# American Institute of Mathematical Sciences

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. Google Scholar [2] W. Bao, X. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [14] I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅲ, Math. Japan, 24 (1979), 307-321. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, New York, 1995.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] J. Hong, S. 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. Google Scholar [23] L. Kong, R. 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. Google Scholar [24] L. Kong, J Zhang, Y. Cao, Y. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] V. Petviashvili and O. Pokhotelov, Solitary Waves in Plasmas and in The Atmosphere, Gordon and Breach, Philadelphia, 1992. Google Scholar [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. Google Scholar [33] Y. R. Shen, Principles of Nonlinear Optics, Wiley, New York, 1984.Google Scholar [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. Google Scholar [35] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1997. doi: 10.1007/3-540-33122-0. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [40] H. Yukawa, On the interaction of elementary particles, Ⅰ, Proc. Phys. Math. Soc. Japan, 17 (1935), 48-57. Google Scholar [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. Google Scholar

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. Google Scholar [2] W. Bao, X. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [14] I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations Ⅲ, Math. Japan, 24 (1979), 307-321. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, New York, 1995.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] J. Hong, S. 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. Google Scholar [23] L. Kong, R. 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. Google Scholar [24] L. Kong, J Zhang, Y. Cao, Y. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] V. Petviashvili and O. Pokhotelov, Solitary Waves in Plasmas and in The Atmosphere, Gordon and Breach, Philadelphia, 1992. Google Scholar [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. Google Scholar [33] Y. R. Shen, Principles of Nonlinear Optics, Wiley, New York, 1984.Google Scholar [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. Google Scholar [35] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1997. doi: 10.1007/3-540-33122-0. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [40] H. Yukawa, On the interaction of elementary particles, Ⅰ, Proc. Phys. Math. Soc. Japan, 17 (1935), 48-57. Google Scholar [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. Google Scholar
Diagram of different limits of the KGS system (1.3).
The temporal oscillation (a) and rapid outspreading wave in space (b) of the KGS system (1.10).
