Semi-linear Wave Equation The Long Time Behavior Of Spectral Methods

One-dimensional damped semilinear wave equations is considered in this paper.u_tt +αu_t - u_xx + g(u) = f, (x, t)∈Ω×R⁺, with Dirichlet boundary conditionu(-1) = u(1) = 0, and initial conditionsu(x,0) = u₀(x), u_t(x, 0) = u₁(x).WhereΩ= (-1,1),α> 0, f∈L²(Ω).Firstly, a semidiscrete Legendre spectral scheme for the one-dimensional damped semilinear wave equation with initial condition and Dirichlet boundary cinditions is constructed. We get a uniform priori estimate for the approximate solution of the semidiscrete Legendre spectral scheme in time. The stability, the convergence and error estimate of the semidiscrete Legendre spectral scheme over a finite time interval (0, T] are obtained under some conditions.Secondly, we construct a completely discrete Legendre sepectral scheme for the one-dimensional damped semilinear wave equation with initial condition and Dirichlet boundary cinditions. The solvability of the fully discrete Legendre sepectral scheme is discussed by Leray-Schauder fixed point theorem. We get a uniform priori estimate for the approximate solution of the completely discrete Legendre spectral scheme in time with g(u) =βsin u and g(u) = |u|^γu. The stability, the convergence and error estimate of the completely discrete Legendre spectral scheme over a finite time interval (0, T] are obtained under some conditions.Finally, we discuss discrete dynamical systems (the finite-dimensional dynamical systems) which were generated by the semidiscrete Legendre spectral scheme and the completely discrete Legendre spectral scheme. The existence of global attractors are proved for the discrete dynamical systems.