A Froude-adaptive upwinding strategy for the explicit CPR scheme on unstructured meshes.
Abstract
Several years of application-driven experience with the fully upwind Centred Potential Regularization (CPR) scheme have shown that, although robust and effective in low-Froude regimes, it may introduce noticeable transport dissipation over long-time simulations of low-to-moderate Froude coastal flows. We therefore introduce and analyze a Froude-adaptive discretization of the momentum transport in the CPR finite-volume scheme for the shallowwater equations with stationary topography. The transport flux contains an edgewise parameter that continuously connects a centered form in low-Froude regions to the fully upwind form when the local Froude number reaches one. The main result is a fully discrete nonlinear mechanical-energy estimate on general polygonal meshes. Exact cellwise identities are first derived for the potential and kinetic energies, and the associated production terms are controlled by the pressure stabilization and the dissipation generated by the parametrized transport. From this estimate, we derive local edgewise sufficient conditions for the gravity and velocity time-step restrictions and combine them into a single Froude-uniform CFL condition ensuring energy stability. For the second-order Heun discretization, retaining the exact convexity defect of the final averaging step yields a refined mechanical-energy estimate in which only a reduced fraction of the Forward Euler production remains to be controlled, allowing smaller stabilization parameters up to an O(∆t 4 ) consistency defect. The resulting method preserves the lake-at-rest equilibrium and retains the favorable low-Froude stability properties of the CPR regularization, while reducing the momentum-transport dissipation at moderate Froude numbers -typically in the range 0.1 ≤ Fr ≤ 0.8 -spanning energetic nearshore currents and locally near-critical flows in shallow or constricted coastal zones. Numerical experiments assess well-balancedness, accuracy, long-time dissipation, and the smooth transition between low-and order-one Froude regimes. The present analysis is restricted to fully wet configurations and does not address shock capturing or wetting-anddrying fronts.
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BibTeX
@misc{duran2026froudeadaptive,
title = {{A Froude-adaptive upwinding strategy for the explicit CPR scheme on unstructured meshes.}},
author = {Arnaud Duran},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05726403v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Froude-adaptive upwinding strategy for the explicit CPR scheme on unstructured meshes.
AU - Arnaud Duran
PY - 2026
DA - 2026/09/20
PB - HAL
UR - https://hal.science/hal-05726403v2
AB - Several years of application-driven experience with the fully upwind Centred Potential Regularization (CPR) scheme have shown that, although robust and effective in low-Froude regimes, it may introduce noticeable transport dissipation over long-time simulations of low-to-moderate Froude coastal flows. We therefore introduce and analyze a Froude-adaptive discretization of the momentum transport in the CPR finite-volume scheme for the shallowwater equations with stationary topography. The transport flux contains an edgewise parameter that continuously connects a centered form in low-Froude regions to the fully upwind form when the local Froude number reaches one. The main result is a fully discrete nonlinear mechanical-energy estimate on general polygonal meshes. Exact cellwise identities are first derived for the potential and kinetic energies, and the associated production terms are controlled by the pressure stabilization and the dissipation generated by the parametrized transport. From this estimate, we derive local edgewise sufficient conditions for the gravity and velocity time-step restrictions and combine them into a single Froude-uniform CFL condition ensuring energy stability. For the second-order Heun discretization, retaining the exact convexity defect of the final averaging step yields a refined mechanical-energy estimate in which only a reduced fraction of the Forward Euler production remains to be controlled, allowing smaller stabilization parameters up to an O(∆t 4 ) consistency defect. The resulting method preserves the lake-at-rest equilibrium and retains the favorable low-Froude stability properties of the CPR regularization, while reducing the momentum-transport dissipation at moderate Froude numbers -typically in the range 0.1 ≤ Fr ≤ 0.8 -spanning energetic nearshore currents and locally near-critical flows in shallow or constricted coastal zones. Numerical experiments assess well-balancedness, accuracy, long-time dissipation, and the smooth transition between low-and order-one Froude regimes. The present analysis is restricted to fully wet configurations and does not address shock capturing or wetting-anddrying fronts.
ER -