doi: 10.3934/dcdsb.2020043

On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

1. 

Fakultät für Mathematik, Karlsruhe Institute of Technology, Germany

2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

3. 

Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China

Received  May 2018 Revised  January 2019 Published  February 2020

This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation and long-time behaviour of the remainder.

Citation: Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020043
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

D. D. Ba${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$nov and E. Minchev, Nonexistence of global solutions of the initial-boundary value problem for the nonlinear Klein-Gordon equation, J. Math. Phys., 36 (1995), 756-762.  doi: 10.1063/1.531154.  Google Scholar

[3]

W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 50 (2012), 492-521.  doi: 10.1137/110830800.  Google Scholar

[4]

W. Bao and Y. Cai, Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 52 (2014), 1103-1127.  doi: 10.1137/120866890.  Google Scholar

[5]

W. BaoY. Cai and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon equation in the nonrelativistic limit regime, SIAM J. Numer. Anal., 52 (2014), 2488-2511.  doi: 10.1137/130950665.  Google Scholar

[6]

W. Bao and X. Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.  Google Scholar

[7]

W. BaoX. Dong and X. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study, 47 (2014), 111-150.  doi: 10.4208/jms.v47n2.14.01.  Google Scholar

[8]

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

[9]

W. Bao and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon-Zakharov system in the high-plasma-frequency limit regime, J. Comput. Phys., 327 (2016), 270-293.  doi: 10.1016/j.jcp.2016.09.046.  Google Scholar

[10]

W. Bao and X. Zhao, A uniform second-order in time multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, preprint, 2019. Google Scholar

[11]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ: Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[12]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅱ: Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[13]

S. BaumstarkE. Faou and K. Schratz, Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to the classical NLS splitting, Math. Comp., 87 (2018), 1227-1254.  doi: 10.1090/mcom/3263.  Google Scholar

[14]

M. BennouneM. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803.  doi: 10.1016/j.jcp.2007.11.032.  Google Scholar

[15]

F. CastellaP. ChartierF. Méhats and A. Murua, Stroboscopic averaging for the nonlinear Schrödinger equation, Found. Comput. Math., 15 (2015), 519-559.  doi: 10.1007/s10208-014-9235-7.  Google Scholar

[16] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, London, 1970.   Google Scholar
[17]

P. ChartierA. Murua and J. M. Sanz-Serna, A formal series approach to averaging: Exponentially small error estimates, Discrete Contin. Dyn. Syst., 32 (2012), 3009-3027.  doi: 10.3934/dcds.2012.32.3009.  Google Scholar

[18]

P. ChartierN. CrouseillesM. Lemou and F. Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schrödinger equations, Numer. Math., 129 (2015), 211-250.  doi: 10.1007/s00211-014-0638-9.  Google Scholar

[19]

P. Chartier, M. Lemou, F. Méhats and G. Vilmart, A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations, preprint, arXiv: 1712.06371. Google Scholar

[20]

D. CohenE. Hairer and C. Lubich, Modulated Fourier expansions of highly oscillatory differential equations, Found. Comput. Math., 3 (2003), 327-345.  doi: 10.1007/s10208-002-0062-x.  Google Scholar

[21]

A. CrestettoN. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equations using particles, Kinet. Relat. Models., 5 (2012), 787-816.  doi: 10.3934/krm.2012.5.787.  Google Scholar

[22]

N. CrouseillesM. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations, J. Comput. Phys., 248 (2013), 287-308.  doi: 10.1016/j.jcp.2013.04.022.  Google Scholar

[23]

A. S. Davydov, Quantum Mechanics, International Series in Natural Philosophy, 1, Pergamon Press, Oxford-New York-Toronto, Ont., 1976. doi: 10.1016/C2013-0-05735-0.  Google Scholar

[24]

P. DegondM. Lemou and M. Picasso, Viscoelastic fluid models derived from kinetic equations for polymers, SIAM J. Appl. Math., 62 (2002), 1501-1519.  doi: 10.1137/S0036139900374404.  Google Scholar

[25]

X. DongZ. Xu and X. Zhao, On time-splitting pseudospectral discretization for nonlinear Klein-Gordon equation in nonrelativistic limit regime, Commun. Comput. Phys., 16 (2014), 440-466.  doi: 10.4208/cicp.280813.190214a.  Google Scholar

