Collaborative Research: Efficient, Stable and Accurate Numerical Algorithms for a class of Gradient Flow Systems and their Applications

本项目致力于开发高效、稳定和精确的数值算法，用于模拟现实世界现象中普遍存在的梯度流系统。期望这一研究不仅能为一类当前感兴趣的问题提供高效的数值算法，而且有助于通过数值模拟更好地理解材料科学、生物技术等相关领域的一些基本问题。这个项目还将为相关的研究生和本科生提供机会，学习计算和应用数学的关键技能，并为科学和工程应用开发最先进的数值算法。梯度流系统的自由能通常由各种复数格式的非线性势能组成，这给构造高效、准确的时间离散格式带来了巨大的挑战。该项目旨在通过使用灵活而稳健的IEQ方法来克服这一挑战，该方法使人们能够为大类梯度流系统开发时间离散化方案。这项建议的目标有三个方面：(I)开发一个统一的时间推进格式的数值框架，用于求解具有高度非线性的一般梯度流模型；(Ii)为一些具有挑战性的当前感兴趣的梯度流模型(例如，非线性耦合多变量模型、非局部模型、各向异性模型、遵循各种物理原理的非线性耦合系统、基于张量的液晶模型)开发有效的数值格式；(Iii)利用开发的预测数值工具研究衬底上的粘弹性液滴和活跃的液晶液滴的一些基本问题。所提出的方案将产生数值预测工具，扩展数学和实验分析的能力，并有助于更好地理解与多相复杂流体有关的一些紧迫的科学和工程问题。

This project focuses on the development of efficient, stable and accurate numerical algorithms for gradient flow systems which are ubiquitous in modeling of real-world phenomena. It is expected that the research will not only lead to efficient numerical algorithms for a class of problems of current interests, but also contribute to better understandings of some fundamental issues in materials science, biotechnology, and other related fields through numerical simulations. This project will also provide opportunities for the involved graduate and undergraduate students to learn critical skills of computational and applied mathematics and to develop state-of-the-art numerical algorithms for science and engineering applications. The free energies of gradient flow systems usually consist of various nonlinear potentials formulated in diverse complex formats which present a major challenge in the construction of efficient and accurate time discretization schemes. The project aims at overcoming this challenge by using a flexible and robust IEQ approach that enables one to develop time discretization schemes for a large class of gradient flow systems. The goals of this proposal are three folds: (i) to develop a unified numerical framework of time-marching schemes for solving general gradient flow models with high nonlinearity; (ii) to develop efficient numerical schemes for a number of challenging gradient flow models of current interests (e.g., nonlinear coupled multivariable models, nonlocal models, anisotropic models, nonlinear coupled systems that follow various physical principles, tensor based liquid crystal models); (iii) to investigate some fundamental issues of viscoelastic drops on substrates and active liquid crystal droplets using the developed predictive numerical tools. The proposed schemes will lead to numerical predictive tools that extend the capability of mathematical and experimental analysis, and contribute to better understanding of some pressing science and engineering issues related to multi-phase complex fluids.