{"title":"Tamed stability of finite difference schemes for the transport equation on the half-line","authors":"Lucas Coeuret","doi":"10.1090/mcom/3901","DOIUrl":null,"url":null,"abstract":"In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l Superscript 1\"> <mml:semantics> <mml:msup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\ell ^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable but <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l Superscript q\"> <mml:semantics> <mml:msup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\ell ^q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-unstable for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">q>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=\"application/x-tex\">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> embedded into the essential spectrum.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"2012 1","pages":"0"},"PeriodicalIF":2.2000,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mcom/3901","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are ℓ1\ell ^1-stable but ℓq\ell ^q-unstable for any q>1q>1. The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus 11 embedded into the essential spectrum.
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All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are.
This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.
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