{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T01:00:15Z","timestamp":1776474015588,"version":"3.51.2"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,7,23]],"date-time":"2020-07-23T00:00:00Z","timestamp":1595462400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,23]],"date-time":"2020-07-23T00:00:00Z","timestamp":1595462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["JBK1901027"],"award-info":[{"award-number":["JBK1901027"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006469","name":"Fundo para o Desenvolvimento das Ci\u00eancias e da Tecnologia","doi-asserted-by":"publisher","award":["0005\/2019\/A"],"award-info":[{"award-number":["0005\/2019\/A"]}],"id":[{"id":"10.13039\/501100006469","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004733","name":"Universidade de Macau","doi-asserted-by":"publisher","award":["MYRG2018-00047-FST"],"award-info":[{"award-number":["MYRG2018-00047-FST"]}],"id":[{"id":"10.13039\/501100004733","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004733","name":"Universidade de Macau","doi-asserted-by":"publisher","award":["MYRG2017-00098-FST"],"award-info":[{"award-number":["MYRG2017-00098-FST"]}],"id":[{"id":"10.13039\/501100004733","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,5]]},"DOI":"10.1007\/s11075-020-00971-0","type":"journal-article","created":{"date-parts":[[2020,7,22]],"date-time":"2020-07-22T23:15:02Z","timestamp":1595459702000},"page":"381-408","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A fast linearized numerical method for nonlinear time-fractional diffusion equations"],"prefix":"10.1007","volume":"87","author":[{"given":"Pin","family":"Lyu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3017-3346","authenticated-orcid":false,"given":"Seakweng","family":"Vong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,23]]},"reference":[{"key":"971_CR1","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"A Alikhanov","year":"2015","unstructured":"Alikhanov, A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424\u2013438 (2015)","journal-title":"J. Comput. Phys."},{"key":"971_CR2","doi-asserted-by":"publisher","first-page":"1524","DOI":"10.1137\/18M1189750","volume":"57","author":"M Al-Maskari","year":"2019","unstructured":"Al-Maskari, M., Karaa, S.: Numerical approximation of semilinear subdiffusion equations with nonsmooth initial data. SIAM J. Numer. Anal. 57, 1524\u20131544 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR3","doi-asserted-by":"publisher","first-page":"A3070","DOI":"10.1137\/16M1070323","volume":"38","author":"W Cao","year":"2016","unstructured":"Cao, W., Zeng, F., Zhang, Z., Karniadakis, G.E.: Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput. 38, A3070\u2013A3093 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"971_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-017-0403-1","volume":"73","author":"G Gao","year":"2017","unstructured":"Gao, G., Alikhanov, A., Sun, Z.Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73, 1\u201329 (2017)","journal-title":"J. Sci. Comput."},{"key":"971_CR5","doi-asserted-by":"publisher","first-page":"650","DOI":"10.4208\/cicp.OA-2016-0136","volume":"21","author":"S Jiang","year":"2017","unstructured":"Jiang, S., Zhang, J., Zhang, Q., Zhang, Z.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650\u2013678 (2017)","journal-title":"Commun. Comput. Phys."},{"key":"971_CR6","first-page":"197","volume":"36","author":"B Jin","year":"2016","unstructured":"Jin, B., Lazarov, R., Zhou, Z.: An analysis of the l1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197\u2013221 (2016)","journal-title":"IMA J. Numer. Anal."},{"key":"971_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/16M1089320","volume":"56","author":"B Jin","year":"2018","unstructured":"Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56, 1\u201323 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR8","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1016\/j.jcp.2016.04.039","volume":"316","author":"C Li","year":"2016","unstructured":"Li, C., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316, 614\u2013631 (2016)","journal-title":"J. Comput. Phys."},{"key":"971_CR9","first-page":"86","volume":"24","author":"D Li","year":"2018","unstructured":"Li, D., Liao, H., Sun, W., Wang, J., Zhang, J.: Analysis of L1-Galerkin FEMs for time fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86\u2013103 (2018)","journal-title":"Commun. Comput. Phys."},{"key":"971_CR10","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/s10915-019-00943-0","volume":"80","author":"D Li","year":"2019","unstructured":"Li, D., Wu, C., Zhang, Z.: Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction. J. Sci. Comput. 80, 403\u2013419 (2019)","journal-title":"J. Sci. Comput."},{"key":"971_CR11","doi-asserted-by":"publisher","first-page":"848","DOI":"10.1007\/s10915-018-0642-9","volume":"76","author":"D Li","year":"2018","unstructured":"Li, D., Zhang, J., Zhang, Z.: Unconditionally optimal error estimates of a linearized Galerkin method for nonlinear time fractional reaction-subdiffusion equations. J. Sci. Comput. 76, 848\u2013866 (2018)","journal-title":"J. Sci. Comput."},{"key":"971_CR12","doi-asserted-by":"publisher","first-page":"1223","DOI":"10.1007\/s11075-019-00722-w","volume":"83","author":"X Li","year":"2019","unstructured":"Li, X., Zhang, L., Liao, H.L.: Sharp h1-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations. Numer. Algorithms 83, 1223\u20131248 (2019)","journal-title":"Numer. Algorithms"},{"key":"971_CR13","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1002\/num.22423","volume":"36","author":"Y Liang","year":"2019","unstructured":"Liang, Y., Yao, Z., Wang, Z.: Fast high order difference schemes for the time fractional telegraph equation. Numer. Meth. Part Differ. Equ. 36, 154\u2013172 (2019)","journal-title":"Numer. Meth. Part Differ. Equ."