Design and Analysis of Highly Efficient Algorithms for Complex Nonlinear Systems

该项目专注于科学和工程应用中复杂非线性系统的创新、结构保留算法的开发和分析。该项目不仅将为当前关注的一大类复杂非线性系统提供高效的数值算法，而且通过数值模拟有助于更好地理解材料科学、流体力学和其他相关领域的一些基本问题。该项目还将为参与的学生提供机会，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。该项目每年将资助一名研究生。具有耗散或守恒性质的复杂非线性系统在现实世界现象的建模中无处不在。如何构建有效而准确的数值格式，以保持重要的耗散或守恒特性，并在某些情况下保持物理变量的正性，是一个重大挑战。该项目将通过扩展灵活且鲁棒的标量辅助变量（SAV）方法来克服这些挑战，该方法已被证明对梯度流非常有效。特别是，将扩展SAV方法以处理其他困难，例如(i)具有物理限制的系统，例如质量和/或表面积保护；（ii）高度各向异性系统和具有非线性运动的系统；（iii）具有保正性原则或最大值原则的系统；（iv）系统耦合梯度流动与其他守恒定律，以及(v)优化。提出的方法将导致数值预测工具，扩展数学和实验分析的适用性，并有助于更好地理解复杂的物理系统。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project focuses on the development and analysis of innovative, structure preserving algorithms for complex nonlinear systems in science and engineering applications. This project will not only lead to efficient numerical algorithms for a large class of complex nonlinear systems of current interests, but also contribute through numerical simulation to a better understanding of some fundamental issues in materials science, fluid mechanics and other related fields. This project will also provide opportunities for the involved students to learn critical skills of computational and applied mathematics and to develop state-of-the-art numerical algorithms for science and engineering applications. This project will support one graduate student per year.Complex nonlinear systems that possess dissipative or conservation properties are ubiquitous in modeling of real-world phenomena. It is a major challenge to construct efficient and accurate numerical schemes that can preserve important dissipative or conservation properties, and in certain cases, positivity of physical variables. This project will overcome these challenges by extending the flexible and robust scalar auxiliary variable or SAV approach, which has proven to be highly effective for gradient flows. In particular, the SAV approach will be extended to deal with additional difficulties such as those in (i) systems with physical constraints such as mass and/or surface area conservations; (ii) highly anisotropic systems and systems with nonlinear mobilities; (iii) systems with positivity preserving or maximum principles; (iv) systems coupling gradient flows with other conservation laws, and (v) optimizations. The proposed methodology will lead to numerical predictive tools that extend the applicability of mathematical and experimental analysis, and contribute to better understanding of complex physical systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.