Solutions and Grids from Genuinely Multidimensional Residual Distribution Schemes

求解一阶偏微分方程的波动分裂方法是一种基于求分段线性单纯形元残差的分配格式。分布阶跃的性质尽可能地与局部物理相匹配，从而达到最小的数值耗散。我们知道如何在二维空间中实现这个程序，但是在三维空间中出现了新的物理。在超音速流动中，这是由于双特征面取代了特征线，在超音速和亚音速流动中，出现了螺旋度，即涡度的流向成分，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，这是展示这些特征的最简单的模型问题，并且嵌入在欧拉方程中。对欧拉方程本身的应用将是通过建立良好的局部线性化。自然界和技术中的许多现象都已经被很好地理解，描述它们的数学方程可以被写下来，但要真正解决这些方程需要大量的计算机资源。可以引用的例子包括飞机周围的气流、星系的演化或天气的未来。特别是在过去十年中，可用于此类项目的计算机资源已经大大增加，但科学对计算能力的需求仍然没有得到满足。当新机器可用时，它们最初可能会闲置一小段时间，但很快就会被充分利用，通常在一个月内。为了从给定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），无论是从有效算术的角度还是从将数学方程转换为算术形式的过程（方案）的角度来看。存在着许多不同复杂程度的方案，但新的方案不断被寻求，而这项研究试图扩展一个特定方案的范围，该方案基于尽可能直接地模仿物理，而不需要多少武断的决定。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的数量。重点将放在复杂的流体流动中，其中旋转漩涡起着重要作用。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起飞或降落时的流动，以及计算直升机周围的流动。

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.