-
Previous Article
Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions
- DCDS Home
- This Issue
-
Next Article
Hermodynamic formalism and k-bonacci substitutions
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 |
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.
Keywords:
Nonlinear wave equation,
trigonometric integrators,
symplectic integrators,
long-time behaviour,
modulated Fourier expansion.
Mathematics Subject Classification:
Primary: 65P10, 65P40; Secondary: 65M70, 35L05.
Citation:
Ludwig Gauckler, Daniel Weiss. Metastable energy strata in numerical discretizations of weakly nonlinear wave equations. Discrete and Continuous Dynamical Systems,
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
L. Gauckler,
Error analysis of trigonometric integrators for semilinear wave equations, SIAM J. Numer. Anal., 53 (2015), 1082-1106.
doi: 10.1137/140977217. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.pdf |
| [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. |
| [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. |
| [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. |
show all references
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [14] |
L. Gauckler,
Error analysis of trigonometric integrators for semilinear wave equations, SIAM J. Numer. Anal., 53 (2015), 1082-1106.
doi: 10.1137/140977217. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.pdf |
| [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. |
| [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. |
| [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. |
Figure 2.
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
Figure 3.
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)
Figure 4.
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
Figure 5.
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)
| [1] |
A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185 |
| [2] |
Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675 |
| [3] |
Xinmin Xiang. The long-time behaviour for nonlinear Schrödinger equation and its rational pseudospectral approximation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 469-488. doi: 10.3934/dcdsb.2005.5.469 |
| [4] |
A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373 |
| [5] |
Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455 |
| [6] |
Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185 |
| [7] |
H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119 |
| [8] |
Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625 |
| [9] |
Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873 |
| [10] |
Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273 |
| [11] |
Lingbing He, Claude Le Bris, Tony Lelièvre. Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic and Related Models, 2012, 5 (2) : 357-382. doi: 10.3934/krm.2012.5.357 |
| [12] |
Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265 |
| [13] |
Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557 |
| [14] |
Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110 |
| [15] |
Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks and Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767 |
| [16] |
C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139 |
| [17] |
Igor Chueshov, Stanislav Kolbasin. Long-time dynamics in plate models with strong nonlinear damping. Communications on Pure and Applied Analysis, 2012, 11 (2) : 659-674. doi: 10.3934/cpaa.2012.11.659 |
| [18] |
Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations and Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023 |
| [19] |
Eduard Feireisl, Françoise Issard-Roch, Hana Petzeltová. Long-time behaviour and convergence towards equilibria for a conserved phase field model. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 239-252. doi: 10.3934/dcds.2004.10.239 |
| [20] |
Tamara Fastovska. Long-time behaviour of a radially symmetric fluid-shell interaction system. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1315-1348. doi: 10.3934/dcds.2018054 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]