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1.
The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1?d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes to zero. Then we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. Our method of proof relies essentially on discrete multiplier techniques. 相似文献
2.
Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves 总被引:2,自引:1,他引:1
In this paper we investigate the efficiency of the method of perfectly matched layers (PML) for the 1-d wave equation. The
PML method furnishes a way to compute solutions of the wave equation for exterior problems in a finite computational domain
by adding a damping term on the matched layer. In view of the properties of solutions in the whole free space, one expects
the energy of solutions obtained by the PML method to tend to zero as t → ∞, and the rate of decay can be understood as a measure of the efficiency of the method. We prove, indeed, that the exponential
decay holds and characterize the exponential decay rate in terms of the parameters and damping potentials entering in the
implementation of the PML method. We also consider a space semi-discrete numerical approximation scheme and we prove that,
due to the high frequency spurious numerical solutions, the decay rate fails to be uniform as the mesh size parameter h tends to zero. We show however that adding a numerical viscosity term allows us to recover the property of exponential decay
of the energy uniformly on h. Although our analysis is restricted to finite differences in 1-d, most of the methods and results apply to finite elements
on regular meshes and to multi-dimensional problems.
This work started while the first author was visiting the Department of Mathematics of the Universidad Autónoma de Madrid,
in the frame of the European program “New materials, adaptive systems and their nonlinearities: modeling, control and numerical
simulation” HPRN-CT-2002-00284. The work was finished while both authors visited the Isaac Newton Institute of Cambridge within
the Program “Highly Oscillatory Problems”. 相似文献
3.
We study initial–boundary value problems for strongly damped nonlinear wave equations. By using improved integral estimates, it is proven that the solutions of the problems decay to zero exponentially as time t approaches infinity, under a very simple and general assumption regarding the nonlinear term. 相似文献
4.
This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force $f$ . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on $f$ , the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term $f$ is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms. 相似文献
5.
In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity‐loss type, which causes the difficulty in high‐frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for n≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for n≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for n = 1. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
6.
Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square 总被引:1,自引:0,他引:1
7.
Aissa Guesmia Salim A. Messaoudi 《Mathematical Methods in the Applied Sciences》2009,32(16):2102-2122
In this paper we consider the following Timoshenko system: with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ρ1, ρ2, k1, k2 and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
8.
In this paper we study well‐posedness of the damped nonlinear wave equation in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ?2; ω?0, ωλ1+µ>0 with λ1 being the first eigenvalue of ?Δ under zero boundary condition. Under the assumptions that g(·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow‐up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
9.
Dietmar Saupe 《Acta Appl Math》1988,13(1-2):59-80
We consider the damped Newton's method N
h
(z) = z – hp(z)/p(z), 0<h<1 for polynomialsp(z) with complex coefficients. For the usual Newton's method (h=1) and polynomialsp(z), it is known that the method may fail to converge to a root ofp and rather leads to an attractive periodic cycle.N
h(z) may be interpreted as an Euler step for the differential equation =–p(z)/p(z) with step sizeh. In contrast to the possible failure of Newton's method, we have that for almost all initial conditions to the differential equation that the solutions converge to a root ofp. We show that this property generally carries over to Newton's methodN
h(z) only for certain nondegenerate polynomials and for sufficiently small step sizesh>0. Further we discuss the damped Newton's method applied to the family of polynomials of degree 3. 相似文献
10.
ABSTRACTThis paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399]. 相似文献
11.
Yan Lv 《Journal of Differential Equations》2008,244(1):1-23
In this paper, relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered. By introducing almost surely D-α-contracting property for random dynamical systems, we obtain a global random attractor of the stochastic wave equation endowed with Dirichlet boundary condition for any 0<ν?1. The upper semicontinuity of this global random attractor and the global attractor of the heat equation zt−Δz+f(z)=0 with Dirichlet boundary condition as ν goes to zero is investigated. Furthermore we show the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation. 相似文献
12.
The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful. 相似文献
13.
The paper deals with the numerical approximation of the HUM control of the 2D wave equation. Most of the discrete models obtained with classical finite difference or finite element methods do not produce convergent sequences of discrete controls, as the mesh size h and the time step Δt go to zero. We introduce a family of fully-discrete schemes, nondispersive, stable under the condition \(\Delta t\leq h\slash\sqrt{2}\) and uniformly controllable with respect to h and Δt. These implicit schemes differ from the usual explicit one (obtained with leapfrog time approximation and five point spatial approximations) by the addition of terms proportional to h 2 and Δt 2. Numerical experiments for nonsmooth initial conditions on the unit square using a conjugate gradient algorithm indicate the excellent performance of the schemes. 相似文献
14.
We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ? 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.
相似文献
$$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$
15.
We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data: We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the Lp–Lq estimates of fundamental solutions for linear damped wave equations [K. Nishihara, Lp–Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The Lp–Lq estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and Lp–Lq estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, Lp–Lq estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701]. 相似文献
16.
In this paper, we analyze the exponential decay property of solutions of the semilinear wave equation in with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions on the nonlinearity we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p<5. The method of proof combines classical energy estimates for the linear wave equation allowing to estimate the total energy of solutions in terms of the energy localized in the exterior of a ball, Strichartz's estimates and results by P. Gérard on microlocal defect measures and linearizable sequences. We also give an application to the stabilization and controllability of the semilinear wave equation in a bounded domain under the same growth condition on the nonlinearity but provided the nonlinearity has been cut-off away from the boundary. 相似文献
17.
Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions 下载免费PDF全文
A variable coefficient viscoelastic wave equation with acoustic boundary conditions and nonlinear source term is considered. Under suitable conditions on the initial data and the relaxation function g, we show the polynomial decay of the energy solution and the blow up of solutions by energy methods. The estimates for the lifespan of solutions are also given. 相似文献
18.
19.
We consider the Cauchy problem in R n for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L1,1( R n) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L2‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference. 相似文献
20.
Abbes Benaissa Salim A. Messaoudi 《NoDEA : Nonlinear Differential Equations and Applications》2006,12(4):391-399
The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning
energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation
a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponentially
even if m > 2. 相似文献