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A Finite-Volume Discretization of Viscoelastic Saint-Venant Equations for FENE-P Fluids

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 200)


Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in Bouchut and Boyaval (M3AS 23(08): 1479–1526, 2013, [6]), which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solutions to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but the scheme is fully computable. And, using numerical simulations, it may help understand the famous High-Weissenberg number problem (HWNP) well-known in computational rheology.


  • Saint-venant equations
  • Fene-p viscoelastic fluids
  • Finite-volume
  • Simple Riemann solver
  • Suliciu relaxation scheme

MSC (2010):

  • 65M08
  • 65N08
  • 35Q30

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Correspondence to Sébastien Boyaval .

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Boyaval, S. (2017). A Finite-Volume Discretization of Viscoelastic Saint-Venant Equations for FENE-P Fluids. In: Cancès, C., Omnes, P. (eds) Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems. FVCA 2017. Springer Proceedings in Mathematics & Statistics, vol 200. Springer, Cham.

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