As computer simulations are playing an ever increasing role in many branches of science and engineering and are rapidly replacing much of the expensive prototyping and testing phases in manufacturing and in science explorations, fast and robust numerical methods are becoming an indispensable tool for many scientists and engineers. The focus of this project is to design fast and robust spectral-Galerkin methods for solving a large class of partial differential equations, and apply them to investigate several important problems of current interest. The proposed research will result in fast and accurate numerical algorithms for a class of partial differential equations with applications in acoustic and electromagnetic scattering, fluid dynamics and materials science.
It is expected that the proposed numerical algorithms will allow simulations of three-dimensional time-dependent problems within a reasonable turn-over time, and provide capabilities to numerically attack challenging problems arising from emerging engineering and scientific applications. In particular, the proposed numerical simulations will contribute towards better understandings of the complex physical and mathematical problems, and provide valuable information for the design of advanced materials and on the rheological and hydrodynamic properties of complex fluids. The proposed research will also generate opportunities for undergraduate research projects such as software development, parallel computation and numerical simulation.
