# A Finite-Volume discretization of viscoelastic Saint-Venant equations for FENE-P fluids

## TL;DR

This paper extends a finite-volume discretization method for viscoelastic shallow flow equations to FENE-P fluids, demonstrating practical numerical stability and applicability despite some unresolved theoretical stability guarantees.

## Contribution

It introduces an extension of the finite-volume scheme to FENE-P fluids, broadening the applicability of the method to more complex viscoelastic models.

## Key findings

- Numerical simulations showed smooth solutions in practical parameter ranges.
- The method preserves physical free-energy dissipation properties.
- Stability conditions are not fully guaranteed theoretically but are observed numerically.

## Abstract

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.