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Numerical attractors and approximations for stochastic or deterministic sine-Gordon lattice equations

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  • Yang, Shuang
  • Li, Yangrong

Abstract

First, we apply the implicit Euler scheme to discretize the sine-Gordon lattice equation (possessing a global attractor) and prove the existence of a numerical attractor for the time-discrete sine-Gordon lattice system with small step sizes. Second, we establish the upper semi-convergence from the numerical attractor towards the global attractor when the step size tends to zero. Third, we establish the upper semi-convergence from the random attractor of the stochastic sine-Gordon lattice equation to the global attractor when the intensity of noise goes to zero. Fourth, we show the finitely dimensional approximations of the three (numerical, random and global) attractors as the dimension of the state space goes to infinity. In a word, we establish four paths of convergence of finitely dimensional (numerical and random) attractors towards the global attractor.

Suggested Citation

  • Yang, Shuang & Li, Yangrong, 2022. "Numerical attractors and approximations for stochastic or deterministic sine-Gordon lattice equations," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    • Handle: RePEc:eee:apmaco:v:413:y:2022:i:c:s0096300321007244
    DOI: 10.1016/j.amc.2021.126640
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    References listed on IDEAS

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    1. Wang, Renhai & Li, Yangrong, 2019. "Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 86-102.
    2. Zhao, Wenqiang & Zhang, Yijin, 2016. "Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space ℓρp," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 226-243.
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    Cited by:

    1. Li, Fuzhi & Xu, Dongmei, 2022. "Backward regularity of attractors for lattice FitzHugh-Nagumo system with double random coefficients," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    2. Li, Yangrong & Wang, Fengling & Xia, Huan, 2024. "Continuity-sets of pullback random attractors for discrete porous media equations with colored noise," Applied Mathematics and Computation, Elsevier, vol. 465(C).

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    2. Li, Yangrong & Wang, Fengling & Xia, Huan, 2024. "Continuity-sets of pullback random attractors for discrete porous media equations with colored noise," Applied Mathematics and Computation, Elsevier, vol. 465(C).
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