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Published in:
2020 | OriginalPaper | Chapter
Note on the Convergence of a Finite Volume Scheme for a Second Order Hyperbolic Equation with a Time Delay in Any Space Dimension
Authors : Fayssal Benkhaldoun, Abdallah Bradji
Published in: Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples
Publisher: Springer International Publishing
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Abstract
In this note, we establish a finite volume scheme for a model of a second order hyperbolic equation with a time delay in any space dimension. This model is considered in [10, 11] where some exponential stability estimates and oscillatory behaviour are proved. The scheme we shall present uses, as space discretization, the general class of nonconforming finite volume meshes of [5]. In addition to the proof of the existence and uniqueness of the discrete solution, we develop a new discrete a priori estimate. Thanks to this a priori estimate, we prove error estimates in discrete seminorms of \(L^\infty (H^1_0)\), \(L^\infty (L^2)\), and \(W^{1,\infty }(L^2)\). This work can be viewed as extension to the previous ones [2, 4] which dealt with the analysis of finite volume methods for respectively semilinear parabolic equations with a time delay and the wave equation.