# 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:

show all references

##### References:
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)
 [1] A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & 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 & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373 [5] H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119 [6] Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455 [7] Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185 [8] Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873 [10] Lingbing He, Claude Le Bris, Tony Lelièvre. Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic & Related Models, 2012, 5 (2) : 357-382. doi: 10.3934/krm.2012.5.357 [11] Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273 [12] Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete & 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 & Continuous Dynamical Systems - A, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557 [14] 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 & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767 [15] 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 [16] Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations & Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023 [17] Igor Chueshov, Stanislav Kolbasin. Long-time dynamics in plate models with strong nonlinear damping. Communications on Pure & Applied Analysis, 2012, 11 (2) : 659-674. doi: 10.3934/cpaa.2012.11.659 [18] Eduard Feireisl, Françoise Issard-Roch, Hana Petzeltová. Long-time behaviour and convergence towards equilibria for a conserved phase field model. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 239-252. doi: 10.3934/dcds.2004.10.239 [19] Tamara Fastovska. Long-time behaviour of a radially symmetric fluid-shell interaction system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1315-1348. doi: 10.3934/dcds.2018054 [20] Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367

2018 Impact Factor: 1.143