Abstract
The aim of this paper is two-folds. Firstly we study first order stationary systems of PDEs of the form \(\sum _k A_k\partial _k U + KU=0\) with \(K\ngtr 0\) on \(\mathbb {R}^d\). We prove that the classical assumption \(K>0\) is not necessary for the well-posedness of the system and is violated in the particular case of the first order Poisson problem. Secondly we prove the \(L^2\) stability of the finite volume discretisations provided the term KU is appropriately discretised on faces. Our result relies on a discrete Gagliardo-Nirenberg-Sobolev inequality to be submitted [15].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Audusse, E., Minh, H.D., Omnes, P., Penel, Y.: Analysis of a modified godunov scheme for the linear wave equation with coriolis source term on cartesian meshes. J. Comput. Phys. 373, 91–129 (2018)
Bermudez, A., Vazquez, E.: Upwind methods for hyperbolic conservation laws with source terms. Comp. Fluids 23(8), 1049–1071 (1994)
Bessemoulin-Chatard, M., Chainais-Hillairet, C., Filbet, F.: On discrete functional inequalities for some finite volume schemes. IMA J. Numer. Anal. 35(3), 1125–1149 (2015)
Bouchut, F.: Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. In: Frontiers in Mathematics Series. Birkhäuser (2004)
Bouchut, F., Eymard, R., Prignet, A.: Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data. J. Evol. Equ. 11, 687–724 (2011)
Bouchut, F., Ounaissa, H., Perthame, B.: Upwinding of the source term at interfaces for euler equations with high friction. Comput. Math. Appl. 53(3–4), 361–375 (2007)
Brezis, H.: Functional Analysis, Sobolev Spaces and PDEs, Universitext. Springer, New York (2010)
Coudière, Y., Gallouët, T., Herbin, R.: Discrete sobolev inequalities and Lp error estimates for approximate finite volume solutions of convection diffusion equations. M2AN Math. Model. Numer. Anal 35, 767–778 (2001)
Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)
Eymard, R., Gallouët, T., Herbin, R.: Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal. 30, 1009–1043 (2010)
Friedrichs, K.O.: Symmetric positive linear differential equations. Commun. Pure Appl. Math. 11, 333–418 (1958)
Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118. Springer, New York (1996)
Gosse, L.: Computing qualitatively correct approximations of balance laws. SIMAI Springer Series, vol. 2. Springer, Mailand (2013)
Greenberg, J.M., Leroux, A.Y.: A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996)
Ndjinga, M.: A discrete gagliardo-nirenberg-sobolev inequality. To be submitted
Ndjinga, M.: \( {L}^2\) stability of nonlinear finite volume schemes for linear hyperbolic systems. C. R. Acad. Sci. Paris 351, 707–711 (2013)
Serre, D.: Systems of Conservation Laws I: Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge (1999)
Vila, J.P., Villedieu, P.: Convergence de la méthode des volumes finis pour les systèmes de friedrichs. C. R. Acad. Sci. Paris 325 (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Ndjinga, M., Ngwamou, S.K. (2020). On the \(L^2\) Stability of Finite Volumes for Stationary First Order Systems. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_38
Download citation
DOI: https://doi.org/10.1007/978-3-030-43651-3_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43650-6
Online ISBN: 978-3-030-43651-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)