{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T20:01:03Z","timestamp":1774382463217,"version":"3.50.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T00:00:00Z","timestamp":1617580800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T00:00:00Z","timestamp":1617580800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Nature Science Foundation of China","doi-asserted-by":"crossref","award":["No. 12071061"],"award-info":[{"award-number":["No. 12071061"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,5]]},"DOI":"10.1007\/s10915-021-01466-3","type":"journal-article","created":{"date-parts":[[2021,4,5]],"date-time":"2021-04-05T17:02:55Z","timestamp":1617642175000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["An Analysis on the Finite Volume Schemes and the Discrete Lyapunov Inequalities for the Chemotaxis System"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6196-9451","authenticated-orcid":false,"given":"Guanyu","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,4,5]]},"reference":[{"key":"1466_CR1","doi-asserted-by":"publisher","first-page":"4015","DOI":"10.1016\/j.cam.2011.02.023","volume":"235","author":"B Andreianov","year":"2011","unstructured":"Andreianov, B., Bendahmane, M., Saad, M.: Finite volume methods for degenerate chemotaxis model. J. Comput. Appl. Math. 235, 4015\u20134031 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"1466_CR2","doi-asserted-by":"publisher","first-page":"1125","DOI":"10.1093\/imanum\/dru032","volume":"35","author":"M Bessemoulin-Chatard","year":"2015","unstructured":"Bessemoulin-Chatard, M., Chainais-Hilliairet, C., Filbet, F.: On discrete functional inequalities for some finite volume schemes. IMA J. Numer. Anal. 35, 1125\u20131149 (2015)","journal-title":"IMA J. Numer. Anal."},{"key":"1466_CR3","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1093\/imanum\/drs061","volume":"34","author":"M Bessemoulin-Chatard","year":"2014","unstructured":"Bessemoulin-Chatard, M., J\u00fcngel, A.: A finite volume scheme for a Keller\u2013Segel model with additional cross-diffusion. IMA. J. Numer. 34, 96\u2013122 (2014)","journal-title":"IMA. J. Numer."},{"key":"1466_CR4","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/BF01385651","volume":"58","author":"Z Cai","year":"1991","unstructured":"Cai, Z.: On the finite volume element method. Numer. Math. 58, 713\u2013735 (1991)","journal-title":"Numer. Math."},{"key":"1466_CR5","doi-asserted-by":"publisher","first-page":"1030","DOI":"10.1002\/num.21858","volume":"30","author":"G Chamoun","year":"2014","unstructured":"Chamoun, G., Saad, M., Talhouk, R.: Monotone combined edge finite volume-finite element scheme for anisotropic Keller\u2013Segel model. Numer. Methods Partial Differ. Equ. 30, 1030\u20131065 (2014)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"1466_CR6","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s10444-017-9545-9","volume":"44","author":"A Chertock","year":"2018","unstructured":"Chertock, A., Epshteyn, Y., Hu, H., Kurganov, A.: High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems. Adv. Comput. Math. 44, 327\u2013350 (2018)","journal-title":"Adv. Comput. Math."},{"key":"1466_CR7","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/s00211-008-0188-0","volume":"111","author":"A Chertock","year":"2008","unstructured":"Chertock, A., Kurganov, A.: A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169\u2013205 (2008)","journal-title":"Numer. Math."},{"key":"1466_CR8","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/0025-5564(81)90055-9","volume":"56","author":"S Childress","year":"1981","unstructured":"Childress, S., Percus, J.K.: Nolinear aspects of chemotaxis. Math. Biosci. 56, 217\u2013237 (1981)","journal-title":"Math. Biosci."