Time evolution of $e_\chi ^\varepsilon \left( t \right), e_\psi ^\varepsilon \left( t \right)$ and $e_\infty ^\varepsilon \left( t \right)$ .
Spatial errors for Case Ⅱ (a) and temporal errors of $\psi^\varepsilon$ for Case Ⅰ (b).
Convergence behavior between the KGS equations (1.10) and the SY equations (2.12) for different initial data.
Convergence behavior between the KGS equations (1.10) and the SY-OP (2.24) with ill-prepared initial data, i.e., $\alpha = 0$, $\beta = -1$.
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-2 5.48E-3 1.39E-3 3.49E-4 8.75E-5 2.19E-5 5.48E-6 1.38E-6 rate - 1.97 1.98 1.99 2.00 2.00 2.00 1.99 $\varepsilon=1/2^{2}$ 4.72E-2 1.57E-2 4.19E-3 1.07E-3 2.68E-4 6.72E-5 1.68E-5 4.21E-6 rate - 1.59 1.91 1.97 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{3}$ 2.38E-2 1.36E-2 4.60E-3 1.24E-3 3.15E-4 7.92E-5 1.98E-5 4.96E-6 rate - 0.81 1.56 1.89 1.97 1.99 2.00 2.00 $\varepsilon=1/2^{4}$ 2.19E-2 8.12E-3 4.79E-3 2.16E-3 6.21E-4 1.59E-4 3.99E-5 1.00E-5 rate - 1.43 0.76 1.15 1.80 1.97 1.99 2.00 $\varepsilon=1/2^{5}$ 2.45E-2 5.22E-3 1.83E-3 1.37E-3 9.03E-4 3.11E-4 8.20E-5 2.07E-5 rate - 2.23 1.51 0.42 0.60 1.54 1.92 1.99 $\varepsilon=1/2^{6}$ 2.57E-2 7.93E-3 1.70E-3 4.97E-4 3.25E-4 3.06E-4 1.44E-4 4.16E-5 rate - 1.70 2.22 1.77 0.61 0.09 1.09 1.79 $\varepsilon=1/2^{7}$ 2.61E-2 6.58E-3 1.90E-3 4.20E-4 1.25E-4 7.70E-5 8.50E-5 5.75E-5 rate - 1.99 1.79 2.18 1.75 0.69 -0.14 0.57 $\varepsilon=1/2^8$ 2.62E-2 6.26E-3 1.75E-3 3.85E-4 1.05E-4 3.12E-5 1.97E-5 1.94E-5 rate - 2.07 1.84 2.19 1.87 1.75 0.67 0.02 $\varepsilon=1/2^9$ 2.62E-2 6.21E-3 1.54E-3 5.00E-4 1.06E-4 2.63E-5 7.80E-6 4.92E-6 rate - 2.08 2.01 1.62 2.24 2.01 1.75 0.67 $\varepsilon=1/2^{10}$ 2.62E-2 6.19E-3 1.50E-3 3.97E-4 1.17E-4 2.62E-5 6.58E-6 1.95E-6 rate - 2.08 2.04 1.92 1.76 2.16 2.00 1.75
 $e_\phi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$ $\varepsilon=1/2$ 2.15E-2 5.48E-3 1.39E-3 3.49E-4 8.75E-5 2.19E-5 5.48E-6 1.38E-6 rate - 1.97 1.98 1.99 2.00 2.00 2.00 1.99 $\varepsilon=1/2^{2}$ 4.72E-2 1.57E-2 4.19E-3 1.07E-3 2.68E-4 6.72E-5 1.68E-5 4.21E-6 rate - 1.59 1.91 1.97 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{3}$ 2.38E-2 1.36E-2 4.60E-3 1.24E-3 3.15E-4 7.92E-5 1.98E-5 4.96E-6 rate - 0.81 1.56 1.89 1.97 1.99 2.00 2.00 $\varepsilon=1/2^{4}$ 2.19E-2 8.12E-3 4.79E-3 2.16E-3 6.21E-4 1.59E-4 3.99E-5 1.00E-5 rate - 1.43 0.76 1.15 1.80 1.97 1.99 2.00 $\varepsilon=1/2^{5}$ 2.45E-2 5.22E-3 1.83E-3 1.37E-3 9.03E-4 3.11E-4 8.20E-5 2.07E-5 rate - 2.23 1.51 0.42 0.60 1.54 1.92 1.99 $\varepsilon=1/2^{6}$ 2.57E-2 7.93E-3 1.70E-3 4.97E-4 3.25E-4 3.06E-4 1.44E-4 4.16E-5 rate - 1.70 2.22 1.77 0.61 0.09 1.09 1.79 $\varepsilon=1/2^{7}$ 2.61E-2 6.58E-3 1.90E-3 4.20E-4 1.25E-4 7.70E-5 8.50E-5 5.75E-5 rate - 1.99 1.79 2.18 1.75 0.69 -0.14 0.57 $\varepsilon=1/2^8$ 2.62E-2 6.26E-3 1.75E-3 3.85E-4 1.05E-4 3.12E-5 1.97E-5 1.94E-5 rate - 2.07 1.84 2.19 1.87 1.75 0.67 0.02 $\varepsilon=1/2^9$ 2.62E-2 6.21E-3 1.54E-3 5.00E-4 1.06E-4 2.63E-5 7.80E-6 4.92E-6 rate - 2.08 2.01 1.62 2.24 2.01 1.75 0.67 $\varepsilon=1/2^{10}$ 2.62E-2 6.19E-3 1.50E-3 3.97E-4 1.17E-4 2.62E-5 6.58E-6 1.95E-6 rate - 2.08 2.04 1.92 1.76 2.16 2.00 1.75
