# The fast scalar auxiliary variable approach with unconditional energy stability for nonlocal Cahn-Hilliard equation

## TL;DR

This paper introduces a fast, energy-stable scalar auxiliary variable method for the nonlocal Cahn-Hilliard equation, improving computational efficiency and stability for modeling phase transitions with nonlocal effects.

## Contribution

It develops an unconditionally energy-stable scheme for the nonlocal Cahn-Hilliard equation and proposes a fast computational method leveraging BTTB matrix structure.

## Key findings

- Proved unconditional energy stability of the scheme.
- Reduced computational work using BTTB matrix structure.
- Numerical simulations confirm accuracy and efficiency.

## Abstract

Comparing with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal Cahn-Hilliard equations equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. In this paper, we construct an accurate and efficient scalar auxiliary variable approach for the nonlocal Cahn-Hilliard equation with general nonlinear potential. The first contribution is that we have proved the unconditional energy stability for nonlocal Cahn-Hilliard model and its semi-discrete schemes carefully and rigorously. Secondly, what we need to focus on is that the non-locality of the nonlocal diffusion term will lead the stiffness matrix to be almost full matrix which generates huge computational work and memory requirement. For spatial discretizaion by finite difference method, we find that the discretizaition for nonlocal operator will lead to a block-Toeplitz-Toeplitz-block (BTTB) matrix by applying four transformation operators. Based on this special structure, we present a fast procedure to reduce the computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.