[26]

R. S. Ellis, Chapman-Enskog-Hilbert expansion for a Markovian model of the Boltzmann equation, Comm. Pure Appl. Math., 26 (1973), 327-359.  doi: 10.1002/cpa.3160260304.  Google Scholar

[27]

E. FaouL. Gauckler and C. Lubich, Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus, Comm. Partial Differential Equations, 38 (2013), 1123-1140.  doi: 10.1080/03605302.2013.785562.  Google Scholar

[28]

E. Faou and K. Schratz, Asympotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime, Numer. Math., 126 (2014), 441-469.  doi: 10.1007/s00211-013-0567-z.  Google Scholar

[29]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.  Google Scholar

[30]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286.  doi: 10.1017/S0962492910000048.  Google Scholar

[31]

K. Huang, C. Xiong and X. Zhao, Scalar-field theory of dark matter, Inter. J. Modern Physics A, 29 (2014). doi: 10.1142/S0217751X14500742.  Google Scholar

[32]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153.  doi: 10.1090/S0025-5718-08-02101-7.  Google Scholar

[33]

S. MachiharaK. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations, Math. Ann., 322 (2002), 603-621.  doi: 10.1007/s002080200008.  Google Scholar

[34]

N. Masmoudi and K. Nakanishi, From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrödinger equations, Math. Ann., 324 (2002), 359-389.  doi: 10.1007/s00208-002-0342-4.  Google Scholar

[35]

A. Murua and J. M. Sanz-Serna, Word series for dynamical systems and their numerical integrators, Found. Comput. Math., 17 (2017), 675-712.  doi: 10.1007/s10208-015-9295-3.  Google Scholar

[36]

B. Najman, The nonrelativistic limit of the nonlinear Klein-Gordon equation, Nonlinear Anal., 15 (1990), 217-228.  doi: 10.1016/0362-546X(90)90158-D.  Google Scholar

[37]

A. Ostermannm and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), 731-755.  doi: 10.1007/s10208-017-9352-1.  Google Scholar

[38]

S. Pasquali, Dynamics of the nonlinear Klein-Gordon equation in the nonrelativistic limit, Ann. Mat. Pura Appl. (4), 198 (2019), 903–972. doi: 10.1007/s10231-018-0805-1.  Google Scholar

[39]

C. Xiong, M. R. R. Good, Y. Guo, X. Liu and K. Huang, Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation, Phys. Rev. D, 90 (2014). doi: 10.1103/PhysRevD.90.125019.  Google Scholar

[40]

X. Zhao, A combination of multiscale time integrator and two-scale formulation for the nonlinear Schrödinger equation with wave operator, J. Comput. Appl. Math., 326 (2017), 320-336.  doi: 10.1016/j.cam.2017.06.006.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

D. D. Ba${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$nov and E. Minchev, Nonexistence of global solutions of the initial-boundary value problem for the nonlinear Klein-Gordon equation, J. Math. Phys., 36 (1995), 756-762.  doi: 10.1063/1.531154.  Google Scholar

[3]

W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 50 (2012), 492-521.  doi: 10.1137/110830800.  Google Scholar

[4]

W. Bao and Y. Cai, Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 52 (2014), 1103-1127.  doi: 10.1137/120866890.  Google Scholar

[5]

W. BaoY. Cai and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon equation in the nonrelativistic limit regime, SIAM J. Numer. Anal., 52 (2014), 2488-2511.  doi: 10.1137/130950665.  Google Scholar

[6]

W. Bao and X. Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.  Google Scholar

[7]

W. BaoX. Dong and X. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study, 47 (2014), 111-150.  doi: 10.4208/jms.v47n2.14.01.  Google Scholar

[8]

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

[9]

W. Bao and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral method for the Klein-Gordon-Zakharov system in the high-plasma-frequency limit regime, J. Comput. Phys., 327 (2016), 270-293.  doi: 10.1016/j.jcp.2016.09.046.  Google Scholar

[10]

W. Bao and X. Zhao, A uniform second-order in time multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, preprint, 2019. Google Scholar

[11]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅰ: Formal derivations, J. Statist. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608.  Google Scholar

[12]

C. BardosF. Golse and D. Levermore, Fluid dynamic limits of kinetic equations Ⅱ: Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math., 46 (1993), 667-753.  doi: 10.1002/cpa.3160460503.  Google Scholar

