1,900 research outputs found
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
Transient dynamics and structure of optimal excitations in thermocapillary spreading: Precursor film model
Linearized modal stability theory has shown that the thermocapillary spreading of a liquid film on a homogeneous, completely wetting surface can produce a rivulet instability at the advancing front due to formation of a capillary ridge. Mechanisms that drain fluid from the ridge can stabilize the flow against rivulet formation. Numerical predictions from this analysis for the film speed, shape, and most unstable wavelength agree remarkably well with experimental measurements even though the linearized disturbance operator is non-normal, which allows transient growth of perturbations. Our previous studies using a more generalized nonmodal stability analysis for contact lines models describing partially wetting liquids (i.e., either boundary slip or van der Waals interactions) have shown that the transient amplification is not sufficient to affect the predictions of eigenvalue analysis. In this work we complete examination of the various contact line models by studying the influence of an infinite and flat precursor film, which is the most commonly employed contact line model for completely wetting films. The maximum amplification of arbitrary disturbances and the optimal initial excitations that elicit the maximum growth over a specified time, which quantify the sensitivity of the film to perturbations of different structure, are presented. While the modal results for the three different contact line models are essentially indistinguishable, the transient dynamics and maximum possible amplification differ, which suggests different transient dynamics for completely and partially wetting films. These differences are explained by the structure of the computed optimal excitations, which provides further basis for understanding the agreement between experiment and predictions of conventional modal analysis
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme
We discuss the numerical solution of nonlinear parabolic partial differential
equations, exhibiting finite speed of propagation, via a strongly implicit
finite-difference scheme with formal truncation error . Our application of interest is the spreading of
viscous gravity currents in the study of which these type of differential
equations arise. Viscous gravity currents are low Reynolds number (viscous
forces dominate inertial forces) flow phenomena in which a dense, viscous fluid
displaces a lighter (usually immiscible) fluid. The fluids may be confined by
the sidewalls of a channel or propagate in an unconfined two-dimensional (or
axisymmetric three-dimensional) geometry. Under the lubrication approximation,
the mathematical description of the spreading of these fluids reduces to
solving the so-called thin-film equation for the current's shape . To
solve such nonlinear parabolic equations we propose a finite-difference scheme
based on the Crank--Nicolson idea. We implement the scheme for problems
involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or
spherically-symmetric three-dimensional currents) on an equispaced but
staggered grid. We benchmark the scheme against analytical solutions and
highlight its strong numerical stability by specifically considering the
spreading of non-Newtonian power-law fluids in a variable-width confined
channel-like geometry (a "Hele-Shaw cell") subject to a given mass
conservation/balance constraint. We show that this constraint can be
implemented by re-expressing it as nonlinear flux boundary conditions on the
domain's endpoints. Then, we show numerically that the scheme achieves its full
second-order accuracy in space and time. We also highlight through numerical
simulations how the proposed scheme accurately respects the mass
conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements
and corrections; to appear as a contribution in "Applied Wave Mathematics II
Time integration and steady-state continuation for 2d lubrication equations
Lubrication equations allow to describe many structurin processes of thin
liquid films. We develop and apply numerical tools suitable for their analysis
employing a dynamical systems approach. In particular, we present a time
integration algorithm based on exponential propagation and an algorithm for
steady-state continuation. In both algorithms a Cayley transform is employed to
overcome numerical problems resulting from scale separation in space and time.
An adaptive time-step allows to study the dynamics close to hetero- or
homoclinic connections. The developed framework is employed on the one hand to
analyse different phases of the dewetting of a liquid film on a horizontal
homogeneous substrate. On the other hand, we consider the depinning of drops
pinned by a wettability defect. Time-stepping and path-following are used in
both cases to analyse steady-state solutions and their bifurcations as well as
dynamic processes on short and long time-scales. Both examples are treated for
two- and three-dimensional physical settings and prove that the developed
algorithms are reliable and efficient for 1d and 2d lubrication equations,
respectively.Comment: 33 pages, 16 figure
Recommended from our members
Thermodynamische Materialtheorien
The meeting was focused on research in the broad field of thermodynamic constitutive theories. It provided a contact between physicists, engineers and mathematicians, whose talks led to lively and interesting discussions. The debate concentrated on the physical motivation of the models subjected to mathematical analysis
Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
International audienceWe study the spectral stability of roll-wave solutions of the viscous St. Venant equationsmodeling inclined shallow-water flow, both at onset in the small-Froude number or âweakly unstableâlimit F â 2+ and for general values of the Froude number F , including the limit F â +â. In the former,F â 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV)equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate thisformal limit, showing that stability as F â 2+ is equivalent to stability of the corresponding KdV-KSwaves in the KdV limit. Together with recent results obtained for KdV-KS by JohnsonâNobleâRodriguesâZumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St.Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainderof the paper, we investigate numerically and analytically the evolution of the stability diagram as Froudenumber increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable todifferent, large-F behavior, with stability determined by simple power law relations. The latter stabilitycriteria are potentially useful in hydraulic engineering applications, for which typically 2.5 †F †6.0
Linear Asymptotic Stability and Modulation Behavior near Periodic Waves of the Korteweg-de Vries Equation
We provide a detailed study of the dynamics obtained by linearizing the
Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal
wave. In a suitable sense, linearly analogous to space-modulated stability, we
prove global-in-time bounded stability in any Sobolev space, and asymptotic
stability of dispersive type. Furthermore, we provide both a leading-order
description of the dynamics in terms of slow modulation of local parameters and
asymptotic modulation systems and effective initial data for the evolution of
those parameters. This requires a global-in-time study of the dynamics
generated by a non normal operator with non constant coefficients. On the road
we also prove estimates on oscillatory integrals particularly suitable to
derive large-time asymptotic systems that could be of some general interest
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