10 research outputs found
Linear stability of the 1D Saint-Venant equations and drag parameterizations
The stability of the homogeneous and steady flow based on the one-dimensional Saint-Venant equations for free surface and shallow water flows of constant slope is derived and displayed through graphs. With a suitable choice of units, the small and large drag limits, respectively, correspond to the small and large spatio-temporal scales of a linear system only controlled by the Froude number and two other dimensionless numbers associated with the bottom drag parameterization. Between the small drag limit, with the two families of marginal and non-dispersive shallow water waves, and the large drag limit, with the marginal and non-dispersive waves of the kinematic wave approximation, dispersive roll waves are detailed. These waves are damped or amplified, depending on the value of the three control parameters. The spatial generalized dispersion relations are also derived indicating that the roll-wave instability is of the convective type for all drag parameterizations
Structural Properties of the Stability of Jamitons
It is known that inhomogeneous second-order macroscopic traffic models can
reproduce the phantom traffic jam phenomenon: whenever the sub-characteristic
condition is violated, uniform traffic flow is unstable, and small
perturbations grow into nonlinear traveling waves, called jamitons. In
contrast, what is essentially unstudied is the question: which jamiton
solutions are dynamically stable? To understand which stop-and-go traffic waves
can arise through the dynamics of the model, this question is critical. This
paper first presents a computational study demonstrating which types of
jamitons do arise dynamically, and which do not. Then, a procedure is presented
that characterizes the stability of jamitons. The study reveals that a critical
component of this analysis is the proper treatment of the perturbations to the
shocks, and of the neighborhood of the sonic points.Comment: 22 page, 6 figure
Nonlinear Stability of Viscous Roll Waves
This is the published version, also available here: http://dx.doi.org/10.1137/100785454.Extending results of Oh and Zumbrun and of Johnson and Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant equations for shallow water flow down an inclined ramp. The main new issues to be overcome are incomplete parabolicity and the nonconservative form of the equations, which lead to undifferentiated quadratic source terms that cannot be handled using the estimates of the conservative case. The first is resolved by treating the equations in the more favorable Lagrangian coordinates, for which one can obtain large-amplitude nonlinear damping estimates similar to those carried out by Mascia and Zumbrun in the related shock wave case, assuming only symmetrizability of the hyperbolic part. The second is resolved by the observation that, similarly as in the relaxation and detonation cases, sources occurring in nonconservative components experience decay that is greater than expected, comparable to that experienced by a differentiated source
Multidimensional stability and transverse bifurcation of hydraulic shocks and roll waves in open channel flow
We study by a combination of analytical and numerical methods
multidimensional stability and transverse bifurcation of planar hydraulic shock
and roll wave solutions of the inviscid Saint Venant equations for inclined
shallow-water flow, both in the whole space and in a channel of finite width,
obtaining complete stability diagrams across the full parameter range of
existence. Technical advances include development of efficient multi-d Evans
solvers, low- and high-frequency asymptotics, explicit/semi-explicit
computation of stability boundaries, and rigorous treatment of channel flow
with wall-type physical boundary. Notable behavioral phenomena are a novel
essential transverse bifurcation of hydraulic shocks to invading planar
periodic roll-wave or doubly-transverse periodic herringbone patterns, with
associated metastable behavior driven by mixed roll- and herringbone-type waves
initiating from localized perturbation of an unstable constant state; and
Floquet-type transverse ``flapping'' bifurcation of roll wave patterns.Comment: 99 page
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Hyperbolic systems with supercharacteristic relaxations and roll waves
We study a distinguished limit for general 2 x 2 hyperbolic systems with relaxation, which is valid in both the subcharacteristic and supercharacteristic cases. This is a weakly nonlinear limit, which leads the underlying relaxation systems into a Burgers equation with a source term; the sign of the source term depends on the characteristic interleaving condition. In the supercharacteristic case, the problem admits a periodic solution known as the roll wave, generated by a small perturbation around equilibrium constants. Such a limit is justified in the presence of artificial viscosity, using the energy method