[13]

S. BaumstarkE. Faou and K. Schratz, Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to the classical NLS splitting, Math. Comp., 87 (2018), 1227-1254.  doi: 10.1090/mcom/3263.  Google Scholar

[14]

M. BennouneM. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys., 227 (2008), 3781-3803.  doi: 10.1016/j.jcp.2007.11.032.  Google Scholar

[15]

F. CastellaP. ChartierF. Méhats and A. Murua, Stroboscopic averaging for the nonlinear Schrödinger equation, Found. Comput. Math., 15 (2015), 519-559.  doi: 10.1007/s10208-014-9235-7.  Google Scholar

[16] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, London, 1970.   Google Scholar
[17]

P. ChartierA. Murua and J. M. Sanz-Serna, A formal series approach to averaging: Exponentially small error estimates, Discrete Contin. Dyn. Syst., 32 (2012), 3009-3027.  doi: 10.3934/dcds.2012.32.3009.  Google Scholar

[18]

P. ChartierN. CrouseillesM. Lemou and F. Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schrödinger equations, Numer. Math., 129 (2015), 211-250.  doi: 10.1007/s00211-014-0638-9.  Google Scholar

[19]

P. Chartier, M. Lemou, F. Méhats and G. Vilmart, A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations, preprint, arXiv: 1712.06371. Google Scholar

[20]

D. CohenE. Hairer and C. Lubich, Modulated Fourier expansions of highly oscillatory differential equations, Found. Comput. Math., 3 (2003), 327-345.  doi: 10.1007/s10208-002-0062-x.  Google Scholar

[21]

A. CrestettoN. Crouseilles and M. Lemou, Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equations using particles, Kinet. Relat. Models., 5 (2012), 787-816.  doi: 10.3934/krm.2012.5.787.  Google Scholar

[22]

N. CrouseillesM. Lemou and F. Méhats, Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations, J. Comput. Phys., 248 (2013), 287-308.  doi: 10.1016/j.jcp.2013.04.022.  Google Scholar

[23]

A. S. Davydov, Quantum Mechanics, International Series in Natural Philosophy, 1, Pergamon Press, Oxford-New York-Toronto, Ont., 1976. doi: 10.1016/C2013-0-05735-0.  Google Scholar

[24]

P. DegondM. Lemou and M. Picasso, Viscoelastic fluid models derived from kinetic equations for polymers, SIAM J. Appl. Math., 62 (2002), 1501-1519.  doi: 10.1137/S0036139900374404.  Google Scholar

[25]

X. DongZ. Xu and X. Zhao, On time-splitting pseudospectral discretization for nonlinear Klein-Gordon equation in nonrelativistic limit regime, Commun. Comput. Phys., 16 (2014), 440-466.  doi: 10.4208/cicp.280813.190214a.  Google Scholar

[26]

R. S. Ellis, Chapman-Enskog-Hilbert expansion for a Markovian model of the Boltzmann equation, Comm. Pure Appl. Math., 26 (1973), 327-359.  doi: 10.1002/cpa.3160260304.  Google Scholar

[27]

E. FaouL. Gauckler and C. Lubich, Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus, Comm. Partial Differential Equations, 38 (2013), 1123-1140.  doi: 10.1080/03605302.2013.785562.  Google Scholar

[28]

E. Faou and K. Schratz, Asympotic preserving schemes for the Klein-Gordon equation in the non-relativistic limit regime, Numer. Math., 126 (2014), 441-469.  doi: 10.1007/s00211-013-0567-z.  Google Scholar

[29]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-30666-8.  Google Scholar

[30]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286.  doi: 10.1017/S0962492910000048.  Google Scholar

[31]

K. Huang, C. Xiong and X. Zhao, Scalar-field theory of dark matter, Inter. J. Modern Physics A, 29 (2014). doi: 10.1142/S0217751X14500742.  Google Scholar

[32]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153.  doi: 10.1090/S0025-5718-08-02101-7.  Google Scholar

[33]

S. MachiharaK. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations, Math. Ann., 322 (2002), 603-621.  doi: 10.1007/s002080200008.  Google Scholar

[34]

