Fourier Analysis of Numerical Approximations of Hyperbolic Equations
Fourier Analysis of Numerical Approximations of Hyperbolic Equations
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DOI：
10.1137/1.9781611970876
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
R. Vichnevetsky;J. Bowles
中科院分区：
文献类型：
--
作者：
R. Vichnevetsky;J. Bowles
在过去十年左右的时间里，人们对使用傅立叶分析来检验双曲型方程数值方法的精度和稳定性问题越来越感兴趣。最初在美国和欧洲发表的一系列独立的、互不相关的结果，正在成为更普遍的理论的可识别的单一实体。大量材料以研究论文和报告的形式出现，本书的初衷之一是将这些结果集中在一个地方，以便于参考。就像通常的情况一样，整体被证明比部分的总和更多：在组装过程中出现了几个新的结果，并在本文中得到了适当的发展。双曲型方程数值方法的一个独特方面是，它们引入了扭曲所研究现象的物理性质的误差。通过引用起源于数学物理的概念来描述这些错误，例如能量传播、色散和扩散，因此被证明是一种最具启发性的方法。在这方面，傅立叶分析提供了一个不可或缺的工具。这并不令人惊讶：在过去的两个半世纪里，三角级数和积分的分析发展往往是由物理学偏微分方程组的同时发展所推动的，并与之密切相关。在应用傅立叶方法研究双曲方程的数值离散化时，人们会有这样的感觉，并非巧合，这种分析不是一种发明，而是对两者之间存在的一种自然关系的重新发现。这本书应该为那些从事计算流体力学的多个方面之一的人提供有用的参考材料。它是为物理学家和工程师提供的，他们使用计算机分析不同领域的问题，如水力学、气体动力学、等离子体物理、数值天气预报以及化学和土木工程中的运输过程，并希望或需要了解他们所使用的近似的含义。
There has been a growing interest, over the past decade or so, in the use of Fourier analysis to examine questions of accuracy and stability in numerical methods for hyperbolic equations. What had started at first as a set of individual, disconnected results published in the U.S. and Europe was becoming the identifiable single body of a more general theory. A great deal of material had appeared in the form of research papers and reports, and one of the original intents of this book was to bring together those results in a single place for easy reference. As is often the case, the whole turned out to be more than the sum of the parts: several new results have come to light during the assembly process, and have been suitably developed in the text.A distinctive aspect of numerical methods for hyperbolic equations is that they introduce errors which distort the physical nature of the phenomena under study. Describing those errors by invoking concepts which originated in mathematical physics, such as energy propagation, dispersion and diffusion thus proves to be a most enlightening approach. In this respect, Fourier analysis provides an indispensable tool. This comes as no surprise: the analytical development of trigonometric series and integrals through the past two and a half centuries was often motivated by, and closely related to, the concurrent development of the partial differential equations of physics. In applying Fourier methods to the study of numerical discretizations of hyperbolic equations, one gets the feeling, by no coincidence, that the analysis is not an invention, but rather the rediscovery of a natural relationship that exists between the two.This book should provide useful reference material to those who are engaged in one of the multiple aspects of computational fluid dynamics. It is intended for physicists and engineers who work with computers in the analysis of problems in such diverse fields as hydraulics, gas dynamics, plasma physics, numerical weather prediction and transport processes in chemical and civil engineering and who want or need to understand the implications of the approximations which they have used.