A new interface capturing method for Allen-Cahn type equations based on a flow dynamic approach in Lagrangian coordinates, I. One-dimensional case

TL;DR

This paper introduces a Lagrangian flow dynamic approach based on EnVarA to efficiently capture thin interfaces in Allen-Cahn equations, especially in one-dimensional cases, with fewer points needed for accurate resolution.

Contribution

It presents a novel Lagrangian method using EnVarA for Allen-Cahn equations, improving interface capturing efficiency over traditional Eulerian numerical methods.

Findings

Fewer points are sufficient to resolve very thin interfaces.

The proposed schemes are energy dissipative and respect variational structures.

Numerical results demonstrate the effectiveness of the Lagrangian approach.

Abstract

We develop a new Lagrangian approach --- flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by Rayleigh and Onsager \cite{onsager1931reciprocal,onsager1931reciprocal2}. Its main advantage, comparing with numerical methods in Eulerian coordinates, is that thin interfaces can be effectively captured with few points in the Lagrangian coordinate. We concentrate in the one-dimensional case and construct numerical schemes for the trajectory equation in Lagrangian coordinate that obey the variational structures, and as a consequence, are energy dissipative. Ample numerical results are provided to show that only a fewer points are enough to resolve very thin interfaces by using our Lagrangian approach.