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First and second order error estimates for the Upwind Source at Interface method

Authors: Theodoros Katsaounis and Chiara Simeoni
Journal: Math. Comp. 74 (2005), 103-122
MSC (2000): Primary 65N15, 35L65, 74S10
Published electronically: April 22, 2004
MathSciNet review: 2085404
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Abstract: The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced by Perthame and Simeoni is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove $L^p$-error estimates, $1\lep<+\infty$, in the case of a uniform spatial mesh, for which an optimal result can be obtained. We thus conclude that the same convergence rates hold as for the corresponding homogeneous problem. To improve the numerical accuracy, we develop two different approaches of dealing with the source term and we discuss the question of deriving second order error estimates. Numerical evidence shows that those techniques produce high resolution schemes compatible with the U.S.I. method.

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Additional Information

Theodoros Katsaounis
Affiliation: Department of Applied Mathematics, University of Crete, GR 71409 Heraklion, Crete, Greece; I.A.C.M.–F.O.R.T.H., GR 71110 Heraklion, Crete, Greece

Chiara Simeoni
Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, 45, rue d’Ulm, 75230 Paris Cedex 05, France; I.A.C.M.–F.O.R.T.H., GR 71110 Heraklion, Crete, Greece

Keywords: Scalar conservation laws, source terms, finite volume schemes, upwind interfacial methods, consistency, error estimates.
Received by editor(s): March 20, 2003
Received by editor(s) in revised form: July 8, 2003
Published electronically: April 22, 2004
Additional Notes: This work is partially supported by HYKE European programme HPRN-CT-2002-00282 ( The authors would like to thank Professor B. Perthame for his valuable help and Professor Ch. Makridakis for helpful discussions.
Article copyright: © Copyright 2004 American Mathematical Society