# American Institute of Mathematical Sciences

August  2017, 37(7): 3721-3747. doi: 10.3934/dcds.2017158

## Metastable energy strata in numerical discretizations of weakly nonlinear wave equations

 1 Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany, Present address: Institut für Mathematik, FU Berlin, Arnimallee 9, D-14195 Berlin, Germany 2 Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie (KIT), Englerstr. 2, D-76131 Karlsruhe, Germany

* Corresponding author

Received  June 2016 Revised  February 2017 Published  April 2017

Fund Project: We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173, CRC 1114 and project GA 2073/2-1

The quadratic nonlinear wave equation on a one-dimensional torus with small initial values located in a single Fourier mode is considered. In this situation, the formation of metastable energy strata has recently been described and their long-time stability has been shown. The topic of the present paper is the correct reproduction of these metastable energy strata by a numerical method. For symplectic trigonometric integrators applied to the equation, it is shown that these energy strata are reproduced even on long time intervals in a qualitatively correct way.

Citation: Ludwig Gauckler, Daniel Weiss. Metastable energy strata in numerical discretizations of weakly nonlinear wave equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3721-3747. doi: 10.3934/dcds.2017158
##### References:
 [1] 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 [2] B. Cano, Conservation of invariants by symmetric multistep cosine methods for second-order partial differential equations, BIT, 53 (2013), 29-56. doi: 10.1007/s10543-012-0393-1. Google Scholar [3] B. Cano and A. González-Pachón, Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation, J. Comp. Math., 34 (2016), 385-406. doi: 10.4208/jcm.1601-m4541. Google Scholar [4] B. Cano, Conserved quantities of some Hamiltonian wave equations after full discretization, Numer. Math., 103 (2006), 197-223. doi: 10.1007/s00211-006-0680-3. Google Scholar [5] D. Cohen, L. Gauckler, E. Hairer and C. Lubich, Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions, BIT, 55 (2015), 705-732. doi: 10.1007/s10543-014-0527-8. Google Scholar [6] D. Cohen, E. Hairer and C. Lubich, Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations, Numer. Math., 110 (2008), 113-143. doi: 10.1007/s00211-008-0163-9. Google Scholar [7] M. Dahlby and B. Owren, Plane wave stability of some conservative schemes for the cubic Schrödinger equation, M2AN Math. Model. Numer. Anal., 43 (2009), 677-687. doi: 10.1051/m2an/2009022. Google Scholar [8] P. Deuflhard, A study of extrapolation methods based on multistep schemes without parasitic solutions, Z. Angew. Math. Phys., 30 (1979), 177-189. doi: 10.1007/BF01601932. Google Scholar [9] X. Dong, Stability and convergence of trigonometric integrator pseudospectral discretization for N-coupled nonlinear Klein-Gordon equations, Appl. Math. Comput., 232 (2014), 752-765. doi: 10.1016/j.amc.2014.01.144. Google Scholar [10] E. Faou, L. Gauckler and C. Lubich, Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation, Forum Math. Sigma, 2 (2014), e5, 45pp. doi: 10.1017/fms.2014.4. Google Scholar [11] E. Faou and B. Grébert, Hamiltonian interpolation of splitting approximations for nonlinear PDEs, Found. Comput. Math., 11 (2011), 381-415. doi: 10.1007/s10208-011-9094-4. Google Scholar [12] E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization, Numer. Math., 114 (2010), 429-458. doi: 10.1007/s00211-009-0258-y. Google Scholar [13] B. Garcıa-Archilla, J. M. Sanz-Serna and R. D. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput., 20 (1999), 930-963. doi: 10.1137/S1064827596313851. Google Scholar [14] L. Gauckler, Error analysis of trigonometric integrators for semilinear wave equations, SIAM J. Numer. Anal., 53 (2015), 1082-1106. doi: 10.1137/140977217. Google Scholar [15] L. Gauckler, E. Hairer, C. Lubich and D. Weiss, Metastable energy strata in weakly nonlinear wave equations, Comm. Partial Differential Equations, 37 (2012), 1391-1413. doi: 10.1080/03605302.2012.683503. Google Scholar [16] H. Grubmüller, H. Heller, A. Windemuth and K. Schulten, Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Mol. Sim., 6 (1991), 121-142. doi: 10.1080/08927029108022142. Google Scholar [17] E. Hairer and C. Lubich, Spectral semi-discretisations of weakly nonlinear wave equations over long times, Found. Comput. Math., 8 (2008), 319-334. doi: 10.1007/s10208-007-9014-9. Google Scholar [18] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations, vol. 31 of Springer Series in Computational Mathematics, second edition, Springer-Verlag, Berlin, 2006. doi: 10. 1007/3-540-30666-8. Google Scholar [19] M. Khanamiryan, O. Nevanlinna and T. Vesanen, Long-term behavior of the numerical solution of the cubic non-linear schrödinger equation using strang splitting method, preprint, 2012. URL http://www.damtp.cam.ac.uk/user/na/people/Marianna/papers/NLS.pdfGoogle Scholar [20] T. I. Lakoba, Instability of the split-step method for a signal with nonzero central frequency, J. Opt. Soc. Am. B, 30 (2013), 3260-3271. doi: 10.1364/JOSAB.30.003260. Google Scholar [21] M. Tuckerman, B. J. Berne and G. J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys., 97 (1992), 1990-2001. doi: 10.1063/1.463137. Google Scholar [22] J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 23 (1986), 485-507. doi: 10.1137/0723033. Google Scholar

