Approximation of pseudodifferential operators in absorbing boundary conditions for hyperbolic equations

Summary Engquist and Majda [3] proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. (In the case of the wave equation this boundary condition is valid at all frequencies.) Here, least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. It is shown that for the wave equation this approach leads to Kreiss well-posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of Taylor- and Padé-approximations [3, 4]. Numerical examples indicate that this method works even more effectively for hyperbolic systems. The least-squares approach may be used to generate the boundary conditions automatically.     

Long-time behaviour of absorbing boundary conditions

A new class of computational far-field boundary conditions for hyperbolic partial differential equations was recently introduced by the authors. These boundary conditions combine properties of absorbing conditions for transient solutions and properties of far-field conditions for steady states. This paper analyses the properties of the wave equation coupled with these new boundary conditions: well-posedness, dissipativity and convergence in time.     

Attractor for a conserved phase‐field system with hyperbolic heat conduction

We consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.     

Oscillation theorems for nonlinear hyperbolic systems with impulses

In this paper, we are mainly concerned with the oscillation problems of nonlinear hyperbolic differential systems with impulses. Some sufficient conditions for oscillations of the solutions of nonlinear impulsive hyperbolic systems with Robin boundary conditions are obtained and the criteria of oscillation of the systems are established.     

声波方程吸收边界条件的稳定性分析

引言对于无界区域中波动现象的数值模拟，必需引进人工边界将计算限制在一个有界区域上．为了确定解，需要在人工边界上加适当的边界条件．对于声波和弹性波方程，这样的一组人工边界条件，也叫吸收边界条件，在［1，2］中被系统地构造出来．对于声波方程，这些吸收边界条件恰好是单程波方程的近似．如山中所指出，减少边界反射，便于在计算中应用和稳定性是构造吸收边界条件的三点关键．Ellgqllist和Maid。用模态分析方法15］证明，带有［IJ中构造的吸收边界条件的波动方程初边值问题是适定的，并且估计了人工边界所产生的误差．对于更广…     

Viscous boundary value problems for symmetric systems with variable multiplicities

Extending investigations of Métivier and Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to “real” or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the Zumbrun-Serre-Rousset theorem, valid for variable multiplicities, characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system. The new effects of viscosity are seen to be surprisingly subtle; in particular, viscous coupling of crossing hyperbolic modes may induce a destabilizing effect. We illustrate the theory with applications to magnetohydrodynamics.     

Exact Controllability for Nonautonomous First Order Quasilinear Hyperbolic Systems

By means of the theory on the semi-global C1 solution to the mixed initial-boundary value problem (IBVP) for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary conditions.     

Exact boundary observability for a kind of second order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C² solution, we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary observability for a kind of first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled.     

Asymptotic expansions for hyperbolic–parabolic systems with a small parameter

In this paper we consider initial-boundary value problems for systems with a small parameter ?. The problems are mixed hyperbolic–parabolic when ? > 0 and hyperbolic when ? = 0. Often the solution can be expanded asymptotically in ? and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O(1) boundary layers.     

一类具有吸收边界条件的二阶非线性双曲方程的混合元法分析

1　引　　言关于二阶双曲型方程的有限元解的收敛性问题 ,目前已经有不少结果 .Dupont[1 ] 给出了一类线性双曲方程 Galerkin解的 L2 误差估计 ,Baker[2 ] 对此作了改进 ,用的是一种所谓“非标准的能量方法”.这一方法为 Cowsar,Dupont,Wheeler[3] 所采用 ,分析了一类具有吸收边界条件的线性双曲方程的混合元格式的 L2收敛性 .对于非线性双曲型问题 ,袁益让 ,王宏[4,5] 等给出了标准有限元方法的 H1 与 L2 误差估计 .本文试图把 [3]的工作更进一步研究 ,我们考虑如下非线性双曲问题 :φ(x) utt= mi,j=1 xi(aij(x) p(x,u) u xj) + mi=1…     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.     

Some sufficient conditions for oscillation of impulsive delay hyperbolic systems with Robin boundary conditions

This paper deals with the oscillation problems of delay hyperbolic systems with impulses. Some sufficient conditions for oscillations of impulsive delay hyperbolic systems with Robin boundary conditions are obtained and the criteria of oscillation of the systems are established.     

Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals

In this paper we establish the exact boundary controllability for quasilinear hyperbolic systems with interface conditions. As an application, we get the exact boundary controllability of unsteady flows in a string‐like network of open canals. Copyright © 2004 John Wiley & Sons, Ltd.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

高阶拟线性双曲型方程的精确边界能控性

By means of the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues ,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.     

Existence of solutions for linear hyperbolic initial-boundary value problems in a rectangle

In a recent article, we achieved the well-posedness of linear hyperbolic initial and boundary value problems (IBVP) in a rectangle via semigroup method, and we found that there are only two elementary modes called hyperbolic and elliptic modes in the system. It seems that, there is only one set of boundary conditions for the hyperbolic mode, while there are infinitely many sets of boundary conditions for the elliptic mode, which can lead to well-posedness. In this article, we continue to consider linear hyperbolic IBVP in a rectangle in the constant coefficients case and we show that there are also infinitely many sets of boundary conditions for hyperbolic mode which will lead to the existence of a solution. We also have uniqueness in some special cases. The boundary conditions satisfy the reflection conditions introduced in Section 3, which turn out to be equivalent to the strictly dissipative conditions.     

Analysis of mixed problems of matching in hyperbolic systems of different orders

A mixed problem of matching in hyperbolic systems of different orders with irregular boundary conditions is considered. In particular, these systems include hyperbolic systems with coefficients having discontinuities of the first kind. Under certain conditions, an analytical representation of the solution of the problem is obtained by means of a finite integral transform.     

Contrôlabilité et observabilité unilatérales de systèmes hyperboliques quasi-linéaires

The known theory on the one-side exact boundary controllability and the one-side exact boundary observability for first-order quasilinear hyperbolic systems requires that the unknown variables should be suitably coupled in the boundary conditions at the non-control or non-observation side. In this Note we illustrate, with an inspiring example, that the one-side exact boundary controllability and the one-side exact boundary observability can still be realized by means of a suitable coupling among the unknown variables in the quasilinear hyperbolic system itself. To cite this article: T. Li et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Exact boundary observability for nonautonomous first-order quasilinear hyperbolic systems

By means of the theory on the semiglobal C¹ solution to the mixed initial-boundary value problem for first-order quasilinear hyperbolic systems, we establish the local exact boundary observability for general nonautonomous first-order quasilinear hyperbolic systems without zero eigenvalues and reveal the essential difference between nonautonomous hyperbolic systems and autonomous hyperbolic systems. Copyright © 2008 John Wiley & Sons, Ltd.