CAREER: Analysis and Numerics for the Dynamics of Fluids under Magnetic Forces

虽然肉眼看不见，但磁场对自然界和日常生活的许多现象都有影响，比如潮汐、太阳耀斑或电动机。此外，它们还用于操纵工业应用中的材料，如精密传感器、核反应堆的液态金属冷却或磁性药物靶向。这个项目的目标是研究几个涉及磁场强迫流体的数学模型，并开发在计算机上模拟它们的数值算法。这将有利于工程、物理和生物医学科学的实际应用。针对高中，本科和研究生水平的学生的教育组成部分与研究活动相结合。研究计划包括适合研究生和本科生研究的项目。将进行计算数学本科课程的课程开发，并以跨学科数学和工程夏令营的形式向高中生推广。这个项目的目标是对科学和工程领域的三个重要问题进行数学研究。首先是用于液晶显示屏的液晶在磁场作用下的数值模拟。第二部分涉及磁流体动力学湍流的数学方面，以便更深入地了解地球磁场的出现和存在。第三个问题涉及实验数据如何影响数学模型的问题：概率工具将被用于开发可压缩流动应用中的不确定性量化算法。这些问题用非线性混合型偏微分方程系统在数学上进行了描述。这些系统的数学处理需要开发新的分析技术和创新的算法来模拟它们。本文将利用能量估计、补偿紧性和相对熵方法等数学工具对有限差分和有限体积格式进行分析，以证明鲁棒性和收敛性。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。

While invisible to the eye, magnetic fields contribute to many phenomena in nature and aspects of daily life, such as ocean tides, solar flares, or electric motors. Additionally, they are used to manipulate materials for industrial applications like precision sensors, liquid-metal cooling of nuclear reactors, or magnetic drug targeting. The goal of this project is to study several mathematical models involving fluids forced by magnetic fields, and develop numerical algorithms for their simulation on a computer. This will benefit practical applications in engineering and in physical and biomedical sciences. Educational components targeting students at the high school, undergraduate and graduate level, are integrated with the research activities. The research program includes projects suitable for graduate and undergraduate research. Curriculum development for undergraduate courses in computational mathematics, and outreach to high school students in the form of an interdisciplinary math and engineering summer camp, will be undertaken. The objective of this project is the mathematical investigation of three important problems motivated from science and engineering. The first is the numerical simulation of liquid crystals subjected to magnetic fields, which are used in LCD screens. The second deals with mathematical aspects of magnetohydrodynamics turbulence in order to gain more insight into the emergence and existence of the magnetic field of the earth. The third problem concerns the question of how experimental data affects mathematical models: Probabilistic tools will be employed to develop algorithms for uncertainty quantification in compressible flows applications. These problems are described mathematically by nonlinear systems of mixed type partial differential equations (PDEs). The mathematical treatment of these systems requires the development of new analytical techniques and innovative algorithms for their simulation. The finite difference and finite volume schemes constructed in this project will be analyzed with mathematical tools such as energy estimates, compensated compactness and relative entropy methods to prove robustness and convergence.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.