[Paper Review] Convergence of a fully discrete variational scheme for a thin-film equation

Horst Osberger, Daniel Matthes|arXiv (Cornell University)|Sep 4, 2015

Lattice Boltzmann Simulation Studies4 citations

TL;DR

This paper presents a fully discrete Lagrangian finite-volume scheme for the one-dimensional Hele-Shaw flow (a fourth-order thin-film equation with linear mobility), which preserves mass, non-negativity, and dissipates both Dirichlet energy and logarithmic entropy. The scheme is proven to converge to a weak solution in the discrete-to-continuous limit, leveraging the interplay between energy and entropy dissipation without requiring a CFL condition due to its time-implicit formulation.

ABSTRACT

This paper is concerned with a rigorous convergence analysis of a fully discrete Lagrangian scheme for the Hele-Shaw flow, which is the fourth order thin-film equation with linear mobility in one space dimension. The discretization is based on the equation's gradient flow structure in the $L^2$-Wasserstein metric. Apart from its Lagrangian character --- which guarantees positivity and mass conservation --- the main feature of our discretization is that it dissipates both the Dirichlet energy and the logarithmic entropy. The interplay between these two dissipations paves the way to proving convergence of the discrete approximations to a weak solution in the discrete-to-continuous limit. Thanks to the time-implicit character of the scheme, no CFL-type condition is needed. Numerical experiments illustrate the practicability of the scheme.

Motivation & Objective

• To develop a fully discrete numerical scheme for the one-dimensional Hele-Shaw flow that preserves key physical and structural properties of the continuous equation.
• To ensure convergence of discrete solutions to a weak solution in the limit of vanishing mesh size and time step, without requiring a CFL condition.
• To simultaneously preserve both the Dirichlet energy and logarithmic entropy as Lyapunov functionals in the discrete setting.
• To provide a robust numerical method that maintains positivity and mass conservation, especially for solutions with compact support or near-zero initial data.

Proposed method

• The scheme is based on the gradient flow structure of the Hele-Shaw equation in the $L^2$-Wasserstein metric, leveraging the variational formulation of the equation.
• A Lagrangian finite-volume discretization is employed, where particles represent mass points and their positions evolve in time via a variational time discretization.
• The discrete energy and entropy are explicitly shown to decrease over time, ensuring stability and enabling convergence analysis.
• The time-implicit nature of the scheme avoids the need for a CFL-type stability condition, allowing large time steps.
• The method uses a discrete Wasserstein metric to define the variational time stepping, ensuring geometric consistency with the continuous gradient flow structure.
• The convergence proof relies on the interplay between discrete energy and entropy dissipation, combined with compactness arguments and weak* convergence in $L^ ho$ spaces.

Experimental results

Research questions

• RQ1 Can a fully discrete, structure-preserving scheme be constructed for the one-dimensional Hele-Shaw equation that preserves mass and non-negativity?
• RQ2 Does a discrete version of the logarithmic entropy dissipation hold in the numerical scheme, and can it be used to control regularity?
• RQ3 Can convergence of the discrete solution to a weak solution be rigorously proven without a CFL condition?
• RQ4 How does the scheme perform numerically in comparison to standard finite-difference methods, especially for initial data with small or zero values?

Key findings

• The proposed scheme is proven to converge to a weak solution of the Hele-Shaw equation in the limit of vanishing mesh size and time step, under appropriate assumptions on the initial data.
• The scheme preserves both the Dirichlet energy and the logarithmic entropy as Lyapunov functionals, with discrete dissipation rates matching the continuous ones.
• The time-implicit formulation eliminates the need for a CFL condition, enabling stable and efficient time stepping even with large time steps.
• Numerical experiments confirm that the scheme maintains positivity and mass conservation, even for initial data with zero or near-zero values, where standard finite-difference schemes fail.
• The scheme outperforms standard finite-difference methods in terms of physical consistency, especially when high accuracy in mass and positivity is required.
• The convergence rate of the scheme is observed to be optimal in $L^p$-norms, with the error decreasing as expected with mesh refinement.