Some second order time accurate finite volume method for the wave equation using a spatial multidimensional generic mesh

Abstract

The present work is an extension of the previous one [1] which dealt with error analysis of a nite volume scheme of rst 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 nite 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  ux 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 L1(0; T;H1 0 ()) and W1;1(0; T;L2( )) ! under the regularity assumption u 2 C4([0; T]; C2( )) for the exact solution u. These error estimates are useful because they allow to obtain approximations to the exact solution and its rst derivatives of order k2 + hD.