Convergence analysis of some high–order time accurate schemes for a finite volume method for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

The present work is an extension of our previous work (Bradji, Numer Methods Partial Differ Equations, to appear) which dealt with error analysis of a finite volume scheme of a first convergence order (both in time and space) for second‐order hyperbolic equations on general nonconforming multidimensional spatial meshes introduced recently in (Eymard et al. IMAJ Numer Anal 30(2010), 1009–1043). We aim in this article to get some higher‐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 derive a family of finite volume schemes approximating the wave equation, as a model for second‐order hyperbolic equations, in which the discretization in time is performed using a one‐parameter scheme of the Newmark's method. We prove that the error estimate of these finite volume schemes is of order two (or four) in time and it is of optimal order in space. These error estimates are analyzed in several norms which allow us to derive approximations for the exact solution and its first derivatives whose the convergence order is two (or four) in time and it is optimal in space. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}k^2+h_\mathcal{D}\end{align*} \end{document} or \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}k^4+h_\mathcal{D}\end{align*} \end{document} , where \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}h_\mathcal{D}\end{align*} \end{document} (resp. k ) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}u\in C^4(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*} \end{document} , when the schemes are second‐order accurate in time, and \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}u\in C^6(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*} \end{document} , when the schemes are four‐order accurate in time for the exact solution u . The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013