The main objective of this project is to develop, theoretically as well as computationally, efficient numerical methods for solving problems in which convection dominates diffusion. The solutions of such equations typically exhibit shocks, whose accurate and efficient computation is difficult. Special attention will be paid to nonlinear hyperbolic systems, to models of semiconductor devices, and to compressible Navier-Stokes equations for viscous flows. Study of numerical methods will center on the so-called Runge-Kutta Discontinuous Galerkin method, a parallelizable method that adopts from several finite-difference methods their techniques for evaluating cell fluxes. This work will extend to higher dimensions earlier results on one-dimensional problems. Applications of this research arise in studying and designing semiconductor devices when using the drift-diffusion and hydrodynamic models to describe the devices, and in solving flow problems that come from aerodynamics.
