Mathematical Sciences: Numerical Methods for Convection Dominated Problems

科克伯恩研究人员在理论上和计算上都研究了大规模并行的高效方法，用于数值解决对流起主要作用的问题。特别值得关注的是半导体器件的漂移-扩散和流体动力学模型，以及粘性流动的可压缩Navier-Stokes方程(具有高雷诺数)。研究人员得到了与所考虑的数值格式无关的非线性标量守恒律的新的后验误差估计，据此可以设计自适应策略。他还研究了先验误差估计，以加深对所考虑的数值方法的理解。该研究项目的目的是从理论和计算两个方面发展提高大规模并行方法的速度和精度的方法，以模拟几个以对流为中心的问题。在半导体模拟和高速流体流动的高性能计算中的应用是本项目成果的直接应用。对这类问题的算法有深刻的数学理解，对于有效地获得准确的结果是必不可少的，但不幸的是，这方面的理解是不够的。研究人员研究提供这种理解的理论，以获得所需的实用高性能计算算法。

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.