Structure-preserving approximation for non-isothermal phase-field models in melt flow

TL;DR

This paper introduces a finite-element scheme for non-isothermal phase-field models that preserves entropy and energy properties, with convergence tests and practical examples demonstrating its effectiveness.

Contribution

It develops a conforming finite-element method for the coupled non-isothermal Allen-Cahn-Navier-Stokes system that exactly preserves entropy production and nearly conserves energy.

Findings

The scheme preserves entropy production exactly.

It maintains total energy conservation up to numerical dissipation.

Convergence tests confirm the scheme's accuracy.

Abstract

This work presents a conforming finite-element scheme for the non-isothermal Allen-Cahn-Navier-Stokes system, incorporating periodic, closed, and thermal boundary conditions. The system comprises the incompressible Navier-Stokes equations coupled with the non-isothermal Allen-Cahn equation, which includes a non-conserved phase-field equation and a temperature equation. The proposed numerical scheme preserves entropy production exactly and maintains total energy conservation up to a negative numerical dissipation. Convergence tests in both space and time are conducted, and representative examples are provided to demonstrate the scheme's effectiveness.