• DOI:10.1137/16M1084882
• Corpus ID: 37987138

Convergence Rates for Upwind Schemes with Rough Coefficients

@article{Schlichting2017ConvergenceRF,
  title={Convergence Rates for Upwind Schemes with Rough Coefficients},
  author={Andr{\'e} Schlichting and Christian Seis},
  journal={SIAM J. Numer. Anal.},
  year={2017},
  volume={55},
  pages={812-840}
}

• A. Schlichting, Christian Seis
• Published 29 June 2016
• Mathematics
• SIAM J. Numer. Anal.

This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for numerically solving continuity equations. We are interested in the case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable. We estimate the error between approximate solutions constructed by the upwind scheme and distributional solutions of the continuous problem in a Kantorovich--Rubinstein distance, which was recently used for stability estimates for the continuity equation [C. Seis, Ann. Inst. H. Poincare Anal. Non Lineaire, https://doi.org/10.1016/j.anihpc.2017.01.001]. Restricted to Cartesian meshes, our estimate shows that the rate of weak convergence is at least 1/2 in the mesh size. The proof relies on a probabilistic interpretation of the upwind scheme [F. Delarue and F. Lagoutiere, Arch. Ration. Mech. Anal., 199 (2011), pp. 229--268]. We complement the weak convergence result with an example that illustrates that for rough initial data no rates...