Many physical, biological, and engineering problems involve linear or nonlinear wave phenomena in adaptive, multiscale, and uncertain environments. The focus of this proposal is to design, analyze and apply high-order accurate and highly efficient numerical algorithms, including high-order weighted essentially non-oscillatory methods, high-order discontinuous Galerkin finite element methods, and spectral methods, for effective simulations of such wave phenomena using computers. Mathematical tools will be used to guide the design of such algorithms so that they will be able to produce reliable and accurate results for such complicated wave phenomena with a high speed and hence a fast turnaround time. This will in turn allow a deeper understanding of the physical and biological phenomena and also to help in many engineering designs, such as the design of aircrafts and semiconductor devices. In this project, problems in applications will motivate the design of new algorithms or new features in existing algorithms; mathematical tools will be used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues for the computation on massively parallel computers will be addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists will enable the efficient application of these new algorithms or new features in existing algorithms. The training of young researchers in this area will also be an important component of this project.