},{"key":"971_CR14","doi-asserted-by":"publisher","first-page":"1112","DOI":"10.1137\/17M1131829","volume":"56","author":"HL Liao","year":"2018","unstructured":"Liao, H.L., Li, D., Zhang, J.: Sharp error estimate of a nonuniform l1 formula for time-fractional reaction-subdiffusion equations. SIAM J. Numer. Anal. 56, 1112\u20131133 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR15","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1137\/16M1175742","volume":"57","author":"HL Liao","year":"2019","unstructured":"Liao, H.L., McLean, W., Zhang, J.: A discrete Gr\u00f6nwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218\u2013237 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR16","unstructured":"Liao, H.L., McLean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-subdiffusion problem. arXiv:1803.09873v2[math.NA]"},{"key":"971_CR17","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1002\/num.20414","volume":"26","author":"HL Liao","year":"2010","unstructured":"Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Meth. Part Differ. Equ. 26, 37\u201360 (2010)","journal-title":"Numer. Meth. Part Differ. Equ."},{"key":"971_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-019-00927-0","volume":"80","author":"HL Liao","year":"2019","unstructured":"Liao, H.L., Yan, Y., Zhang, J.: Unconditional convergence of a two-level linearized fast algorithm for semilinear subdiffusion equations. J. Sci. Comput. 80, 1\u201325 (2019)","journal-title":"J. Sci. Comput."},{"key":"971_CR19","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","volume":"225","author":"Y Lin","year":"2007","unstructured":"Lin, Y., Xu, C.: Finite difference\/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533\u20131552 (2007)","journal-title":"J. Comput. Phys."},{"key":"971_CR20","doi-asserted-by":"publisher","first-page":"1195","DOI":"10.1007\/s11075-017-0419-5","volume":"95","author":"Y Liu","year":"2018","unstructured":"Liu, Y., Roberts, J., Yan, Y.: Detailed error analysis for a fractional Adams method with graded meshes. Numer. Algorithms 95, 1195\u20131216 (2018)","journal-title":"Numer. Algorithms"},{"key":"971_CR21","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1016\/j.apnum.2019.11.012","volume":"151","author":"P Lyu","year":"2020","unstructured":"Lyu, P., Liang, Y., Wang, Z.: A fast linearized finite difference method for the nonlinear multi-term time fractional wave equation. Appl. Numer. Math. 151, 448\u2013471 (2020)","journal-title":"Appl. Numer. Math."},{"key":"971_CR22","doi-asserted-by":"publisher","first-page":"1607","DOI":"10.1007\/s10915-019-00991-6","volume":"80","author":"P Lyu","year":"2019","unstructured":"Lyu, P., Vong, S.: A high-order method with a temporal nonuniform mesh for a time-fractional Benjamin-Bona-Mahony equation. J. Sci. Comput. 80, 1607\u20131628 (2019)","journal-title":"J. Sci. Comput."},{"key":"971_CR23","doi-asserted-by":"publisher","first-page":"2153","DOI":"10.1002\/num.22282","volume":"34","author":"P Lyu","year":"2018","unstructured":"Lyu, P., Vong, S.: A linearized and second-order unconditionally convergent scheme for coupled time fractional Klein-Gordon-Schr\u00f6dinger equation. Numer. Meth. Part Differ. Equ. 34, 2153\u20132179 (2018)","journal-title":"Numer. Meth. Part Differ. Equ."},{"key":"971_CR24","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s11075-017-0385-y","volume":"78","author":"P Lyu","year":"2018","unstructured":"Lyu, P., Vong, S.: A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations. Numer Algorithms 78, 485\u2013511 (2018)","journal-title":"Numer Algorithms"},{"key":"971_CR25","unstructured":"Oldham, K., Spanier, J.: The fractional calculus. Academic Press, New York London (1974)"},{"key":"971_CR26","unstructured":"Ren, J., Liao, H.L., Zhang, J., Zhang, Z.: Sharp H1-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems. arXiv:1811.08059v1[math.NA]"},{"key":"971_CR27","doi-asserted-by":"publisher","first-page":"1057","DOI":"10.1137\/16M1082329","volume":"55","author":"M Stynes","year":"2017","unstructured":"Stynes, M., O\u2019Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057\u20131079 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"971_CR28","volume-title":"Numerical Methods of Partial Differential Equations","author":"ZZ Sun","year":"2012","unstructured":"Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2nd Ed. Science Press, Beijing (2012). (in Chinese)","edition":"2nd Ed"},{"key":"971_CR29","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"ZZ Sun","year":"2006","unstructured":"Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193\u2013209 (2006)","journal-title":"Appl. Numer. Math."},{"key":"971_CR30","doi-asserted-by":"publisher","first-page":"1252","DOI":"10.1007\/s10915-018-0659-0","volume":"76","author":"S Vong","year":"2018","unstructured":"Vong, S., Lyu, P.: Unconditional convergence in maximum-norm of a second-order linearized scheme for a time-fractional Burgers-type equation. J. Sci. Comput. 76, 1252\u20131273 (2018)","journal-title":"J. Sci. Comput."},{"key":"971_CR31","doi-asserted-by":"publisher","first-page":"1028","DOI":"10.4208\/cicp.OA-2017-0019","volume":"22","author":"Y Yan","year":"2017","unstructured":"Yan, Y., Sun, Z.Z., Zhang, J.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations: a second-order scheme, C. Comput. Phys. 22, 1028\u20131048 (2017)","journal-title":"Comput. Phys."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00971-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-00971-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-00971-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,23]],"date-time":"2021-07-23T00:31:05Z","timestamp":1627000265000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-00971-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,23]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["971"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-00971-0","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,23]]},"assertion":[{"value":"9 December 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 June 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}