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

该项目的重点是开发高效、稳定和精确的梯度流系统数值算法，这些算法在现实世界现象的建模中无处不在。期望本研究不仅能为当前关注的一类问题提供有效的数值算法，而且通过数值模拟有助于更好地理解材料科学、生物技术和其他相关领域的一些基本问题。该项目还将为参与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杂格式的非线性势组成，这对构建高效、准确的时间离散化方案提出了重大挑战。该项目旨在通过使用灵活而强大的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方案。本提案的目标有三个方面：(i)为求解具有高非线性的一般梯度流模型建立统一的时间推进格式数值框架；（ii）为当前感兴趣的一些具有挑战性的梯度流模型（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）开发有效的数值方案；(3)利用所开发的预测数值工具研究了基材粘弹性液滴和活性液晶液滴的一些基本问题。所提出的方案将导致数值预测工具，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相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.