Dual formulations of geometric curvature flows and their discretizations

TL;DR

This paper introduces dual formulations for geometric curvature flows that enable the development of linearly implicit, energy-stable numerical schemes, improving surface evolution simulations with better mesh quality control.

Contribution

The authors develop a novel dual formulation framework that explicitly reveals energy-dissipation structures, facilitating the design of stable, linearly implicit schemes for curvature-driven surface evolution.

Findings

Schemes are proven to be energy-stable and convergent.

The dual-MDR scheme effectively maintains mesh quality during evolution.

Numerical experiments validate the advantages of the proposed methods.

Abstract

We propose new formulations of geometric curvature flows -- referred to as \emph{dual formulations} -- that are equivalent to the original formulations but provide a novel framework for constructing linearly implicit and energy-stable schemes for curvature-driven surface evolution, including mean curvature flow, surface diffusion, and solid-state dewetting on a substrate with a moving contact line. The dual formulations are derived by introducing, at the continuous level, an additional unknown in the form of a dual multiplier. This augmentation does not alter the continuous dynamics but makes the underlying energy-dissipation structure explicit and, in turn, enables a systematic design of linearly implicit discretizations that inherit energy stability. A key feature of this framework is that it accommodates a broad class of artificial tangential motions which can be used to maintain good mesh quality of the computed surfaces. As an illustration, we combine the framework with the minimal-deformation-rate (MDR) tangential motion, leading to what we call the \emph{dual-MDR} scheme. The resulting method is linearly implicit and energy-stable, while retaining the MDR tangential motion to maintain good mesh quality. Extensive numerical experiments demonstrate the convergence of the proposed schemes, their structure-preserving properties, and advantages on representative benchmark problems.