Numerical methods for conservation laws with rough flux

  • H. HoelEmail author
  • K. H. Karlsen
  • N. H. Risebro
  • E. B. Storrøsten


Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with flux functions driven by low-regularity paths. For a convex flux, it is demonstrated that driving path oscillations may lead to “cancellations” in the solution. Making use of this property, we show that for \(\alpha \)-Hölder continuous paths the convergence rate of the numerical methods can improve from \(\mathcal {O}(\text {COST}^{-\gamma })\), for some \(\gamma \in \left[ \alpha /(12-8\alpha ), \alpha /(10-6\alpha )\right] \), with \(\alpha \in (0, 1)\), to \(\mathcal {O}(\text {COST}^{-\min (1/4,\alpha /2)})\). Numerical examples support the theoretical results.


Stochastic conservation law Rough time-dependent flux Pathwise entropy solution Finite difference method Convergence Stochastic numerics 

Mathematics Subject Classification

Primary: 35L65 65M06 Secondary: 60H15 65C30 



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Authors and Affiliations

  1. 1.Mathematics Institute of Computational Science and EngineeringÉcole polytechnique fédérale de LausanneLausanneSwitzerland
  2. 2.Department of mathematicsUniversity of OsloOsloNorway

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