Functional Underpinnings of Summation-By-Parts Finite Differences

工程设计中经常使用计算机模拟，例如预测特定设计的桥梁可以承受的重量，或者如何优化汽车的形状以将阻力降至最低。计算机模拟也被用于整个科学领域，例如，根据地震波的传播和反射方式来推断地球中心的结构。大多数计算机模拟从一组控制方程开始，然后使用几种不同的数值技术之一来模拟这些方程。两种特定于该项目的数值技术被称为有限元和有限差分。每个问题都需要正确的技术来解决。有限元经常被用来模拟实体，例如桥梁如何支撑重量。有限差分更常用于模拟波浪，特别是这个项目考虑到了用于降低飞机发动机噪音的空气声学模拟。一类特殊的有限差分，称为分部分求和(SBP)有限差分，似乎在这方面表现得特别好。有趣的是，在我的研究中，我注意到这类有限差分与有限元有许多以前没有注意到的相似之处。本项目的目的是从理论上研究这一点，以了解它们的相似之处和不同之处(获得新知识)，然后看看如何利用这些相似之处来产生更好的计算机模拟(新技术)。本项目的最后部分旨在与使用计算机模拟的公司接触，以及谁可以从这些新技术中受益，试图帮助他们开始使用新技术。

Computer simulations are regularly used in engineering design, for example to predict the weight a particular design of bridge can support, or how to optimize the shape of a car to minimize drag. Computer simulations are also used throughout science, for example inferring the structure of the centre of the earth from the way seismic waves are transmitted and reflected. Most computer simulations start with a set of governing equations, and then simulate those equations using one of a few different numerical techniques. Two numerical techniques specific to this project are called finite elements and finite differences.The right technique is needed to solve each problem. Finite elements are often used to simulate solids, such as how bridges support weight. Finite differences are more commonly used to simulate waves, and in particular this project has in mind aeroacoustics simulations used to reduce noise in aircraft engines. A special class of finite differences, called Summation-By-Parts (SBP) finite differences, seem to perform particularly well at this. Interestingly, in my research I have noticed that this class of finite differences has a number of similarities to finite elements that have not been noticed before. The purpose of this project is to investigate this theoretically to see what the similarities and differences are (gaining new knowledge), and then to see how these similarities can be taken advantage of in order to produce better computer simulations (new techniques).The final part of this project aims to engage with companies who use computer simulations and who could benefit from these new techniques, to try to help them start using the new techniques.