Novel Computational Approaches to the Transport Equation

这个项目的方向是线性粒子输运方程的计算解，通常被简单地称为“输运方程”。“从历史上看，这个方程对任何时候最先进的计算机硬件都是一个重大挑战。更具体地说，该项目是针对这个方程的空间近似，因为在这个领域的新方法和更好的基本理解似乎是必不可少的，以便在复杂的材料安排中解决这个方程的一些新应用（例如，为环境恢复识别地下废物位置）。本文对粒子输运中一维近似的抽象模型（CLOF方法）进行了扩展和应用，主要包括：开发了一类一维空间近似的大规模并行代码；该模型被扩展到多个（两个或更多）空间维度；a，允许计算确定任意二维CLOF的精度顺序；对具有满足扩散极限的理想性能的CLOF方法进行了表征。角平衡方法是一种新颖的空间近似方法，似乎具有多维计算非常需要的几个特性，它正在扩展到三维设置，最初用于平行六面体细胞，随后用于由一般（凸）多面体细胞组成的网格。

This project is directed toward computational solution of the linear particle transport equation, which usually is termed simply the ``transport equation.'' This equation historically has provided a major challenge to the state-of-the-art computer hardware available at any given time. More specifically, the project is directed toward spatial approximations to this equation, because new approaches and better basic understanding in this area seem essential to permit the solution of the transport equation in the complex material arrangements underlying some of the emerging applications of this equation (e.g., identification of underground waste locations for environmental restoration). A certain abstract model (CLOF methods) of one-dimensional approximations in particle transport is being extended and applied, as follows: a massively parallel code is being developed for a general class of one-dimensional spatial approximations; the model is being extended to multiple (two or more) spatial dimensions; a that permits computational determination of the order of accuracy of an arbitrary two-dimensional CLOF is being developed; the CLOF methods that have the very desirable property of satisfying the diffusion limit are being characterized. The corner-balance method, which is a novel spatial approximation that seems to enjoy several properties that are highly desirable for multidimensional calculations, is being extended to the three-dimensional setting, initially for parallelepiped cells, and subsequently for meshes composed of cells that are general (convex) polyhedra.