Functional Underpinnings of Summation-By-Parts Finite Differences

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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.