Efficient Conservative High-Order Solution-Flux Domain Decomposition Methods and Local Refinements for Flows in Porous Media and Electromagnetics

污染物在多孔介质中的流动模拟对于理解和预测复杂的物理、化学和流动过程，以及地下水模拟和环境保护中的污染防治具有重要作用。对电磁波的传播进行建模对于电磁科学的许多应用也是至关重要的。近年来，电磁学在材料和超材料中的应用迅速增长，包括生物医学成像和处理、射频识别（RFID）、无线电力传输、全息处理、纳米光刻和隐身设备等。描述这些过程的数学模型很复杂，在开发精确的方法以保持重要的自然物理特性方面存在挑战。 在模拟多孔介质中的污染流时，数学模型是耦合偏微分方程（PDE）的非线性系统，涉及具有挑战性的特征，如传输优势，移动陡峭的前沿，非线性，吸附，界面和异质性。这些问题的解决推动了高效、准确和保守的区域分解方法的发展和分析，这些方法用于并行计算中的长期、大规模场预测问题。 由于电磁场建模面临着微观尺度、局部解行为、复杂精细结构和长时间响应等挑战，因此发展高效、保守的电磁场局部网格精化方法至关重要。本论文的主要研究内容包括：（1）发展污染多孔介质流动的时间二阶守恒特征区域分解方法;（2）发展污染多孔介质流动的守恒四阶块中心紧致差分通量区域分解方法;（3）发展麦克斯韦方程的能量守恒局部网格四阶S-FDTD方法;（4）发展电磁场的能量守恒曲线协调局部网格细化S-FDTD方法;（5）发展超材料电磁场的全局能量守恒局部网格细化S-FDTD方法。该研究计划将通过建立方法，理论，算法和应用程序的保守域分解和局部细化技术的多孔介质和计算电磁学中的计算流体动力学受益于加拿大。它还将培养学生和博士后，以满足环境和电磁行业以及计算机技术领域对高素质人才日益增长的需求。

Modeling contamination flow in porous media plays a crucial role in understanding and predicting complex physical, chemical and flow processes, as well as in preventing and controlling pollution in groundwater modeling and environmental protection. Modeling the propagation of electromagnetic waves is also critically important for many applications of electromagnetic science. Recent rapid growth in the use of electromagnetics in materials and metamaterials includes biomedical imaging and processing, radio-frequency identification (RFID), wireless power transmission, holographic processing, nanolithography, and cloaking devices, etc. The mathematical models describing these processes are complex and there are challenges in the development of accurate methods that preserve the important natural physical properties.  When modeling contamination flows in porous media, mathematical models are nonlinear systems of coupled partial differential equations (PDEs) that involve challenging features such as transport dominance, moving steep fronts, nonlinearity, adsorption, interface and heterogeneity. Solving these problems is driving the development and analysis of efficient, accurate, and conservative domain decomposition methods for long-term, large-scale field prediction problems in parallel computing.  Since modeling faces with the challenges of micro-scale, local solution behavior, complex and refined structure, and long-time response, it is of utmost importance to develop efficient and conservative local mesh-refined methods for electromagnetics. The proposed research program includes: (1) Develop time second-order conservative characteristic domain decomposition methods for contamination porous media flows; (2) Develop conservative fourth-order block-centered compact-difference solution-flux domain decomposition method for contamination flows; (3) Develop energy-preserving local mesh-refined fourth-order S-FDTD schemes for Maxwell's equations; (4) Develop energy-preserving curvilinear-coordinated local mesh-refined S-FDTD schemes for electromagnetics; and (5) Develop global energy-preserving local mesh-refined S-FDTD schemes for metamaterial electromagnetics. The research program will benefit Canada through the establishment of methods, theories, algorithms and applications of conservative domain decomposition and local refinement techniques for computational fluid dynamics in porous media and computational electromagnetics. It will also train students and postdocs to meet the growing demand for highly qualified personnel in the environmental and electromagnetic industries and in the computing technology.