Recovery of high frequency wave fields, kinetic theory of photons and entropy satisfying methods

该研究项目开发数学工具和数值算法，以研究由不同应用中出现的一组偏微分方程控制的多尺度问题。提出的工作包括三个项目：恢复由各种波动方程控制的高频波场，包括双曲方程（如声波方程）、色散方程（如薛定谔方程）和时间无关的亥姆霍兹方程；能量输运中光子动力学理论的发展设计了满足不连续Galerkin方法的动力学Fokker-Planck方程的熵，重点是聚合物流体中的动力学模型和生物学中的集体运动。每一个拟议的项目都有可能对具有根本性和技术重要性的问题产生重大影响。高频波场的恢复是高频波传播中的一个基本问题，对其研究可以加深对各种应用中的高频波动力学的理解。对光子凝聚现象的研究将为设计新型光源提供可能的方法。满足熵的方法对于在长时间数值模拟中捕捉正确的物理现象是很重要的。这种方法的设计有可能阐明物理过程的许多方面。该理论将应用于确定的实际应用并由其驱动。

This research project develops mathematical tools and numerical algorithms to study multiscale problems governed by a selected set of partial differential equations arising in diverse applications. The proposed work consists of three projects: recovery of high frequency wave fields governed by a variety of wave equations, including hyperbolic equations such as the acoustic wave equation, dispersive equations such as the Schroedinger equation, and the time independent Helmholtz equation; development of kinetic theory of photons in the transport of energy; and design of entropy satisfying discontinuous Galerkin methods for kinetic Fokker-Planck equations, with emphasis on kinetic models arising in polymeric fluids and collective motions in biology. Each of the proposed projects has the potential to have a significant impact on problems that are both fundamental and technologically important. Recovery of high frequency wave fields is a fundamental problem in high frequency wave propagation, the study of which can provide a deeper understanding of high frequency wave dynamics occurring in various applications. The study of condensation of photons would result in methods which may potentially be suitable for designing novel light sources. Entropy satisfying methods are important in capturing the right physics in long time numerical simulations. The design of such methods has the potential to elucidate many aspects of the physical process. The theory will be applied to and driven by identified practical applications.