},{"key":"1466_CR9","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1002\/num.10059","volume":"19","author":"S Chou","year":"2003","unstructured":"Chou, S., Kwak, D.Y., Li, Q.: $${L}^p$$ error estimates and superconvergence for covolume or finite volume element methods. Numer. Methods Partial Differ. Equ. 19, 463\u2013486 (2003)","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"1466_CR10","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1090\/S0025-5718-99-01192-8","volume":"69","author":"S Chou","year":"1999","unstructured":"Chou, S., Li, Q.: Error estimates in $${L}^2$$, $${H}^1$$ and $${L}^\\infty $$ in covolume methods for elliptic and parabolic problems: a unified approach. Math. Comput. 69, 103\u2013120 (1999)","journal-title":"Math. Comput."},{"issue":"1","key":"1466_CR11","doi-asserted-by":"publisher","first-page":"3","DOI":"10.2478\/cmam-2001-0001","volume":"1","author":"M Crouzeix","year":"2001","unstructured":"Crouzeix, M., Thom\u00e9e, V.: Resolvent estimates in $$l^p$$ for discrete Laplacians on irregular meshes and maximum-norm stability of parabolic finite difference schemes. Comput. Methods Appl. Meth. 1(1), 3\u201317 (2001)","journal-title":"Comput. Methods Appl. Meth."},{"key":"1466_CR12","volume-title":"Some Gronwall Type Inequalities and Applications","author":"SS Dragomir","year":"2003","unstructured":"Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers Inc, Hauppauge, NY (2003)"},{"key":"1466_CR13","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1007\/s10915-012-9599-2","volume":"53","author":"Y Epshteyn","year":"2012","unstructured":"Epshteyn, Y.: Upwind-difference potentials method Patlak\u2013Keller\u2013Segel chemotaxis model. J. Sci. Comput. 53, 689\u2013713 (2012)","journal-title":"J. Sci. Comput."},{"key":"1466_CR14","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s10915-009-9281-5","volume":"40","author":"Y Epshteyn","year":"2009","unstructured":"Epshteyn, Y., Izmirlioglu, A.: Fully discrete analysis of a discontinuous finite element method for the Keller\u2013Segel chemotaxis model. J. Sci. Comput. 40, 211\u2013256 (2009)","journal-title":"J. Sci. Comput."},{"key":"1466_CR15","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1137\/07070423X","volume":"47","author":"Y Epshteyn","year":"2008","unstructured":"Epshteyn, Y., Kurganov, A.: New interior penalty discontinuous Galerkin methods for the Keller\u2013Segel chemotaxis model. SIAM J. Numer. Anal. 47, 386\u2013408 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"1466_CR16","series-title":"Handbook of Numerical Analysis, vii","doi-asserted-by":"publisher","DOI":"10.1016\/S1570-8659(00)07005-8","volume-title":"Finite Volume Methods","author":"R Eymard","year":"2000","unstructured":"Eymard, R., Gallou\u00ebt, T., Herbin, R.: Finite Volume Methods. Handbook of Numerical Analysis, vii, North-Holland, Amsterdam (2000)"},{"key":"1466_CR17","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1088\/0951-7715\/26\/1\/81","volume":"26","author":"I Fatkullin","year":"2013","unstructured":"Fatkullin, I.: A study of blow-ups in the Keller\u2013Segel model of chemotaxis. Nonlinearity 26, 81\u201394 (2013)","journal-title":"Nonlinearity"},{"key":"1466_CR18","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1007\/s00211-006-0024-3","volume":"104","author":"F Filbet","year":"2006","unstructured":"Filbet, F.: A finite volume scheme for the Patlak\u2013Keller\u2013Segel chemotaxis model. Numer. Math. 104, 457\u2013488 (2006)","journal-title":"Numer. Math."},{"key":"1466_CR19","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1002\/mana.19981950106","volume":"195","author":"H Gajewski","year":"1998","unstructured":"Gajewski, H., Zacharias, K.: Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195, 77\u2013114 (1998)","journal-title":"Math. Nachr."},{"key":"1466_CR20","volume-title":"Elliptic Problems in Nonsmooth Domains","author":"P Grisvard","year":"1985","unstructured":"Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)"},{"key":"1466_CR21","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/s10955-009-9717-1","volume":"135","author":"J Ha\u0161kovec","year":"2009","unstructured":"Ha\u0161kovec, J., Schmeiser, C.: Stochastic p approximation for measure valued solutions of the 2D Keller\u2013Segel system. J. Stat. Phys. 135, 133\u2013151 (2009)","journal-title":"J. Stat. Phys."