Continuation for thin film hydrodynamics and related scalar problems

TL;DR

This chapter demonstrates how continuation techniques can analyze nonlinear scalar kinetic equations, including variational and nonvariational types, to find steady and periodic solutions and bifurcation diagrams in thin film and related problems.

Contribution

It introduces a systematic approach to applying numerical continuation to a broad class of scalar equations, including variational and nonvariational, with practical implementation insights.

Findings

Successfully applied continuation to variational equations like Allen-Cahn and Cahn-Hilliard.

Extended analysis to nonvariational equations such as Kuramoto-Sivashinsky and thin-film models.

Provided detailed guidance on boundary conditions and implementation for bifurcation analysis.

Abstract

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.