Note on the convergence of a finite volume scheme for a second order hyperbolic equation with a time delay in any space dimension

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.