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

虽然肉眼看不到磁场，但磁场对自然界的许多现象和日常生活的方方面面都有贡献，如海潮、太阳耀斑或电动马达。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作用下的流体的数学模型，并为它们在计算机上的模拟开发数值算法。这将有利于在工程以及物理和生物医学科学中的实际应用。针对高中生、本科生和研究生的教育部分与研究活动相结合。该研究计划包括适合研究生和本科生研究的项目。将为计算数学本科课程开发课程，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中的三个重要问题进行数学研究。第一个是液晶在磁场作用下的数值模拟，用于LCD屏幕。第二部分涉及磁流体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：将使用概率工具来开发可压缩流动应用中不确定性量化的算法。这些问题可以用混合型偏微分方程组(PDE)来描述。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进行模拟。在这个项目中构建的有限差分和有限体积格式将用数学工具进行分析，如能量估计、补偿紧致性和相对熵方法，以证明稳健性和收敛。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.