Solutions and Grids from Genuinely Multidimensional Residual Distribution Schemes

真正多维残差分布方案的解决方案和网格

• 批准号: 0108540
• 负责人: Philip Roe
• 金额: $ 16万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-15 至 2005-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0108540&HistoricalAwards=false
• 关键词: Solutions; Grids; Genuinely; Multidimensional; Residual

The Fluctuation-splitting method for solving first-order partial differential equations is a distributive scheme based on evaluating residuals over piecewise linear simplex elements. The nature of the distribution step matches as closely as possible the local physics, thereby achieving minimal numerical dissipation. It is known how to implement this program is two space dimensions but new physics arises in three dimensions. In supersonic flow this is due to the replacement of characteristic lines by bicharacteristic surfaces, and in both supersonic and subsonic flow by the appearance of helicity, the streamwise component of vorticity, which interacts with the acoustics. We will attempt to design distribution schemes around the system version of the linear wave equation, which is the simplest model problem to exhibit these features, and which is embedded in the Euler equations. Application to the Euler equations themselves will be by means of a local linearization that is well established.Many phenomena in nature and technology are well enough understood that the mathematical equations describing them can be written down, but to actually solve these equations requires huge computer resources. Examples that could be cited include the flow round an aircraft, the evolution of a galaxy, or the future of the weather. Especially in the last decade, the computer resources available for such projects has grown enormously, but the scientific appetite for computing power remains unsatisfied. When new machines become available, they may initially lie idle for some fraction of the day, but are soon fully utilized, typically within a month. To extract the most information from given resources, it is important that attention be paid to the set of instructions (the algorithm) by which the computer processes the data, both from the viewpoint of efficient arithmetic and also with regard to the process that converts the mathematical equations into arithmetical form (the scheme). Numerous schemes of varying sophistication exist, but new ones are continually sought, and this research seeks to extend the scope of one particular scheme that is based on trying to mimic the physics as directly as possible with few arbitrary decisions. It is hoped that the result of this will be to reduce the amount of computer storage space needed to model a given physical situation with given accuracy. Focus will on complex fluid flows in which an important role is played by rotating vortices. Such flows are at present very difficult to compute efficiently, Advances in this area would be especially beneficial to the prediction of flow round an aircraft in its take-off or landing configuration, and to computing the flows around helicopters.

解一阶偏微分方程组的涨落分裂方法是一种基于计算分段线性单纯形元上残差的分布格式。分布步骤的性质尽可能地与局部物理相匹配，从而实现最小的数值耗散。已经知道如何实施这个计划是二维的，但新的物理学出现在三维空间。在超音速流动中，这是由于双特性表面取代了特征线，而在超音速和亚音速流动中，螺旋度的出现是由于涡度的流向分量，它与声学相互作用。我们将尝试围绕线性波动方程的系统版本设计分布方案，线性波动方程是体现这些特征的最简单的模型问题，并且嵌入在欧拉方程中。欧拉方程本身的应用将通过建立良好的局部线性化来实现。自然和技术中的许多现象都被充分理解，描述它们的数学方程可以写下来，但实际求解这些方程需要大量的计算机资源。可以引用的例子包括绕过飞机的流动，星系的演化，或者天气的未来。特别是在过去的十年里，可用于这类项目的计算机资源有了巨大的增长，但科学对计算能力的胃口仍然没有得到满足。当新机器可用时，它们最初可能会闲置一天中的一小部分时间，但很快就会被充分利用，通常在一个月内。为了从给定的资源中提取最多的信息，重要的是要注意计算机处理数据的指令集(算法)，这既是从有效算术的观点来看的，也是关于将数学方程转换为算术形式的过程(方案)的。存在着许多不同复杂程度的方案，但人们不断地寻找新的方案，这项研究试图扩大一个特定方案的范围，该方案的基础是尽可能直接地模拟物理学，而几乎没有武断的决定。希望这样做的结果将是减少以给定精度模拟给定物理情况所需的计算机存储空间量。重点将放在复杂的流体流动上，旋转的漩涡在其中扮演着重要的角色。这种流动目前很难有效地计算，这一领域的进展将特别有利于预测飞机在起飞或降落时的绕流，以及计算直升机的绕流。