In a broad sense, my interests encompass scientific computing, numerical analysis and analysis of partial differential equations (PDEs), where I contribute to the mathematical foundation of numerical algorithms and develop high fidelity computational tools to solve PDEs, modeling physical and engineering processes. My work thus far has focused on PDEs describing propagating waves and geophysical/atmospheric fluid flow.Waves appear in many shapes and forms, and have a great impact on our daily lives. Examples include the use of elastic waves to image natural resources in the Earth’s subsurface, to detect cracks and faults in structures, to monitor underground explosions, to investigate strong ground motions from earthquakes, and to access risks and hazards from tsunami, as well as numerical solutions of Einstein’s equations which describe processes such as binary black holes, neutron star collision and gravitational waves. Other application areas involve electromagnetic waves and acoustic waves vis-a-vis wireless communication and ground penetrating radar technologies, as well as sonar and aero-acoustics to name a few.Accurate atmospheric modeling is crucial to understanding, predicting, and responding to the effects of anthropengic warming. While the effects on global mean temperatures are well understood, many important regional and high- resolution factors, and changes in climate extremes, are uncertain as they depend on small scale features currently unre- solved by global climate models. This creates a pressing need for high-order accurate and efficient numerical methods to improve atmospheric models to the point where they can resolve these small scale features.Waves and the atmosphere can be accurately described by linear and nonlinear PDEs. But available analytical tools are often limited to simple model problems, and can not be used to solve complex mathematical models of most real-world problems. Furthermore, for nonlinear PDEs, in general it is not even known whether, and under what conditions, the equations possess solutions. It becomes necessary to compute approximate solutions using advanced numerical methods and supercomputers. However, the large range of scales present in realistic modeling applications make their numeri- cal solution computationally expensive, which motivates the development of highly efficient, scalable, and high-order accurate numerical methods which can efficiently leverage the parallelism of modern supercomputers.