In this project, research in the algorithm design and analysis of high order numerical methods, including the finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin finite element methods, for solving hyperbolic and other convection dominated partial differential equations, will be carried out. While the emphasis of this project is on algorithm design and analysis, close attention will be paid to efficient parallel implementation and applications. The intellectual merit of the proposed activity lies in its comprehensive coverage of algorithm development, analysis, implementation and applications. Problems in applications motivate the design of new algorithms or new features in existing algorithms; mathematics tools are used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists enable the efficient application of these new algorithms or new featuresin existing algorithms.
The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications with involving convection dominated partial differential equations, in adaptive, multiscale and uncertain environments. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications. The application areas include (but are not limited to) computational fluid dynamics, traffic flow problems, semiconductor device simulations, and computational biology. Graduate students will be involved in this project, and will get training in performing mathematics research on problems closely related to applications. Special attention will be paid to the recruitment and training of Ph.D. students from under-represented groups including women.