A Finite-Volume discretization of viscoelastic Saint-Venant equations for FENE-P fluids
This provides an incremental extension of a numerical method for shallow flow simulations with a more complex viscoelastic rheology, relevant to geophysical and industrial fluid dynamics.
The paper extends a Finite-Volume discretization for viscoelastic Saint-Venant equations from UCM to FENE-P fluids, demonstrating stable numerical simulations in a practically useful parameter range despite lacking a priori stability guarantees.
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 \& Boyaval, 2013], 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 solution 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 numerical simulations went smoothly in a practically useful range of parameters.