},{"key":"1466_CR22","doi-asserted-by":"publisher","first-page":"940","DOI":"10.1080\/03605302.2010.538783","volume":"36","author":"J Ha\u0161kovec","year":"2011","unstructured":"Ha\u0161kovec, J., Schmeiser, C.: Convergence of a stochastic p approximation for measure solutions of the 2D Keller\u2013Segel system. Commun. Partial Differ. Equ. 36, 940\u2013960 (2011)","journal-title":"Commun. Partial Differ. Equ."},{"key":"1466_CR23","first-page":"103","volume":"105","author":"D Horstmann","year":"2003","unstructured":"Horstmann, D.: From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Deutsch. Math.-Verein. 105, 103\u2013165 (2003)","journal-title":"Jahresber. Deutsch. Math.-Verein."},{"key":"1466_CR24","first-page":"51","volume":"106","author":"D Horstmann","year":"2004","unstructured":"Horstmann, D.: From 1970 until present: the Keller\u2013Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein. 106, 51\u201369 (2004)","journal-title":"Jahresber. Deutsch. Math.-Verein."},{"key":"1466_CR25","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1016\/0022-5193(70)90092-5","volume":"26","author":"FF Keller","year":"1970","unstructured":"Keller, F.F., Segel, L.A.: Initiation on slime mold aggregation viewed as instability. J. Theor. Biol. 26, 399\u2013415 (1970)","journal-title":"J. Theor. Biol."},{"key":"1466_CR26","volume-title":"Numerical Methods for Elliptic and Parabolic Partial Differential Equations","author":"P Knabner","year":"2003","unstructured":"Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)"},{"key":"1466_CR27","doi-asserted-by":"publisher","DOI":"10.1201\/9781482270211","volume-title":"Generalized Difference Methods for Differential Equations, Numerical Analysis of Finite","author":"R Li","year":"2000","unstructured":"Li, R., Chen, Z., Wu, W.: Generalized Difference Methods for Differential Equations, Numerical Analysis of Finite, vol. method. Marcel Dekker Inc, New York (2000)"},{"key":"1466_CR28","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1051\/m2an:2003048","volume":"37","author":"A Marrocco","year":"2003","unstructured":"Marrocco, A.: 2D simulation of chemotaxis bacteria aggregation. M2AN 37, 617\u2013630 (2003)","journal-title":"M2AN"},{"key":"1466_CR29","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1512\/iumj.1971.20.20101","volume":"20","author":"J Moser","year":"1971","unstructured":"Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Math. J. 20, 1077\u20131093 (1971)","journal-title":"Indiana Math. J."},{"key":"1466_CR30","doi-asserted-by":"publisher","first-page":"5381","DOI":"10.1016\/S0362-546X(97)00395-7","volume":"30","author":"T Nagai","year":"1997","unstructured":"Nagai, T.: Global existence of solutions to a parabolic system for chemotaxis in two space dimensions. Nonlinear Anal. Theory Methods Appl. 30, 5381\u20135388 (1997)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"1466_CR31","doi-asserted-by":"publisher","first-page":"463","DOI":"10.32917\/hmj\/1206124609","volume":"30","author":"T Nagai","year":"2000","unstructured":"Nagai, T., Senba, T., Suzuki, T.: Chemotactic collapse in a parabolic system of mathematical biology. Hiroshima Math. J. 30, 463\u2013497 (2000)","journal-title":"Hiroshima Math. J."},{"key":"1466_CR32","first-page":"613","volume":"29","author":"H Okamoto","year":"1982","unstructured":"Okamoto, H.: On the semidiscrete finite element approximation for the nonstationary Navier\u2013Stokes equation. J. Fac. Sci. Univ. Tokyo Sect. IA.Math. 29, 613\u2013651 (1982)","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA.Math."},{"key":"1466_CR33","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1007\/s10492-004-6431-9","volume":"49","author":"B Perthame","year":"2004","unstructured":"Perthame, B.: PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl. Math. 49, 539\u2013364 (2004)","journal-title":"Appl. Math."