A theoretical analysis of a new finite volume scheme for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

Abstract

AbstractWe consider the wave equation, on a multidimensional spatial domain. The discretization of the spatial domain is performed using a general class of nonconforming meshes which has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMAJ Numer Anal 30 (2010), 1009–1043). The discretization in time is performed using a uniform mesh. We derive a new implicit finite volume scheme approximating the wave equation and we prove error estimates of the finite volume approximate solution in several norms which allow us to derive error estimates for the approximations of the exact solution and its first derivatives. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \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. This estimate is valid under the regularity assumption \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}u\in C^3(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*} \end{document} 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