Mathematical Sciences: Numerical Methods for Convection-Dominated Problems

该项目的主要目标是在理论上和计算上发展有效的数值方法来解决对流主导扩散的问题。这类方程的解通常表现为激波，其准确和有效的计算是困难的。将特别关注非线性双曲型系统，半导体器件模型，以及粘性流动的可压缩Navier-Stokes方程。数值方法的研究将集中在所谓的Runge-Kutta间断Galerkin方法上，这是一种可并行的方法，它采用了几种有限差分方法中的计算单元通量的技术。这项工作将扩展到一维问题的早期结果的高维。当使用漂移-扩散模型和流体动力学模型来描述半导体器件时，该研究在研究和设计半导体器件以及解决来自空气动力学的流动问题中得到了应用。

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.