Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case

Abstract

We report the developments of hydrodynamic instabilities in several wellbalanced finite volume schemes that are observed during the computation of the temporal evolution of an out-balance flow which is essentially a kinematic wave. The numerical simulations are based on the one-dimensional shallow-water equations for a uniformly sloping bed with hydraulic resistance. Subsequently, we highlight the need of low dissipative high-order well-balanced filter schemes for non-equilibrium flows with variable cut-off wavenumber to compute the out-balance flow under consideration, i.e. the kinematic wave.

Stochastic-deterministic modeling of bed load transport in shallow water flow over erodible slope: Linear stability analysis and numerical simulation

“…1. In the present study, we use closure equations for λ and κ consistent with (15). The key problem is that the available data do not allow us to fit all of the parameters.…”

“…However, numerical simulations of free surface instabilities in frictional flows down inclined planes have shown that a non-negligible eddy viscosity ν is required to avoid the development of unphysical short wavelength instabilities as well as to reproduce the non-linear steady-state shape of roll waves [11,15,18,64]. In this study the main role of the eddy viscosity is to prevent the development of discontinuities in the free surface, which allows us to evaluate the free surface slope in the presence of jumps and steps of the bed elevation.…”

“…Before describing some numerical results for this apparently simple problem, we present a spatio-temporal stability analysis of the steady uniform base flow [48], which is subsequently employed to verify the outputs of the numerical simulations by comparing the theoretical and numerical growth rates of infinitesimal perturbations. A similar benchmark was proposed in Bohorquez and Rentschler [15] for the flow of kinematic waves down fixed beds, which is associated with the spontaneous formation of natural roll waves [13]. The linear stability properties of the equations at hand have not been explored beforehand.…”

“…The hydrodynamic one has a neutral curve starting at a = 0 and Fr ≈ 2. It is the most intuitive mode and leads to the formation of roll waves as for non-erodible beds [11,15,64]. As a grows, the Froude number required for the onset of roll waves is slightly larger.…”

“…A fractional-step method was applied to split the advection-diffusion equations (30) into a hyperbolic subproblem with source terms and a parabolic subproblem [57]. The numerical strategy for the homogeneous hyperbolic system of equations is similar to the classical q-wave propagation algorithm that was adopted with success to compute roll wave developments for transient, nonuniform kinematic waves in previous contributions [13,15]. The innovative parts of the algorithm are introduced below, in particular the wave decomposition of the Riemann problem, the source term discretization and the summary of the full semidiscrete algorithm used in each time step, appropriate both for deterministic and stochastic partial differential equations [55].…”