A theoretical analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the one dimensional wave equation

This paper details our note [6] and it is an extension of our previous works  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for second order hyperbolic equations with Dirichlet boundary conditions on general nonconforming multidimensional spatial meshes introduced recently in [14]. We aim in this work (and some forthcoming studies) to get higher order (both in time and space) finite volume approximations for the exact solution of hyperbolic equations using the class of spatial generic meshes introduced recently in [14] on low order schemes from which the matrices used to compute the discrete solutions are sparse. We focus in the present contribution on the one dimensional wave equation and on one of its implicit finite volume schemes described in [4]. The implicit finite volume scheme approximating the one dimensional wave equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal, symmetric and definite positive. The finite volume approximate solution of the basic finite volume scheme is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allow us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using linear systems whose matrices are the same ones used to compute the discrete solution of the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximation of some high order spatial derivatives of the exact solution. The computation of the approximation of these high order spatial derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [13] without any condition. To reach the above results, a theoretical framework is developed and some numerical examples supporting the theory are presented. Some of the tools of this framework are new and interesting and they are stated in the one space dimension but they can be extended to several space dimensions. In particular a new and useful a prior estimate for a suitable discrete problem is developed and proved. The proof of this a prior estimate result is based essentially on the decomposition of the solution of the discrete problem into the solutions of two suitable discrete problems. A new technique is used in order to get a convenient finite volume approximation whose discrete time derivatives of order up to order two are also converging towards the solution of the wave equation and their corresponding time derivatives.     

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

We 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 begin{align*} h_mathcal{D}end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption begin{align*}uin C^3(lbrack 0,Trbrack;C^2(overline{Omega}))end{align*} 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     

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 \begin{align*}k^2+h_\mathcal{D}\end{align*} or \begin{align*}k^4+h_\mathcal{D}\end{align*}, where \begin{align*}h_\mathcal{D}\end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption \begin{align*}u\in C^4(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are second‐order accurate in time, and \begin{align*}u\in C^6(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, 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