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-1 7.00E-2 2.19E-2 5.89E-3 1.50E-3 3.76E-4 9.41E-5 2.36E-5 rate - 1.40 1.67 1.90 1.98 1.99 2.00 2.00 $\varepsilon=1/2^{2}$ 3.64E-1 1.99E-1 6.66E-2 1.75E-2 4.40E-3 1.10E-3 2.76E-4 6.90E-5 rate - 0.87 1.58 1.93 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{3}$ 1.31E-1 5.94E-2 3.36E-2 1.62E-2 4.95E-3 1.28E-3 3.23E-4 8.09E-5 rate - 1.14 0.82 1.05 1.71 1.95 1.99 2.00 $\varepsilon=1/2^{4}$ 1.46E-1 4.21E-2 1.12E-2 2.91E-3 7.34E-4 1.84E-4 4.59E-5 1.15E-5 rate - 1.79 1.91 1.95 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{5}$ 1.05E-1 4.15E-2 1.09E-2 2.62E-3 6.38E-4 1.57E-4 3.90E-5 9.75E-6 rate - 1.35 1.93 2.06 2.04 2.02 2.01 2.00 $\varepsilon=1/2^{6}$ 1.00E-1 3.14E-2 9.05E-3 3.02E-3 6.81E-4 1.60E-4 3.86E-5 9.53E-6 rate - 1.67 1.79 1.58 2.15 2.09 2.05 2.02 $\varepsilon=1/2^{7}$ 1.01E-1 3.30E-2 8.75E-3 2.88E-3 9.29E-4 1.93E-4 4.23E-5 9.88E-6 rate - 1.61 1.92 1.61 1.63 2.27 2.19 2.10 $\varepsilon=1/2^8$ 1.00E-1 3.30E-2 9.80E-3 2.59E-3 1.16E-3 3.30E-4 6.16E-5 1.21E-5 rate - 1.61 1.75 1.92 1.17 1.81 2.42 2.35 $\varepsilon=1/2^9$ 1.01E-1 3.31E-2 9.84E-3 3.05E-3 8.71E-4 5.22E-4 1.36E-4 2.31E-5 rate - 1.61 1.75 1.69 1.81 0.74 1.94 2.55 $\varepsilon=1/2^{10}$ 1.01E-1 3.34E-2 9.96E-3 3.11E-3 1.08E-3 3.41E-4 2.50E-4 6.16E-5 rate - 1.59 1.75 1.68 1.52 1.67 0.45 2.02 $e_\phi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$ $\varepsilon=1/2$ 1.71E-2 4.30E-3 1.09E-3 2.74E-4 6.88E-5 1.72E-5 4.31E-6 1.08E-6 rate - 1.99 1.98 1.99 2.00 2.00 2.00 1.99 $\varepsilon=1/2^{2}$ 2.76E-2 9.96E-3 2.63E-3 6.69E-4 1.68E-4 4.21E-5 1.05E-5 2.64E-6 rate - 1.47 1.92 1.97 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{3}$ 9.75E-3 8.65E-3 3.62E-3 1.05E-3 2.71E-4 6.83E-5 1.71E-5 4.28E-6 rate - 0.17 1.26 1.79 1.95 1.99 2.00 2.00 $\varepsilon=1/2^{4}$ 6.62E-3 2.61E-3 2.72E-3 1.58E-3 4.52E-4 1.15E-4 2.90E-5 7.25E-6 rate - 1.34 -0.06 0.78 1.81 1.97 1.99 2.00 $\varepsilon=1/2^{5}$ 3.24E-3 1.64E-3 7.12E-4 6.54E-4 7.69E-4 2.66E-4 6.90E-5 1.73E-5 rate - 0.98 1.20 0.12 -0.23 1.53 1.94 1.99 $\varepsilon=1/2^{6}$ 3.47E-3 1.17E-3 6.10E-4 2.23E-4 1.75E-4 1.47E-4 1.38E-4 3.84E-5 rate - 1.57 0.94 1.45 0.35 0.26 0.09 1.84 $\varepsilon=1/2^{7}$ 3.51E-3 1.12E-3 3.07E-4 2.75E-4 8.62E-5 4.14E-5 4.63E-5 5.33E-5 rate - 1.65 1.86 0.16 1.67 1.06 -0.16 -0.20 $\varepsilon=1/2^8$ 3.53E-3 1.01E-3 3.85E-4 1.19E-4 1.32E-4 3.88E-5 1.21E-5 1.19E-5 rate - 1.80 1.39 1.70 -0.15 1.77 1.69 0.02 $\varepsilon=1/2^9$ 3.56E-3 9.95E-4 3.38E-4 1.45E-4 5.13E-5 6.48E-5 1.86E-5 3.96E-6 rate - 1.84 1.56 1.22 1.50 -0.34 1.80 2.23 $\varepsilon=1/2^{10}$ 3.57E-3 1.01E-3 3.29E-4 1.34E-4 5.64E-5 2.39E-5 3.22E-5 9.21E-6 rate - 1.82 1.62 1.30 1.24 1.24 -0.43 1.81
 $e_\psi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$ $\varepsilon=1/2$ 1.85E-1 7.00E-2 2.19E-2 5.89E-3 1.50E-3 3.76E-4 9.41E-5 2.36E-5 rate - 1.40 1.67 1.90 1.98 1.99 2.00 2.00 $\varepsilon=1/2^{2}$ 3.64E-1 1.99E-1 6.66E-2 1.75E-2 4.40E-3 1.10E-3 2.76E-4 6.90E-5 rate - 0.87 1.58 1.93 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{3}$ 1.31E-1 5.94E-2 3.36E-2 1.62E-2 4.95E-3 1.28E-3 3.23E-4 8.09E-5 rate - 1.14 0.82 1.05 1.71 1.95 1.99 2.00 $\varepsilon=1/2^{4}$ 1.46E-1 4.21E-2 1.12E-2 2.91E-3 7.34E-4 1.84E-4 4.59E-5 1.15E-5 rate - 1.79 1.91 1.95 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{5}$ 1.05E-1 4.15E-2 1.09E-2 2.62E-3 6.38E-4 1.57E-4 3.90E-5 9.75E-6 rate - 1.35 1.93 2.06 2.04 2.02 2.01 2.00 $\varepsilon=1/2^{6}$ 1.00E-1 3.14E-2 9.05E-3 3.02E-3 6.81E-4 1.60E-4 3.86E-5 9.53E-6 rate - 1.67 1.79 1.58 2.15 2.09 2.05 2.02 $\varepsilon=1/2^{7}$ 1.01E-1 3.30E-2 8.75E-3 2.88E-3 9.29E-4 1.93E-4 4.23E-5 9.88E-6 rate - 1.61 1.92 1.61 1.63 2.27 2.19 2.10 $\varepsilon=1/2^8$ 1.00E-1 3.30E-2 9.80E-3 2.59E-3 1.16E-3 3.30E-4 6.16E-5 1.21E-5 rate - 1.61 1.75 1.92 1.17 1.81 2.42 2.35 $\varepsilon=1/2^9$ 1.01E-1 3.31E-2 9.84E-3 3.05E-3 8.71E-4 5.22E-4 1.36E-4 2.31E-5 rate - 1.61 1.75 1.69 1.81 0.74 1.94 2.55 $\varepsilon=1/2^{10}$ 1.01E-1 3.34E-2 9.96E-3 3.11E-3 1.08E-3 3.41E-4 2.50E-4 6.16E-5 rate - 1.59 1.75 1.68 1.52 1.67 0.45 2.02 $e_\phi^\varepsilon(1)$ $\tau_0=0.1$ $\tau_0/2$ $\tau_0/2^2$ $\tau_0/2^3$ $\tau_0/2^4$ $\tau_0/2^5$ $\tau_0/2^6$ $\tau_0/2^7$ $\varepsilon=1/2$ 1.71E-2 4.30E-3 1.09E-3 2.74E-4 6.88E-5 1.72E-5 4.31E-6 1.08E-6 rate - 1.99 1.98 1.99 2.00 2.00 2.00 1.99 $\varepsilon=1/2^{2}$ 2.76E-2 9.96E-3 2.63E-3 6.69E-4 1.68E-4 4.21E-5 1.05E-5 2.64E-6 rate - 1.47 1.92 1.97 1.99 2.00 2.00 2.00 $\varepsilon=1/2^{3}$ 9.75E-3 8.65E-3 3.62E-3 1.05E-3 2.71E-4 6.83E-5 1.71E-5 4.28E-6 rate - 0.17 1.26 1.79 1.95 1.99 2.00 2.00 $\varepsilon=1/2^{4}$ 6.62E-3 2.61E-3 2.72E-3 1.58E-3 4.52E-4 1.15E-4 2.90E-5 7.25E-6 rate - 1.34 -0.06 0.78 1.81 1.97 1.99 2.00 $\varepsilon=1/2^{5}$ 3.24E-3 1.64E-3 7.12E-4 6.54E-4 7.69E-4 2.66E-4 6.90E-5 1.73E-5 rate - 0.98 1.20 0.12 -0.23 1.53 1.94 1.99 $\varepsilon=1/2^{6}$ 3.47E-3 1.17E-3 6.10E-4 2.23E-4 1.75E-4 1.47E-4 1.38E-4 3.84E-5 rate - 1.57 0.94 1.45 0.35 0.26 0.09 1.84 $\varepsilon=1/2^{7}$ 3.51E-3 1.12E-3 3.07E-4 2.75E-4 8.62E-5 4.14E-5 4.63E-5 5.33E-5 rate - 1.65 1.86 0.16 1.67 1.06 -0.16 -0.20 $\varepsilon=1/2^8$ 3.53E-3 1.01E-3 3.85E-4 1.19E-4 1.32E-4 3.88E-5 1.21E-5 1.19E-5 rate - 1.80 1.39 1.70 -0.15 1.77 1.69 0.02 $\varepsilon=1/2^9$ 3.56E-3 9.95E-4 3.38E-4 1.45E-4 5.13E-5 6.48E-5 1.86E-5 3.96E-6 rate - 1.84 1.56 1.22 1.50 -0.34 1.80 2.23 $\varepsilon=1/2^{10}$ 3.57E-3 1.01E-3 3.29E-4 1.34E-4 5.64E-5 2.39E-5 3.22E-5 9.21E-6 rate - 1.82 1.62 1.30 1.24 1.24 -0.43 1.81
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
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