School of Mathematical Sciences, Shenzhen University  

Liyuan Scholar Colloquium No. 144  

Lecture Title: Efficient Numerical Algorithms for Phase-Field Equations  

Lecturer: Professor Qiao Zhonghua (The Hong Kong Polytechnic University)  

Lecture Time: October 16, 2025, 10:30-11:30 (morning)  

Venue: Classroom 1, Huixing Building, Yuehai Campus, Shenzhen University  

Abstract:  

This lecture will systematically elaborate on the latest progress of the phase-field method in the fields of modeling, numerical simulation, and theoretical analysis. As an important mathematical modeling and numerical simulation technique, the phase-field method plays a key role in the study of phase transition processes and interface dynamics in materials science, fluid mechanics, and life sciences. This method describes the interface evolution characteristics between different phases or components by introducing a continuously distributed phase-field variable. This variable maintains a constant value in a single phase region and exhibits continuous and smooth variation in the interface transition region, thereby realizing a continuous description of the traditional sharp interface.  

In terms of numerical solution, the main challenges of phase-field equations stem from their nonlinear nature and the strict time-step constraints caused by high-order differential terms. Meanwhile, long-time scale simulations are often required to obtain steady-state solutions. Notably, phase-field models generally satisfy a nonlinear stability relationship called "energy stability"—that is, the system’s free energy functional decreases monotonically over time. In recent years, the design of numerical schemes that can preserve energy stability has become a research focus. Such algorithms can not only use larger time steps to ensure computational efficiency but also maintain the accuracy of numerical solutions.  

For several classical phase-field models, we will focus on introducing several widely used efficient numerical schemes, with special emphasis on the adaptive time-step strategy based on the instantaneous rate of change of the energy functional. By dynamically adjusting the computational time step, this strategy significantly improves computational efficiency while ensuring the accuracy and stability of numerical solutions, providing an effective solution for large-scale and long-time scale simulations.  

Lecturer’s Biography:  

Professor Qiao Zhonghua obtained his doctoral degree from Hong Kong Baptist University in 2006. He is currently a Chair Professor in the Department of Applied Mathematics at The Hong Kong Polytechnic University, the Hong Kong-side Deputy Director of the Joint Laboratory of Applied Mathematics (Academy of Mathematics and Systems Science, Chinese Academy of Sciences - The Hong Kong Polytechnic University), a Director of the Chinese Society for Industrial and Applied Mathematics, and a Vice Chairman of the Computational Mathematics Branch of the Chinese Mathematical Society.  

Professor Qiao’s main research focuses on the design and analysis of algorithms for numerical differential equations. In recent years, his research work has concentrated on the numerical simulation of phase-field equations and efficient algorithms for computational fluid dynamics. He received the Outstanding Young Researcher Award (2013-2014) issued by the Research Grants Council of Hong Kong, the Young Scholar Award of the Hong Kong Mathematical Society in 2018, and the Research Fellow Award of the Research Grants Council of Hong Kong in 2020.  

Faculty and students are welcome to attend!  

School of Mathematical Sciences  

October 9, 2025