{"title":"Numerical differentiation of the piecewise smooth function by using Fourier extension method","authors":"Zhenyu Zhao , Kai Yu , Xianzheng Jia , Zhihong Dou","doi":"10.1016/j.apnum.2024.09.026","DOIUrl":null,"url":null,"abstract":"<div><div>Numerical differentiation of the piecewise smooth function is considered in this paper. To avoid the large error of numerical differentiation that may occur near potential non-smooth points, we identify the discontinuity points of the first or second derivative of the function. Then we divide the domain of the function into several sub-domains. For each sub-domain, the approximation is constructed by Fourier extension, and the global approximation of the piecewise smooth function is formed by superposition to improve accuracy. Some numerical experiments are conducted to further verify the efficacy of the method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"208 ","pages":"Pages 45-57"},"PeriodicalIF":2.4000,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168927424002630","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/10/1 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
Numerical differentiation of the piecewise smooth function is considered in this paper. To avoid the large error of numerical differentiation that may occur near potential non-smooth points, we identify the discontinuity points of the first or second derivative of the function. Then we divide the domain of the function into several sub-domains. For each sub-domain, the approximation is constructed by Fourier extension, and the global approximation of the piecewise smooth function is formed by superposition to improve accuracy. Some numerical experiments are conducted to further verify the efficacy of the method.
期刊介绍：
The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are:
(i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments.
(ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers.
(iii) Short notes, which present specific new results and techniques in a brief communication.
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