On The Stability of Solutions of Certain Classes of Initial-Boundary-Value Problems in Aerohydroelasticity

2021 ◽  
Author(s):  
P. A. Vel’misov ◽  
A. V. Ankilov ◽  
Yu. V. Pokladova
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Stability Of Solutions ◽  
Initial Boundary Value Problems ◽  
The Stability ◽  
Initial Boundary Value

Generic types and transitions in hyperbolic initial–boundary-value problems

2002 ◽  
Vol 132 (5) ◽  
pp. 1073-1104 ◽  
Author(s):  
Sylvie Benzoni-Gavage ◽  
Frédéric Rousset ◽  
Denis Serre ◽  
K. Zumbrun
Keyword(s):  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Stability Class ◽  
Weakly Stable ◽  
The Stability ◽  
Initial Boundary Value ◽  
Open Class

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

Difference approximations for initial boundary-value problems

1971 ◽  
Vol 323 (1553) ◽  
pp. 255-261 ◽  
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Difference Approximations ◽  
Strengthened Form ◽  
The Stability ◽  
Initial Boundary Value ◽  
Scalar Equations

The problem of the stability of difference approximations for initial boundary-value problems is examined and recent results obtained in this field by Gustafsson, Kreiss & Sundström (1971) are presented. It is shown that even for scalar equations the Ryabenkii-Godunow condition is not sufficient for stability and a procedure is outlined which leads to a slightly strengthened form of this condition which is satisfactory in practice.

Generic types and transitions in hyperbolic initial–boundary-value problems

2002 ◽  
Vol 132 (5) ◽  
pp. 1073-1104
Author(s):  
Sylvie Benzoni-Gavage ◽  
Frédéric Rousset ◽  
Denis Serre ◽  
K. Zumbrun
Keyword(s):  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Stability Class ◽  
Weakly Stable ◽  
The Stability ◽  
Initial Boundary Value ◽  
Open Class

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

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