Abstract
In this paper we present an error estimate for the approximate solution of the nonlinear hyperbolic equation u t + div (f(u(x, t))v(x)) = 0 by an implicit finite volume scheme, using an Engquist-Osher numerical flux. We show that the error is of order \(\sqrt {k + \sqrt h } \) , where h and k are respectively the space and the time steps size parameters. The error estimate shows that the convergence of this scheme is possible with an unbounded CFL condition. This result is extended to other numerical fluxes and explicit scheme in [3].
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 subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Champier, S., Gallouët T. and Herbin, R. (1993) Convergence of an Upstream finite volume Scheme on a Triangular Mesh for a Nonlinear Hyperbolic Equation, Numer. Math. 66, 139–157.
Cockburn, B. and Gremaud, P. A. (1996) Error estimates for finite element methods for scalar conservation laws, SIAM J. Numer. Anal. Vol. 33, N. 2.
Eymard, R., Gallouët, T., Ghilani, M. and Herbin, R., Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes. submitted.
Ghilani, M. (1997) Estimation d’erreur pour une loi de conservation scalaire multidimensionnelle approchée par un schéma implicite de volumes finis. C.R. Acad. Sci. Paris, t. 324, Série I.
Ghilani, M. (1997) Thèse d’Etat, Rabat, Morocco.
Kruzkov, S.N. (1970) First Order quasilinear equations with several space variables, Math. USSR. Sb. 10. 217–243.
Vila, J.P. (1994) Convergence and error estimate in finite volume schemes for general multidimensional conservation laws, I. explicit monotone schemes, M2AN, 28, 3, 267 – 285.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Ghilani, M. (1998). An Error Estimate for the Approximate Solution of a Porous Media Diphasic Flow Equation. In: Crolet, J.M., El. Hatri, M. (eds) Recent Advances in Problems of Flow and Transport in Porous Media. Theory and Applications of Transport in Porous Media, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2856-0_3
Download citation
DOI: https://doi.org/10.1007/978-94-017-2856-0_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4989-6
Online ISBN: 978-94-017-2856-0
eBook Packages: Springer Book Archive