65 research outputs found
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
The cutoff method, which cuts off the values of a function less than a given
number, is studied for the numerical computation of nonnegative solutions of
parabolic partial differential equations. A convergence analysis is given for a
broad class of finite difference methods combined with cutoff for linear
parabolic equations. Two applications are investigated, linear anisotropic
diffusion problems satisfying the setting of the convergence analysis and
nonlinear lubrication-type equations for which it is unclear if the convergence
analysis applies. The numerical results are shown to be consistent with the
theory and in good agreement with existing results in the literature. The
convergence analysis and applications demonstrate that the cutoff method is an
effective tool for use in the computation of nonnegative solutions. Cutoff can
also be used with other discretization methods such as collocation, finite
volume, finite element, and spectral methods and for the computation of
positive solutions.Comment: 19 pages, 41 figure
A waiting time phenomenon for thin film equations
We prove the occurrence of a waiting time phenomenon for solutions to fourth order degenerate parabolic differential equations which model the evolution of thin films of viscous fluids. In space dimension less or equal to three, we identify a general criterion on the growth of initial data near the free boundary which guarantees that for sufficiently small times the support of strong solutions locally does not increase. It turns out that this condition only depends on the smoothness of the diffusion coefficient in its point of degeneracy. Our approach combines a new version of Stampacchia's iteration lemma with weighted energy or entropy estimates. On account of numerical experiments, we conjecture that the
aforementioned growth criterion is optimal
Existence of positive solutions to stochastic thin-film equations
We construct martingale solutions to stochastic thin-film equations by introducing a (spatial) semidiscretization and establishing convergence. The discrete scheme allows for variants of the energy and entropy estimates in the continuous setting as long as the discrete energy does not exceed certain threshold values depending on the spatial grid size . Using a stopping time argument to prolongate high-energy paths constant in time, arbitrary moments of coupled energy/entropy functionals can be controlled. Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments---in particular on Jakubowski's generalization of Skorokhod's theorem---weak convergence methods, and recent tools on martingale convergence
Signatures of slip in dewetting polymer films
Thin liquid polymer films on hydrophobic substrates are susceptable to
rupture and formation of holes, which in turn initiate a complex dewetting
process that eventually evolves into characteristic stationary droplet
patterns. Experimental and theoretical studies suggest that the specific type
of droplet pattern largely depends on the nature of the polymer-substrate
boundary condition. To follow the morphological evolution numerically over
long time scales and for the multiple length scales involved has so far been
a major challenge. In this study a highly adaptive finite-element based
numerical scheme is presented that allows for large-scale simulations to
follow the evolution of the dewetting process deep into the nonlinear regime
of the model equations, capturing the complex dynamics including shedding of
droplets. In addition, the numerical results predict the previouly unknown
shedding of satellite droplets during the destabilisation of liquid ridges,
that form during the late stages of the dewetting process. While the
formation of satellite droplets is well-known in the context of elongating
fluid filaments and jets, we show here that for dewetting liquid ridges this
property can be dramatically altered by the interfacial condition between
polymer and substrate, namely slip
Signatures of slip in dewetting polymer films
Thin liquid polymer films on hydrophobic substrates are susceptable to rupture and formation of holes, which in turn initiate a complex dewetting process that eventually evolves into characteristic stationary droplet patterns. Experimental and theoretical studies suggest that the specific type of droplet pattern largely depends on the nature of the polymer-substrate boundary condition. To follow the morphological evolution numerically over long time scales and for the multiple length scales involved has so far been a major challenge. In this study a highly adaptive finite-element based numerical scheme is presented that allows for large-scale simulations to follow the evolution of the dewetting process deep into the nonlinear regime of the model equations, capturing the complex dynamics including shedding of droplets. In addition, the numerical results predict the previouly unknown shedding of satellite droplets during the destabilisation of liquid ridges, that form during the late stages of the dewetting process. While the formation of satellite droplets is well-known in the context of elongating fluid filaments and jets, we show here that for dewetting liquid ridges this property can be dramatically altered by the interfacial condition between polymer and substrate, namely slip
Structure Preserving Finite Volume Approximation of Cross-Diffusion Systems Coupled by a Free Interface
We propose a two-point flux approximation finite-volume scheme for the
approximation of two cross-diffusion systems coupled by a free interface to
account for vapor deposition. The moving interface is addressed with a cut-cell
approach, where the mesh is locally deformed around the interface. The scheme
preserves the structure of the continuous system, namely: mass conservation,
nonnegativity, volume-filling constraints and decay of the free energy.
Numerical results illustrate the properties of the scheme.Comment: 8 pages, 6 figure
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