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##### References:
 [1] 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 [2] B. Cano, Conservation of invariants by symmetric multistep cosine methods for second-order partial differential equations, BIT, 53 (2013), 29-56. doi: 10.1007/s10543-012-0393-1. Google Scholar [3] B. Cano and A. González-Pachón, Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation, J. Comp. Math., 34 (2016), 385-406. doi: 10.4208/jcm.1601-m4541. Google Scholar [4] B. Cano, Conserved quantities of some Hamiltonian wave equations after full discretization, Numer. Math., 103 (2006), 197-223. doi: 10.1007/s00211-006-0680-3. Google Scholar [5] D. Cohen, L. Gauckler, E. Hairer and C. Lubich, Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions, BIT, 55 (2015), 705-732. doi: 10.1007/s10543-014-0527-8. Google Scholar [6] D. Cohen, E. Hairer and C. Lubich, Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations, Numer. Math., 110 (2008), 113-143. doi: 10.1007/s00211-008-0163-9. Google Scholar [7] M. Dahlby and B. Owren, Plane wave stability of some conservative schemes for the cubic Schrödinger equation, M2AN Math. Model. Numer. Anal., 43 (2009), 677-687. doi: 10.1051/m2an/2009022. Google Scholar [8] P. Deuflhard, A study of extrapolation methods based on multistep schemes without parasitic solutions, Z. Angew. Math. Phys., 30 (1979), 177-189. doi: 10.1007/BF01601932. Google Scholar [9] X. Dong, Stability and convergence of trigonometric integrator pseudospectral discretization for N-coupled nonlinear Klein-Gordon equations, Appl. Math. Comput., 232 (2014), 752-765. doi: 10.1016/j.amc.2014.01.144. Google Scholar [10] E. Faou, L. Gauckler and C. Lubich, Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation, Forum Math. Sigma, 2 (2014), e5, 45pp. doi: 10.1017/fms.2014.4. Google Scholar [11] E. Faou and B. Grébert, Hamiltonian interpolation of splitting approximations for nonlinear PDEs, Found. Comput. Math., 11 (2011), 381-415. doi: 10.1007/s10208-011-9094-4. Google Scholar [12] E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization, Numer. Math., 114 (2010), 429-458. doi: 10.1007/s00211-009-0258-y. Google Scholar [13] B. Garcıa-Archilla, J. M. Sanz-Serna and R. D. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput., 20 (1999), 930-963. doi: 10.1137/S1064827596313851. Google Scholar [14] L. Gauckler, Error analysis of trigonometric integrators for semilinear wave equations, SIAM J. Numer. Anal., 53 (2015), 1082-1106. doi: 10.1137/140977217. Google Scholar [15] L. Gauckler, E. Hairer, C. Lubich and D. Weiss, Metastable energy strata in weakly nonlinear wave equations, Comm. Partial Differential Equations, 37 (2012), 1391-1413. doi: 10.1080/03605302.2012.683503. Google Scholar [16] H. Grubmüller, H. Heller, A. Windemuth and K. Schulten, Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Mol. Sim., 6 (1991), 121-142. doi: 10.1080/08927029108022142. Google Scholar [17] E. Hairer and C. Lubich, Spectral semi-discretisations of weakly nonlinear wave equations over long times, Found. Comput. Math., 8 (2008), 319-334. doi: 10.1007/s10208-007-9014-9. Google Scholar [18] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations, vol. 31 of Springer Series in Computational Mathematics, second edition, Springer-Verlag, Berlin, 2006. doi: 10. 1007/3-540-30666-8. Google Scholar [19] M. Khanamiryan, O. Nevanlinna and T. Vesanen, Long-term behavior of the numerical solution of the cubic non-linear schrödinger equation using strang splitting method, preprint, 2012. URL http://www.damtp.cam.ac.uk/user/na/people/Marianna/papers/NLS.pdfGoogle Scholar [20] T. I. Lakoba, Instability of the split-step method for a signal with nonzero central frequency, J. Opt. Soc. Am. B, 30 (2013), 3260-3271. doi: 10.1364/JOSAB.30.003260. Google Scholar [21] M. Tuckerman, B. J. Berne and G. J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys., 97 (1992), 1990-2001. doi: 10.1063/1.463137. Google Scholar [22] J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 23 (1986), 485-507. doi: 10.1137/0723033. Google Scholar
Illustration of the bound $\epsilon ^{e(j)}$ for $E_j$ in the case $K=5$
Mode energies $E_l^n$ vs. time $t_n$ for the numerical solution with time step-size $\tau=0.05$ as in Theorem 2.1: $E_{1}^{n}$ corresponds to the 'line' at the top, $E_{0}^{n}$ and $E_{2}^{n}$ to the following graphs, and the remaining mode energies in decreasing order
Mode energies $E_l^n$ vs. time $t_n$ for the numerical solutions with time step-size $\tau=2\pi/(\omega _1+\omega _6+\omega _7)$ (left) and time step-size $\tau=2\pi/(-\omega _1+\omega _6+\omega _7)$ (right)
A schematic overview of the proof of Theorem 2.1: How the results on the almost-invariant energies εl are applied to control the mode energies El and vice versa
Approximation error $|{u_j^n-\widehat{u}_j^n}$ vs. time $t_n$ for $j=0$ (left), $j=1$ (middle) and $j=2$ (right). Different lines correspond to different values of $\epsilon =10^{-2}, 10^{-3}, \ldots, 10^{-8}$ in (9)
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