N. Masmoudi and K. Nakanishi, From nonlinear Klein-Gordon equation to a system of coupled nonlinear Schrödinger equations, Math. Ann., 324 (2002), 359-389.  doi: 10.1007/s00208-002-0342-4.  Google Scholar

[35]

A. Murua and J. M. Sanz-Serna, Word series for dynamical systems and their numerical integrators, Found. Comput. Math., 17 (2017), 675-712.  doi: 10.1007/s10208-015-9295-3.  Google Scholar

[36]

B. Najman, The nonrelativistic limit of the nonlinear Klein-Gordon equation, Nonlinear Anal., 15 (1990), 217-228.  doi: 10.1016/0362-546X(90)90158-D.  Google Scholar

[37]

A. Ostermannm and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), 731-755.  doi: 10.1007/s10208-017-9352-1.  Google Scholar

[38]

S. Pasquali, Dynamics of the nonlinear Klein-Gordon equation in the nonrelativistic limit, Ann. Mat. Pura Appl. (4), 198 (2019), 903–972. doi: 10.1007/s10231-018-0805-1.  Google Scholar

[39]

C. Xiong, M. R. R. Good, Y. Guo, X. Liu and K. Huang, Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation, Phys. Rev. D, 90 (2014). doi: 10.1103/PhysRevD.90.125019.  Google Scholar

[40]

X. Zhao, A combination of multiscale time integrator and two-scale formulation for the nonlinear Schrödinger equation with wave operator, J. Comput. Appl. Math., 326 (2017), 320-336.  doi: 10.1016/j.cam.2017.06.006.  Google Scholar

