关于Cahn-Hilliard-Darcy方程的数值格式研究综述

A Review of Numerical Schemes for the Cahn-Hilliard-Darcy System

摘要: Cahn-Hilliard-Darcy (CHD)系统是模拟多孔介质或Hele-Shaw池中两相不可压缩流动的重要相场模型，在岩溶含水层污染物运移、岩溶油藏开采及生物医学等众多领域具有广泛应用。由于该模型具有强非线性、多物理场耦合及刚度特征，发展高效、稳定且高精度的数值方法一直是计算流体力学领域的核心课题。本文系统综述了CHD系统数值方法的研究进展。首先概述了模型的基本方程与能量耗散律。其次，详细评述了一阶时间精度的数值格式，并分析了其稳定性条件。针对精度和效率的更高需求，重点总结了二阶精度的数值方法，如基于压力修正投影法的解耦格式，以及近年来迅速发展的标量辅助变量(SAV)和不变能量二次化(IEQ)方法。这些方法通过引入辅助变量和精巧的算子分裂技术，成功地将原强耦合非线性系统转化为线性、解耦的可求解形式，并在理论上保证了无条件能量稳定性。最后，本文讨论了不同方法的特点、现存挑战以及未来的研究方向，旨在为CHD系统及相关多相流问题的数值模拟提供参考。

Abstract: The Cahn-Hilliard-Darcy (CHD) system is a critical phase-field model for simulating two-phase incompressible flow in porous media or Hele-Shaw cells, with extensive ap- plications in various fields such as contaminant transport in karst aquifers, karst reser- voir recovery, and biomedicine. Due to the model’s strong nonlinearity, multi-physics coupling, and stiffness characteristics, developing efficient, stable, and high-precision numerical methods has long been a core subject in computational fluid dynamics. This paper provides a systematic review of recent advances in numerical methods for the CHD system. First, the basic governing equations and energy dissipation law of the model are outlined. Subsequently, first-order time-accurate numerical schemes are reviewed in detail, along with an analysis of their stability conditions. In response to higher demands for accuracy and efficiency, second-order accurate numerical methods are highlighted, such as decoupled schemes based on pressure-correction projection methods, and the rapidly developing Scalar Auxiliary Variable (SAV) and Invariant Energy Quadratization (IEQ) approaches. By introducing auxiliary variables and em- ploying clever operator-splitting techniques, these methods successfully transform the original strongly coupled nonlinear system into linear, decoupled, and solvable forms while theoretically guaranteeing unconditional energy stability. Finally, this paper discusses the characteristics of different methods, existing challenges, and potential future research directions, aiming to provide a reference for the numerical simulation of the CHD system and related multiphase flow problems.