# Arbitrarily High-order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models

## TL;DR

This paper develops high-order, unconditionally energy stable numerical schemes for gradient flow models using the energy quadratization method, enabling accurate and stable simulations of thermodynamically consistent PDEs.

## Contribution

It introduces two novel classes of energy stable schemes for gradient flows, employing prediction-correction and Gaussian collocation methods, with rigorous stability proofs and broad applicability.

## Key findings

- Schemes are unconditionally energy stable.
- Numerical experiments confirm high accuracy.
- Methods are generalizable to other PDE models.

## Abstract

We present a systematical approach to developing arbitrarily high order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization (EQ) method, We formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models.