},{"key":"1466_CR34","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/BF03167586","volume":"21","author":"N Saito","year":"2004","unstructured":"Saito, N.: Remarks on the rational approximation of holomorphic semigroups with nonuniform partitions. Jpn. J. Ind. Appl. Math. 21, 323\u2013337 (2004)","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"1466_CR35","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1093\/imanum\/drl018","volume":"27","author":"N Saito","year":"2007","unstructured":"Saito, N.: Conservative upwind finite-element method for a simplified Keller\u2013Segel system modelling chemotaxis. IMA J. Numer. Anal. 27, 332\u2013365 (2007)","journal-title":"IMA J. Numer. Anal."},{"key":"1466_CR36","doi-asserted-by":"publisher","first-page":"339","DOI":"10.3934\/cpaa.2012.11.339","volume":"11","author":"N Saito","year":"2012","unstructured":"Saito, N.: Error analysis of a conservative finite-element approximation for the Keller\u2013Segel system of chemotaxis. Commun. Pure Appl. Anal. 11, 339\u2013364 (2012)","journal-title":"Commun. Pure Appl. Anal."},{"key":"1466_CR37","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.amc.2005.01.037","volume":"171","author":"N Saito","year":"2005","unstructured":"Saito, N., Suzuki, T.: Notes on finite difference schemes to a parabolic\u2013elliptic system modelling chemotaxis. Appl. Math. Comput. 171, 72\u201390 (2005)","journal-title":"Appl. Math. Comput."},{"key":"1466_CR38","doi-asserted-by":"publisher","first-page":"219","DOI":"10.2478\/cmam-2010-0013","volume":"10","author":"R Strehl","year":"2010","unstructured":"Strehl, R., Sokolov, A., Kuzmin, D., Turek, S.: A flux-corrected finite element method for chemotaxis problems. Comput. Methods Appl. Meth. 10, 219\u2013232 (2010)","journal-title":"Comput. Methods Appl. Meth."},{"key":"1466_CR39","doi-asserted-by":"publisher","DOI":"10.1007\/0-8176-4436-9","volume-title":"Free Energy and Self-interacting Particles","author":"T Suzuki","year":"2005","unstructured":"Suzuki, T.: Free Energy and Self-interacting Particles. Birkhauser, Boston (2005)"},{"key":"1466_CR40","volume-title":"Galerkin Finite Element Methods for Parabolic Problems","author":"V Thom\u00e9e","year":"2006","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)"},{"key":"1466_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05156-2","volume-title":"Matrix Iterative Analysis","author":"RS Varga","year":"2000","unstructured":"Varga, R.S.: Matrix Iterative Analysis. Springer, Heidelberg (2000)"},{"key":"1466_CR42","doi-asserted-by":"publisher","first-page":"2889","DOI":"10.1016\/j.jde.2010.02.008","volume":"248","author":"M Winkler","year":"2010","unstructured":"Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimentional Keller\u2013Segel model. J. Differ. Equ. 248, 2889\u20132905 (2010)","journal-title":"J. Differ. Equ."},{"key":"1466_CR43","first-page":"241","volume":"45","author":"A Yagi","year":"1997","unstructured":"Yagi, A.: Norm behaviour of solutions to a parabolic system of chemotaxis. Math. Jpn. 45, 241\u2013256 (1997)","journal-title":"Math. Jpn."},{"key":"1466_CR44","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/s00211-016-0793-2","volume":"135","author":"G Zhou","year":"2017","unstructured":"Zhou, G., Saito, N.: Finite volume methods for a Keller\u2013Segel system: discrete energy, error estimates and numerical blow-up analysis. Numer. Math. 135, 265\u2013311 (2017)","journal-title":"Numer. Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01466-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01466-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01466-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,1]],"date-time":"2023-11-01T17:36:29Z","timestamp":1698860189000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01466-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,5]]},"references-count":44,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["1466"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01466-3","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,5]]},"assertion":[{"value":"8 August 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 April 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"54"}}