{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:40:28Z","timestamp":1776836428709,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2001,7,11]],"date-time":"2001-07-11T00:00:00Z","timestamp":994809600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this article, we prove the convergence of a splitting scheme of high order for a reaction-diffusion system of the form <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u Subscript t Baseline minus upper M normal upper Delta u plus upper F left-parenthesis u right-parenthesis equals 0\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>u<\/mml:mi>\n        <mml:mi>t<\/mml:mi>\n      <\/mml:msub>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:mi>M<\/mml:mi>\n      <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n      <mml:mi>u<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>F<\/mml:mi>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>u<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n      <mml:mo>=<\/mml:mo>\n      <mml:mn>0<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">u_t-M\\Delta u +F(u)=0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> where <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M\">\n  <mml:semantics>\n    <mml:mi>M<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">M<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is an <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m times m\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>\u00d7<\/mml:mo>\n      <mml:mi>m<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">m \\times m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> matrix whose spectrum is included in <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace German upper R z greater-than 0 right-brace\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi mathvariant=\"fraktur\">R<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:mrow>\n      <mml:mi>z<\/mml:mi>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mn>0<\/mml:mn>\n      <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\{{\\mathfrak {R}} z &gt; 0 \\}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. This scheme is obtained by applying a Richardson extrapolation to a Strang formula.<\/p>","DOI":"10.1090\/s0025-5718-00-01277-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1481-1501","source":"Crossref","is-referenced-by-count":46,"title":["Convergence of a splitting method of high order for reaction-diffusion systems"],"prefix":"10.1090","volume":"70","author":[{"given":"St\u00e9phane","family":"Descombes","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,7,11]]},"reference":[{"issue":"11","key":"1","first-page":"1521","article-title":"Directions altern\u00e9es d\u2019ordre \u00e9lev\u00e9 en r\u00e9action-diffusion","volume":"321","author":"Descombes, St\u00e9phane","year":"1995","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"2","first-page":"429","article-title":"On Richardson extrapolation of Strang formula for reaction-diffusion equations","author":"Descombes, St\u00e9phane","year":"1998"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1051\/m2an\/1996300303431","article-title":"Commutateurs de certains semi-groupes holomorphes et applications aux directions altern\u00e9es","volume":"30","author":"Dia, Boun Oumar","year":"1996","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1006\/jfan.1996.3024","article-title":"An estimate of the Kac transfer operator","volume":"145","author":"Dia, Boun Oumar","year":"1997","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"key":"5","unstructured":"B. O. Dia, M. Schatzman, On the order of extrapolation of integration formul\u00e6, Preprint no 275, Equipe d\u2019Analyse Num\u00e9rique Lyon-Saint Etienne, UMR 5585, 1998."},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.2307\/2154960","article-title":"Geometrical evolution of developed interfaces","volume":"347","author":"de Mottoni, Piero","year":"1995","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1090\/qam\/1330655","article-title":"A novel method for simulating the complex Ginzburg-Landau equation","volume":"53","author":"Goldman, Daniel","year":"1995","journal-title":"Quart. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-569X","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1137\/0733018","article-title":"\ud835\udc41th-order operator splitting schemes and nonreversible systems","volume":"33","author":"Goldman, Daniel","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"9","first-page":"161","article-title":"Around the transfer operator and the Trotter-Kato formula","author":"Helffer, B.","year":"1995"},{"key":"10","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0089647","volume-title":"Geometric theory of semilinear parabolic equations","volume":"840","author":"Henry, Daniel","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/3540105573"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/BF02885677","article-title":"Estimate of the difference between the Kac operator and the Schr\u00f6dinger semigroup","volume":"186","author":"Ichinose, Takashi","year":"1997","journal-title":"Comm. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3616","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1093\/imanum\/9.2.199","article-title":"Solving linear partial differential equations by exponential splitting","volume":"9","author":"Sheng, Q.","year":"1989","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"13","series-title":"Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0873-0","volume-title":"Shock waves and reaction-diffusion equations","volume":"258","author":"Smoller, Joel","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387942599","edition":"2"},{"key":"14","doi-asserted-by":"publisher","first-page":"392","DOI":"10.1007\/BF00281235","article-title":"Accurate partial difference methods. I. Linear Cauchy problems","volume":"12","author":"Strang, Gilbert","year":"1963","journal-title":"Arch. Rational Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"15","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1137\/0705041","article-title":"On the construction and comparison of difference schemes","volume":"5","author":"Strang, Gilbert","year":"1968","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01277-1\/S0025-5718-00-01277-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01277-1\/S0025-5718-00-01277-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T00:37:20Z","timestamp":1627605440000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-236\/S0025-5718-00-01277-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,7,11]]},"references-count":15,"journal-issue":{"issue":"236","published-print":{"date-parts":[[2001,10]]}},"alternative-id":["S0025-5718-00-01277-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01277-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,7,11]]}}}