Abstract
This proceeding presents an optimal error estimate in the \(L^{1}\)-norm of order 1 / 2 between the exact solution of an initial and boundary value problem for the linear advection equation and its approximation by the explicit upwind scheme. The space domain is bounded and a Dirichlet condition is thus imposed on the entering part of the boundary. This result extends the analysis given in Merlet and Vovelle (Numer. Math. 106(1), 129–155 (2007), [10]) that concerns the case where the equation is posed on the whole space. One of the key point of the proof is the analysis of a suitable regularization by convolution of the exact (weak) solution. Compared to Merlet and Vovelle (Numer. Math. 106(1), 129–155 (2007), [10]) we also relax some hypothesis on the velocity field, which in particular is allowed to be somehow discontinuous in time. This proceeding is a short version of Aguillon and Boyer (IMA J. Numer. Anal. (2017), [1]), aiming to present the steps of the proof and the new intermediate results.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aguillon, N., Boyer, F.: Error estimate for the upwind scheme for the linear transport equation with boundary data. IMA J. Numer. Anal. (2017). https://hal.archives-ouvertes.fr/hal-01328667 (In press)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Boyer, F.: Trace theorems and spatial continuity properties for the solutions of the transport equation. Differ. Integral Equ. 18(8), 891–934 (2005)
Chainais-Hillairet, C.: Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate. M2AN Math. Model. Numer. Anal. 33(1), 129–156 (1999)
Colombo, R., Rossi, E.: Rigorous estimates on balance laws in bounded domains. Acta Mathematica Scientia 35(4), 906–944 (2015)
Coudière, Y., Vila, J.P., Villedieu, P.: Convergence d’un schéma volumes finis explicite en temps pour les systèmes hyperboliques linéaires symétriques en domaines bornés. C. R. Acad. Sci. Paris Sér. I Math. 331(1), 95–100 (2000)
Després, B.: An explicit a priori estimate for a finite volume approximation of linear advection on non-Cartesian grids. SIAM J. Numer. Anal. 42(2), 484–504 (electronic) (2004)
Després, B.: Lax theorem and finite volume schemes. Math. Comput. 73(247), 1203–1234 (2004)
Kuznetsov, N.N.: The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation. Ž. Vyčisl. Mat. i Mat. Fiz. 16(6), 1489–1502, 1627 (1976)
Merlet, B., Vovelle, J.: Error estimate for finite volume scheme. Numer. Math. 106(1), 129–155 (2007)
Vila, J.P., Villedieu, P.: Convergence of an explicit finite volume scheme for first order symmetric systems. Numer. Math. 94(3), 573–602 (2003)
Acknowledgements
This work has been carried out in the framework of Archimède Labex (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Aguillon, N., Boyer, F. (2017). Optimal Order of Convergence for the Upwind Scheme for the Linear Advection on a Bounded Domain. In: Cancès, C., Omnes, P. (eds) Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects . FVCA 2017. Springer Proceedings in Mathematics & Statistics, vol 199. Springer, Cham. https://doi.org/10.1007/978-3-319-57397-7_33
Download citation
DOI: https://doi.org/10.1007/978-3-319-57397-7_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-57396-0
Online ISBN: 978-3-319-57397-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)