Some Second Order Time Accurate for a Finite Volume Method for the Wave Equation Using a Spatial Multidimensional Generic Mesh∗

The present work is an extension of the previous one [1] which dealt with error analysis of a finite volume scheme of first order (both in time and space) for second order hyperbolic equations on general nonconforming multidimensional spatial meshes introduced recently in [4]. The aim of this contribution is to get some second–order time accurate schemes for a finite volume method for second order hyperbolic equations using the same class of spatial generic meshes stated above. We present a family of implicit time schemes to approximate the wave equation. The time discretization is performed using a one–parameter Newmark method. We prove that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is k2 +hD, where hD (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid for discrete norms L∞(0, T ;H1 0 (Ω)) and W1,∞(0, T ;L2(Ω)) ! under the regularity assumption u ∈ C4([0, T ]; C2(Ω)) for the exact solution u. These error estimates are useful because they allow to obtain approximations to the exact solution and its first derivatives of order k2 + hD.