15 research outputs found
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
On the 3D Cahn-Hilliard equation with inertial term
We study the modified Cahn-Hilliard equation proposed by P. Galenko et al. in order to account for rapid spinodal decomposition in certain glasses. This equation contains, as additional term, the second-order time derivative of the (relative) concentration multiplied by a (small) positive coefficient. Thus, in absence of viscosity effects, we are in presence of a Petrovsky type equation and the solutions do not regularize in finite time. Many results are known in one spatial dimension. However, even in two spatial dimensions, the problem of finding a unique solution satisfying given initial and boundary conditions is far from being trivial. A fairly complete analysis of the 2D case has been recently carried out by M. Grasselli, G. Schimperna and S. Zelik. The 3D case is still rather poorly understood but for the existence of energy bounded solutions. Taking advantage of this fact, A. Segatti has investigated the asymptotic behavior of a generalized dynamical system which can be associated with the equation. Here we take a step further by establishing the existence and uniqueness of a global weak solution, provided that the relaxation parameter is small enough. Moreover, we show a regularity result for the attractor by using a decomposition method and we discuss the existence of an exponential attractor
New Time Differencing Methods for Spectral Methods
A new semi-analytical time differencing is applied to spectral methods for partial differential equations which involve higher spatial derivatives. This is developed in Jung and Nguyen (J Sci Comput (2015) 63:355-373) based on the classical integrating factor (IF) and exponential time differencing (ETD) methods. The basic idea is approximating analytically the stiffness (fast part) by the so-called correctors (see 1.3 below) and numerically the non-stiffness (slow part) by the IF and ETD, etc. It turns out that rapid decay and rapid oscillatory modes in the spectral methods are well approximated by our corrector methods, which in turn provides better accuracy in the numerical schemes presented in the text. We investigate some nonlinear problems with a quadratic nonlinear term, which makes all Fourier modes interact with each other. We construct the correctors recursively to accurately capture the stiffness in the mode interactions. Polynomial or other types of nonlinear interactions can be tackled in a similar fashion.close