Figure 1.  Behavior of the remainder $ \|R\|_{H^1}/ \varepsilon^2 $ of the two-term expansion and the one-term expansion with respect to time
Figure 2.  Behavior of the remainder terms $ \|R_{MFo}\|_{H^1}/ \varepsilon^2 $, $ \|R_{MFe}\|_{H^1}/ \varepsilon^2 $ and $ \|R_{CE}\|_{H^1}/ \varepsilon^2 $ with respect to time under two cases: initial smooth localized wave (2.18) in $ {\mathbb R} $ (left) and smooth planewave (2.19) on torus $ {\mathbb T} $ (right)
Figure 3.  Profiles of the expansions $ u_{MFo}(x, 3) $, $ u_{MFe}(x, 3) $, $ u_{CE}(x, 3) $ and exact solution $ u(x, 3) $ at $ t = 3 $ under $ \varepsilon = 0.1 $: the whole space example (2.18) (up) and the torus example (2.19) (down)
Figure 4.  Energy error $ |\mathcal{H}(t)-H(0)|/|H(0)| $ of the three expansions to the leading order under whole space case (up) and torus case (down)
Figure 5.  Behavior of the MFo remainder under lower regularity initial data (2.20): $ \|R_{MFo}\|_{H^1}/ \varepsilon^2 $ (left) under $ H^5 $-data, $ \|R_{MFo}\|_{H^1}/ \varepsilon^2 $ (middle) and $ \|R_{MFo}\|_{L^2}/ \varepsilon^2 $ (right) under $ H^4 $-data
Figure 6.  Behavior of the MFe remainder under lower regularity initial data (2.20): $ \|R_{MFe}\|_{H^1}/ \varepsilon^2 $ (left) under $ H^3 $-data, $ \|R_{MFe}\|_{H^1}/ \varepsilon^2 $ (middle) and $ \|R_{MFe}\|_{L^2}/ \varepsilon^2 $ (right) under $ H^2 $-data
Figure 7.  Behavior of the CE remainder under lower regularity initial data (2.20): $ \|R_{CE}\|_{H^1}/ \varepsilon^2 $ (left) under $ H^3 $-data, $ \|R_{CE}\|_{H^1}/ \varepsilon^2 $ (middle) and $ \|R_{CE}\|_{L^2}/ \varepsilon^2 $ (right) under $ H^2 $-data
Figure 8.  Behavior of the remainder terms $ \|R_{MFo}\|_{H^1}/ \varepsilon^4 $, $ \|R_{MFe}\|_{H^1}/ \varepsilon^4 $ and $ \|R_{CE}\|_{H^1}/ \varepsilon^4 $ with respect to time under two cases: initial smooth localized wave (2.18) in $ {\mathbb R} $ (left) and smooth planewave (2.19) on torus $ {\mathbb T} $ (right)
Figure 9.  Energy error $ |\mathcal{H}(t)-H(0)|/|H(0)| $ of the three expansions to the next order under whole space case (up) and torus case (down)
Figure 10.  Behavior of the remainders under lower regularity initial data (2.20)
Table 1.  Remainders (in $ H^1 $-norm) of the classical two-term expansion and the one-term with perturbation
$t=0.5$ $\varepsilon =0.05$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
two-term 9.00E-3 2.40E-3 5.98E-4 1.51E-4
one-term 9.60E-3 2.50E-3 6.45E-4 1.64E-4
$t=1$ $\varepsilon =0.05$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
two-term 1.79E-2 4.80E-3 1.20E-3 3.05E-4
one-term 1.84E-2 4.90E-3 1.30E-3 3.15E-4
$t=0.5$ $\varepsilon =0.05$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
two-term 9.00E-3 2.40E-3 5.98E-4 1.51E-4
one-term 9.60E-3 2.50E-3 6.45E-4 1.64E-4
$t=1$ $\varepsilon =0.05$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
two-term 1.79E-2 4.80E-3 1.20E-3 3.05E-4
one-term 1.84E-2 4.90E-3 1.30E-3 3.15E-4
Table 2.  Remainders (in $ H^1 $-norm) of the leading order modulated Fourier (MFo) expansion, the multiscale expansion by frequency (MFe) and the Chapman-Enskog (CE) expansion
$t=0.5$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 3.46E-2 9.90E-3 2.60E-3 6.50E-4
MFe 1.00E-2 2.10E-3 5.35E-4 1.31E-4
CE 1.20E-3 2.58E-4 6.59E-5 1.79E-5
$t=1$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 6.15E-2 1.88E-2 5.00E-3 1.30E-3
MFe 9.70E-3 2.40E-3 5.94E-4 1.51E-4
CE 9.29E-4 2.87E-4 7.23E-5 1.74E-5
$t=0.5$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 3.46E-2 9.90E-3 2.60E-3 6.50E-4
MFe 1.00E-2 2.10E-3 5.35E-4 1.31E-4
CE 1.20E-3 2.58E-4 6.59E-5 1.79E-5
$t=1$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 6.15E-2 1.88E-2 5.00E-3 1.30E-3
MFe 9.70E-3 2.40E-3 5.94E-4 1.51E-4
CE 9.29E-4 2.87E-4 7.23E-5 1.74E-5
Table 3.  Time derivative of the remainders: $ \|\partial_tR(\cdot, t)\|_{H^1} $ of the leading order MFo, MFe and CE
$t=0.5$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 4.65 4.72 4.71 4.72
MFe 8.65E-1 1.09 1.10 1.12
CE 7.86E-2 6.68E-2 6.70E-2 6.94E-2
$t=0.5$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 4.65 4.72 4.71 4.72
MFe 8.65E-1 1.09 1.10 1.12
CE 7.86E-2 6.68E-2 6.70E-2 6.94E-2
Table 4.  Remainders (in $ H^1 $-norm) of the next order modulated Fourier (MFo) expansion, the multiscale frequency (MFe) expansion and the Chapman-Enskog (CE) expansion
$t=0.5$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 2.30E-2 2.30E-3 1.82E-4 1.20E-5
MFe 4.99E-6 1.61E-7 1.06E-8 6.69E-10
CE 3.64E-5 1.87E-6 1.20E-7 7.29E-9
$t=1$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 5.68E-2 7.60E-3 6.84E-4 4.61E-5
MFe 3.03E-6 2.42E-7 1.56E-8 1.23E-9
CE 2.20E-5 1.58E-6 1.03E-7 6.48E-9
$t=0.5$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 2.30E-2 2.30E-3 1.82E-4 1.20E-5
MFe 4.99E-6 1.61E-7 1.06E-8 6.69E-10
CE 3.64E-5 1.87E-6 1.20E-7 7.29E-9
$t=1$ $\varepsilon =0.1$ $\varepsilon /2$ $\varepsilon /2^2$ $\varepsilon /2^3$
MFo 5.68E-2 7.60E-3 6.84E-4 4.61E-5
MFe 3.03E-6 2.42E-7 1.56E-8 1.23E-9
CE 2.20E-5 1.58E-6 1.03E-7 6.48E-9
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