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Abstract <p>Local iteration modified (LIM) scheme is an explicit timeintegration method where accuracy and stability for large timesteps are provided by Chebyshev polynomial iterations. The LIMscheme possesses two characteristic features: (1)&#xa0;the Chebyshevpolynomial in the LIM scheme is constructed to give a higheraccuracy for the important low frequency solution modes, (2)&#xa0;thenumber of Chebyshev iterations is chosen depending on time stepsize, to have a stable time integration (rather than to solvearising linear system as done in implicit schemes). The LIM schemecan be seen as a stabilized explicit Runge–Kutta method whereChebyshev iterations play the role of internal stages whichguarantee stability. For parabolic PDEs the number of Chebysheviterations in the LIM scheme is inversely proportional to thespatial grid size&#xa0;<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8134_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(h\)</EquationSource> <!--LobJMat2460840Botchev-m1--> </InlineEquation>. An important property of the LIM schemewhich makes it unique in the class of explicit iterative timeintegration schemes, is positivity. In this work we thoroughlystudy this property of the scheme. Up to now, positivity of theLIM scheme has been shown experimentally in numerical tests aswell as, partially, theoretically. Our aim here is to providefurther theoretical and practical insight in the positivityproperties of the scheme.</p>

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On Positivity of the Local IterationModified Time Integration Scheme

  • 期刊论文

  • 发表日期：

    2025年1月1日

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  • Mikhail A. Botchev,
  • Victor T. Zhukov

摘要

Abstract

Local iteration modified (LIM) scheme is an explicit timeintegration method where accuracy and stability for large timesteps are provided by Chebyshev polynomial iterations. The LIMscheme possesses two characteristic features: (1) the Chebyshevpolynomial in the LIM scheme is constructed to give a higheraccuracy for the important low frequency solution modes, (2) thenumber of Chebyshev iterations is chosen depending on time stepsize, to have a stable time integration (rather than to solvearising linear system as done in implicit schemes). The LIM schemecan be seen as a stabilized explicit Runge–Kutta method whereChebyshev iterations play the role of internal stages whichguarantee stability. For parabolic PDEs the number of Chebysheviterations in the LIM scheme is inversely proportional to thespatial grid size  \(h\) . An important property of the LIM schemewhich makes it unique in the class of explicit iterative timeintegration schemes, is positivity. In this work we thoroughlystudy this property of the scheme. Up to now, positivity of theLIM scheme has been shown experimentally in numerical tests aswell as, partially, theoretically. Our aim here is to providefurther theoretical and practical insight in the positivityproperties of the scheme.