Mathematical Sciences: Numerical Methods for Convection-Dominated Problems

该项目的主要目标是开发， 理论上和计算上，有效的数值 对流占优问题的解法 扩散 这些方程的解通常表现为： 冲击，其准确和有效的计算是困难的。 将特别注意非线性双曲系统， 半导体器件的模型，以及可压缩 粘性流的Navier-Stokes方程。 数值研究 方法将集中在所谓的龙格库塔间断 Galerkin方法是一种可并行化的方法， 有限差分法及其计算单元的技术 剂. 这项工作将更早地扩展到更高的维度 一维问题的结果。 这项研究的应用出现在研究和 设计半导体器件时使用漂移扩散 和流体动力学模型来描述设备，并在解决 来自空气动力学的流动问题。

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.