使用Chebyshev级数和区域分解的ode的严格数值

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2021-01-01 DOI:10.3934/jcd.2021015
J. B. van den Berg, Ray Sheombarsing
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引用次数: 11

摘要

本文用切比雪夫级数和区域分解给出了一种验证非线性微分方程解析解的严格数值方法。其思想是在几何衰减的切比雪夫系数空间上定义一个类牛顿算子,其不动点对应于ODE的解,并使用所谓的半径多项式方法来证明该算子在数值近似的小邻域内有一个孤立的不动点。该方法的新颖之处在于将切比雪夫级数与区域分解相结合。特别地,提出了一种基于解析函数的切比雪夫近似理论的启发式方法来构造有效的网格来验证边值问题的解。通过对Lorenz系统中的长周期轨道和连接轨道的验证,证明了该方法的有效性,而不进行域分解的验证是不可行的。
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Rigorous numerics for ODEs using Chebyshev series and domain decomposition
In this paper we present a rigorous numerical method for validating analytic solutions of nonlinear ODEs by using Chebyshev-series and domain decomposition. The idea is to define a Newton-like operator, whose fixed points correspond to solutions of the ODE, on the space of geometrically decaying Chebyshev coefficients, and to use the so-called radii-polynomial approach to prove that the operator has an isolated fixed point in a small neighborhood of a numerical approximation. The novelty of the proposed method is the use of Chebyshev series in combination with domain decomposition. In particular, a heuristic procedure based on the theory of Chebyshev approximations for analytic functions is presented to construct efficient grids for validating solutions of boundary value problems. The effectiveness of the proposed method is demonstrated by validating long periodic and connecting orbits in the Lorenz system for which validation without domain decomposition is not feasible.
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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