Numerische Mathematik

, Volume 139, Issue 1, pp 155–186 | Cite as

Analysis of the implicit upwind finite volume scheme with rough coefficients

  • André Schlichting
  • Christian SeisEmail author


We study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation in the low regularity DiPerna–Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at \(\hbox {least}~{1}/{2}\). The numerical error is estimated in terms of logarithmic Kantorovich–Rubinstein distances and provides thus a bound on the rate of weak convergence.

Mathematics Subject Classification

65M08 65M12 65M15 



The present work was done when the second author was affiliated with the Universität Bonn.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Institut für Analysis und NumerikUniversität MünsterMünsterGermany

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