E-6
rate-1.34-0.060.781.811.971.992.00
$\varepsilon=1/2^{5}$3.24E-31.64E-37.12E-46.54E-47.69E-42.66E-46.90E-51.73E-5
rate-0.981.200.12-0.231.531.941.99
$\varepsilon=1/2^{6}$3.47E-31.17E-36.10E-42.23E-41.75E-41.47E-41.38E-43.84E-5
rate-1.570.941.450.350.260.091.84
$\varepsilon=1/2^{7}$3.51E-31.12E-33.07E-42.75E-48.62E-54.14E-54.63E-55.33E-5
rate-1.651.860.161.671.06-0.16-0.20
$\varepsilon=1/2^8$3.53E-31.01E-33.85E-41.19E-41.32E-43.88E-51.21E-51.19E-5
rate-1.801.391.70-0.151.771.690.02
$\varepsilon=1/2^9$3.56E-39.95E-43.38E-41.45E-45.13E-56.48E-51.86E-53.96E-6
rate-1.841.561.221.50-0.341.802.23
$\varepsilon=1/2^{10}$3.57E-31.01E-33.29E-41.34E-45.64E-52.39E-53.22E-59.21E-6
rate-1.821.621.301.241.24-0.431.81
$e_\psi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$
$\varepsilon=1/2$1.85E-17.00E-22.19E-25.89E-31.50E-33.76E-49.41E-52.36E-5
rate-1.401.671.901.981.992.002.00
$\varepsilon=1/2^{2}$3.64E-11.99E-16.66E-21.75E-24.40E-31.10E-32.76E-46.90E-5
rate-0.871.581.931.992.002.002.00
$\varepsilon=1/2^{3}$1.31E-15.94E-23.36E-21.62E-24.95E-31.28E-33.23E-48.09E-5
rate-1.140.821.051.711.951.992.00
$\varepsilon=1/2^{4}$1.46E-14.21E-21.12E-22.91E-37.34E-41.84E-44.59E-51.15E-5
rate-1.791.911.951.992.002.002.00
$\varepsilon=1/2^{5}$1.05E-14.15E-21.09E-22.62E-36.38E-41.57E-43.90E-59.75E-6
rate-1.351.932.062.042.022.012.00
$\varepsilon=1/2^{6}$1.00E-13.14E-29.05E-33.02E-36.81E-41.60E-43.86E-59.53E-6
rate-1.671.791.582.152.092.052.02
$\varepsilon=1/2^{7}$1.01E-13.30E-28.75E-32.88E-39.29E-41.93E-44.23E-59.88E-6
rate-1.611.921.611.632.272.192.10
$\varepsilon=1/2^8$1.00E-13.30E-29.80E-32.59E-31.16E-33.30E-46.16E-51.21E-5
rate-1.611.751.921.171.812.422.35
$\varepsilon=1/2^9$1.01E-13.31E-29.84E-33.05E-38.71E-45.22E-41.36E-42.31E-5
rate-1.611.751.691.810.741.942.55
$\varepsilon=1/2^{10}$1.01E-13.34E-29.96E-33.11E-31.08E-33.41E-42.50E-46.16E-5
rate-1.591.751.681.521.670.452.02
$e_\phi^\varepsilon(1)$$\tau_0=0.1$$\tau_0/2$$\tau_0/2^2$$\tau_0/2^3$$\tau_0/2^4$$\tau_0/2^5$$\tau_0/2^6$$\tau_0/2^7$
$\varepsilon=1/2$1.71E-24.30E-31.09E-32.74E-46.88E-51.72E-54.31E-61.08E-6
rate-1.991.981.992.002.002.001.99
$\varepsilon=1/2^{2}$2.76E-29.96E-32.63E-36.69E-41.68E-44.21E-51.05E-52.64E-6
rate-1.471.921.971.992.002.002.00
$\varepsilon=1/2^{3}$9.75E-38.65E-33.62E-31.05E-32.71E-46.83E-51.71E-54.28E-6
rate-0.171.261.791.951.992.002.00
$\varepsilon=1/2^{4}$6.62E-32.61E-32.72E-31.58E-34.52E-41.15E-42.90E-57.25E-6
rate-1.34-0.060.781.811.971.992.00
$\varepsilon=1/2^{5}$3.24E-31.64E-37.12E-46.54E-47.69E-42.66E-46.90E-51.73E-5
rate-0.981.200.12-0.231.531.941.99
$\varepsilon=1/2^{6}$3.47E-31.17E-36.10E-42.23E-41.75E-41.47E-41.38E-43.84E-5
rate-1.570.941.450.350.260.091.84
$\varepsilon=1/2^{7}$3.51E-31.12E-33.07E-42.75E-48.62E-54.14E-54.63E-55.33E-5
rate-1.651.860.161.671.06-0.16-0.20
$\varepsilon=1/2^8$3.53E-31.01E-33.85E-41.19E-41.32E-43.88E-51.21E-51.19E-5
rate-1.801.391.70-0.151.771.690.02
$\varepsilon=1/2^9$3.56E-39.95E-43.38E-41.45E-45.13E-56.48E-51.86E-53.96E-6
rate-1.841.561.221.50-0.341.802.23
$\varepsilon=1/2^{10}$3.57E-31.01E-33.29E-41.34E-45.64E-52.39E-53.22E-59.21E-6
rate-1.821.621.301.241.24-0.431.81
Table 5.3.  Temporal error analysis at time $t = 1$ in the resonance regions for different $\varepsilon$ and $\tau$.
Case Ⅰ ε0 = 1/2 ε0/22 ε0/24 ε0/26
τ = O(ε3/2) τ0 = 0.1 τ0/23 τ0/26 τ0/29
eφε(1) 2.15E-2 1.24E-3 8.20E-5 5.18E-6
rate in time - 4.12/3 3.92/3 3.98/3
Case Ⅱ ε0 = 1/22 ε0/2 ε0/22 ε0/23 ε0/24 ε0/25
τ = O(ε) τ0 = 0.1/22 τ0/2 τ0/22 τ0/23 τ0/24 τ0/25
eφε(1) 2.63E-3 1.05E-3 4.52E-4 2.66E-4 1.38E-4 5.33E-5
rate in time - 1.32 1.21 0.76 0.95 1.37
Case Ⅰ ε0 = 1/2 ε0/22 ε0/24 ε0/26
τ = O(ε3/2) τ0 = 0.1 τ0/23 τ0/26 τ0/29
eφε(1) 2.15E-2 1.24E-3 8.20E-5 5.18E-6
rate in time - 4.12/3 3.92/3 3.98/3
Case Ⅱ ε0 = 1/22 ε0/2 ε0/22 ε0/23 ε0/24 ε0/25
τ = O(ε) τ0 = 0.1/22 τ0/2 τ0/22 τ0/23 τ0/24 τ0/25
eφε(1) 2.63E-3 1.05E-3 4.52E-4 2.66E-4 1.38E-4 5.33E-5
rate in time - 1.32 1.21 0.76 0.95 1.37
[1]

