Cockburn The investigator studies, both theoretically and computationally, massively parallel, efficient methods for numerically solving problems in which convection plays a major role. Of particular interest are the drift-diffusion and hydrodynamic models for semiconductor devices, and the compressible Navier-Stokes equations (with high Reynolds numbers) for viscous flow. The investigator obtains new a posteriori error estimates for nonlinear scalar conservation laws, independent of the numerical scheme under consideration, upon which adaptivity strategies could be devised. He also studies a priori error estimates that allow a deeper understanding of the numerical methods under consideration. The objective of this research project is to develop, both theoretically and computationally, ways to enhance the speed and the accuracy of massively parallel methods for simulating several problems in which convection plays a central role. Applications to the high-performance computing of simulations of semiconductors and high-speed fluid flow are direct applications of the results of this project. A deep mathematical understanding of the algorithms for this type of problem is essential for efficiently obtaining accurate results, but is unfortunately lacking. The investigator studies theories that provide such an understanding to obtain the desired practical high-performance computing algorithms.
