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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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First and second order error estimates for the Upwind Source at Interface method
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by Theodoros Katsaounis and Chiara Simeoni PDF
Math. Comp. 74 (2005), 103-122 Request permission


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, $1\!\le \!p<\!+\infty$, 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
  • Email:
  • 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
  • Email:,
  • 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.
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 103-122
  • MSC (2000): Primary 65N15, 35L65, 74S10
  • DOI:
  • MathSciNet review: 2085404