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

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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.