Semi-Discretized Approximation of Stability of Sine-Gordon System with Average-Central Finite Difference Scheme

Xudong Wang, Sizhe Wang, Xing Qiao and Fu Zheng ()
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Xudong Wang: School of Mathematics and Statistics, Hainan University, Haikou 570228, China
Sizhe Wang: School of Mathematics, Tianjin University, Tianjin 300072, China
Xing Qiao: School of Mathematical Sciences, Daqing Normal University, Daqing 163712, China
Fu Zheng: School of Mathematics and Statistics, Hainan University, Haikou 570228, China

Mathematics, 2024, vol. 12, issue 16, 1-15

Abstract: In this study, the energy control and asymptotic stability of the 1D sine-Gordon equation were investigated from the viewpoint of numerical approximation. An order reduction method was employed to transform the closed-loop system into an equivalent system, and an average-central finite difference scheme was constructed. This scheme is not only energy-preserving but also possesses uniform stability. The discrete multiplier method was utilized to obtain the uniformly asymptotic stability of the discrete systems. Moreover, to cope with the nonlinear term of the model, a discrete Wirtinger inequality suitable for our approximating scheme was established. Finally, several numerical experiments based on the eigenvalue distribution of the linearized approximation systems were conducted to demonstrate the effectiveness of the numerical approximating algorithm.

Keywords: sine-Gordon equation; speed gradient; stability; finite difference; semi-discretization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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