Boling Guo, Yan Lv, Wei Wang. Schrödinger limit of weakly dissipative stochastic Klein--Gordon--Schrödinger equations and large deviations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2795-2818. doi: 10.3934/dcds.2014.34.2795

[2]

Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239

[3]

Marilena N. Poulou, Nikolaos M. Stavrakakis. Finite dimensionality of a Klein-Gordon-Schrödinger type system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 149-161. doi: 10.3934/dcdss.2009.2.149

[4]

Ji Shu. Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1587-1599. doi: 10.3934/dcdsb.2017077

[5]

Caidi Zhao, Gang Xue, Grzegorz Łukaszewicz. Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-24. doi: 10.3934/dcdsb.2018122

[6]

A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097

[7]

Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55

[8]

Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221

[9]

Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695

[10]

Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413

[11]

Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067

[12]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689

[13]

Katherine A. Kime. Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1601-1621. doi: 10.3934/dcdsb.2018063

[14]

Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233

[15]

Tadahiro Oh. Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1563-1580. doi: 10.3934/cpaa.2015.14.1563

[16]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905

[17]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210

[18]

Peng Gao, Yong Li. Averaging principle for the Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089

[19]

Elena Cordero, Fabio Nicola, Luigi Rodino. Schrödinger equations with rough Hamiltonians. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4805-4821. doi: 10.3934/dcds.2015.35.4805

[20]

David Damanik, Zheng Gan. Spectral properties of limit-periodic Schrödinger operators. Communications on Pure & Applied Analysis, 2011, 10 (3) : 859-871. doi: 10.3934/cpaa.2011.10.859

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (25)
  • HTML views (128)
  • Cited by (0)

Other articles
by authors

[Back to Top]