Strong convergence of nonlinear finite volumemethods for linear hyperbolic systems

International audienceUnlike finite elements methods, finite volume methods are far fromhaving a clear functional analytic setting allowing the proof of generalconvergence results. In, compactness methods were used to deriveconvergence results for the Laplace equation on fairly general meshes.The weak convergence of nonlinear finite volume methods for linear hy-perbolic systems was proven in using the Banach-Alaoglu compactnesstheorem. It allowed the use of general $L^2$ initial data which is consistent with the continuous theory based on the $L^2$ Fourier transform.To our knowledge this was the first convergence results applicable to nondifferentiable initial data. However this weak convergence result seems not optimal with regard of numerical simulations. In this paper we prove that the convergence is indeed strong for a wide class of possibly nonlinear upwinding schemes. The context of our study being multidimensional, we cannot use the spaces $L^1$ and $BV$ classically encountered in the study of 1$D$ hyperbolic systems. We propose instead the use of generalised p-variation function, initially introduced by Wiener and first studied by Young. Thesespaces are compactly embedded in $L^p$. They can therefore fitinto the $L^2$ framework imposed by Brenner obstruction result. Usingestimates of the quadratic variation of the finite volume approximationswe prove the compactness of the sequence of approximations and deducethe strong convergence of the numerical method.We finally discuss the applicability of this approach to nonlinear hyper-bolic systems