Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

Numerical and analytical study of the convective Cahn-Hilliard equation

We consider the convective Cahn-Hilliard equation that is used as a model of coarsening dynamics in driven systems and that in two spatial dimensions (x; y) has the form ut + Duux + r2(u u3 + r2u) = 0: Here t denotes time, u = u(x; y; t) is the order parameter and D is the parameter measuring the strength of driving. We primarily consider the case of one spatial dimension, when there is no y-dependence. For the case of no driving, when D = 0, the standard Cahn-Hilliard equation is recovered, and it is known that solutions to this equation are characterised by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters or droplets/holes or islands are obtained) followed by the coarsening process, where the average size of the clusters grows in time and the number of the clusters decreases. Moreover, two main coarsening mechanisms have been identified in the literature, namely, coarsening due to volume and translational modes. On the other hand, for the case of strong driving, when D ! 1, the well-known Kuramoto-Sivashinsky equation is recovered, solutions of which are characterised by complicated chaotic oscillations in both space and time. The primary aim of the present thesis is to perform a detailed and systematic investigation of the transitions in the solutions of the convective Cahn-Hilliard equation for a wide range of parameter values as the driving-force parameter is increased, and, in particular, to understand in detail how the coarsening dynamics is affected by driving. We find that one of the coarsening modes is stabilised at relatively small values of D, and the type of the unstable coarsening mode may change as D increases. In addition, we find that there may be intervals in the driving-force parameter D where coarsening is completely stabilised. On the other hand, there may be intervals where twomode solutions are unstable and the solutions can evolve, for example, into one-droplet/hole solutions, symmetry-broken two-droplet/hole solutions or time-periodic solutions. We present detailed stability diagrams for 2-mode solutions in the parameter planes and corroborate our findings by time-dependent simulations. Finally, we present preliminary results for the case of the (convective) Cahn-Hilliard equation in two spatial dimensions

Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials

We study initial boundary value problems for the convective Cahn-Hilliard equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any $p>0$. In contrast to that, we show that the presence of the convective term u\px u in the Cahn-Hilliard equation prevents blow up at least for $0<p<\frac49$. We also show that the blowing up solutions still exist if $p$ is large enough ($p\ge2$). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered

Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews