# Numerical Approximations for a three components Cahn-Hilliard phase-field Model based on the Invariant Energy Quadratization method

## TL;DR

This paper introduces energy-stable, high-order numerical schemes for a three-component Cahn-Hilliard phase-field model using the Invariant Energy Quadratization method, ensuring efficiency and stability in simulations.

## Contribution

Develops first and second order energy-stable temporal schemes for the three-component Cahn-Hilliard model using IEQ, with rigorous stability proof and practical simulations.

## Key findings

- Schemes are unconditionally energy stable.
- Numerical simulations confirm stability and accuracy.
- Linear systems are well-posed and efficiently solvable.

## Abstract

How to develop efficient numerical schemes while preserving the energy stability at the discrete level is a challenging issue for the three component Cahn-Hilliard phase-field model. In this paper, we develop first and second order temporal approximation schemes based on the "Invariant Energy Quadratization" approach, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to a well-posed linear system with the symmetric positive definite operator to be solved at each time step. We rigorously prove that the proposed schemes are unconditionally energy stable. Various 2D and 3D numerical simulations are presented to demonstrate the stability and the accuracy of the schemes.