Development of High-Order Conservative Numerical Methods for Electromagnetics in Metamaterials and Transport Flows in Environment

该项目的目标是研究、发展和分析超材料电磁学和环境中输运流问题的保守数值方法。 近年来，建模已被认为是电磁学中的一项关键技术。异向介质以其超常的电磁特性在许多应用中发挥着重要的作用。由于超材料的长时间电磁响应和复杂的尺度结构，超材料的电磁学计算面临着巨大的挑战。计算多组分输运问题在大气环境和地下水污染模拟中具有重要意义。描述这一复杂过程的数学模型是非线性偏微分方程组，具有输运优势、移动陡锋或界面、湍流、非线性、多尺度、场尺度大、预报时间长等特点。这些问题的解决一直是发展大规模环境输运流和电磁超材料问题的有效数值方法和计算工具的推动力，其中三维高阶守恒数值方法的发展引起了人们极大的兴趣。 该项目包括：（1）发展和分析三维异向介质电磁学的高阶能量守恒方法;（2）发展和分析三维气溶胶输运问题的高阶质量守恒特征方法;（3）发展和分析环境中多组分气溶胶输运的质量守恒分裂区域分解方法;（4）多孔介质中多组分污染流的控制研究进展。

The objective of the proposed program is to study, develop and analyze the conservative numerical methods for electromagnetics in metamaterials and transport flow problems in environment. Modelling has been recently recognized as a crucial technique in electromagnetics. For their supernormal electromagnetic features, metamaterials play a very important role in many applications. Due to the long-time electromagnetic responses and the complicated scale structures, computations for electromagnetics in metamaterials are confronted with great challenges. Computing multi-component transport problems is of significant importance in the atmospheric environment and groundwater contamination modelling. The mathematical models describing the complex processes are the nonlinear partial differential equations, which are characterized by transport dominance, moving steep front or interface, turbulence, nonlinearity, multi-scales, enormous size of field scale and long time prediction. Solving these problems has been a driving force in developing efficient numerical methods and computational tools for large scale transport flows in environment and for electromagnetics in metamaterials, in which there are great interests in the development of high-order conservative numerical methods in three dimensions. The program includes (1) Develop and analyze high-order energy conservative methods for electromagnetics in three dimensional metamaterials; (2) Develop and analyze high-order mass-preserving characteristic methods for aerosol transport problems in three dimensions; (3) Develop and analyze the mass conservative splitting domain decomposition method for multi-component aerosol transports in environment; (4) Development of controlling multicomponent pollution flows in porous media.