{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T14:24:45Z","timestamp":1773066285059,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,10,16]],"date-time":"2023-10-16T00:00:00Z","timestamp":1697414400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,10,16]],"date-time":"2023-10-16T00:00:00Z","timestamp":1697414400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100010083","name":"Hunan Provincial Innovation Foundation for Postgraduate","doi-asserted-by":"publisher","award":["CX20231110"],"award-info":[{"award-number":["CX20231110"]}],"id":[{"id":"10.13039\/501100010083","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Natural Science Foundation of China Mathematics Tianyuan Foundation","award":["2022JJ50083"],"award-info":[{"award-number":["2022JJ50083"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2024,8]]},"DOI":"10.1007\/s11075-023-01676-w","type":"journal-article","created":{"date-parts":[[2023,10,16]],"date-time":"2023-10-16T10:01:47Z","timestamp":1697450507000},"page":"1533-1551","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["$$H^{1}$$-norm error analysis of a robust ADI method on graded mesh for three-dimensional subdiffusion problems"],"prefix":"10.1007","volume":"96","author":[{"given":"Ziyi","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haixiang","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuehua","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,16]]},"reference":[{"issue":"4","key":"1676_CR1","doi-asserted-by":"publisher","first-page":"872","DOI":"10.1016\/j.nonrwa.2005.06.001","volume":"7","author":"M Tabata","year":"2006","unstructured":"Tabata, M., Eshima, N., Takagi, I.: A geometrical similarity between migration of human population biological and diffusion of particles nonlinear analysis. Real World Applications 7(4), 872\u2013894 (2006)","journal-title":"Real World Applications"},{"key":"1676_CR2","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.physa.2018.12.019","volume":"519","author":"M Abdulhameed","year":"2019","unstructured":"Abdulhameed, M., Muhammad, M.M., Gital, A.Y., Yakubu, D.G., Khan, I.: Effect of fractional derivatives on transient MHD flow and radiative heat transfer in a micro-parallel channel at high zeta potentials. Physica A. 519, 42\u201371 (2019)","journal-title":"Physica A."},{"key":"1676_CR3","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.1007\/s00780-019-00400-8","volume":"23","author":"CD Constantinescu","year":"2019","unstructured":"Constantinescu, C.D., Ramirez, J.M., Zhu, W.R.: An application of fractional differential equations to risk theory. Financ. Stoch. 23, 1001\u20131024 (2019)","journal-title":"Financ. Stoch."},{"key":"1676_CR4","doi-asserted-by":"publisher","first-page":"103707","DOI":"10.1016\/j.cemconcomp.2020.103707","volume":"113","author":"SE Chidiac","year":"2020","unstructured":"Chidiac, S.E., Shafikhani, M.: Electrical resistivity model for quantifying concrete chloride diffusion coefficient. Cem. Concr. Compos. 113, 103707 (2020)","journal-title":"Cem. Concr. Compos."},{"issue":"3","key":"1676_CR5","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1007\/s10915-014-9841-1","volume":"61","author":"H Liao","year":"2014","unstructured":"Liao, H., Zhang, Y., Zhao, Y., Shi, H.: Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractional subdiffusion equations. J. Sci. Comput. 61(3), 629\u2013648 (2014)","journal-title":"J. Sci. Comput."},{"key":"1676_CR6","first-page":"71","volume":"342","author":"Y Wang","year":"2019","unstructured":"Wang, Y., Ren, L.: A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients. Appl. Math. Comput. 342, 71\u201393 (2019)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"1676_CR7","doi-asserted-by":"publisher","first-page":"456","DOI":"10.1016\/j.jcp.2012.08.026","volume":"232","author":"J Ren","year":"2013","unstructured":"Ren, J., Sun, Z.Z., Zhao, X.: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232(1), 456\u2013467 (2013)","journal-title":"J. Comput. Phys."},{"issue":"24","key":"1676_CR8","doi-asserted-by":"publisher","first-page":"8713","DOI":"10.1016\/j.jcp.2011.08.020","volume":"230","author":"Y Zhang","year":"2011","unstructured":"Zhang, Y., Sun, Z.Z.: Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. of Comput. Phys. 230(24), 8713\u20138728 (2011)","journal-title":"J. of Comput. Phys."},{"issue":"3","key":"1676_CR9","doi-asserted-by":"publisher","first-page":"586","DOI":"10.1016\/j.jcp.2010.10.007","volume":"230","author":"G Gao","year":"2011","unstructured":"Gao, G., Sun, Z.Z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230(3), 586\u2013595 (2011)","journal-title":"J. Comput. Phys."},{"key":"1676_CR10","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1007\/s11071-015-1906-7","volume":"80","author":"SY Lukashchuk","year":"2015","unstructured":"Lukashchuk, S.Y.: Conservation laws for time-fractional subdiffusion and diffusion-wave equations. Nonlinear Dyn. 80, 791\u2013802 (2015)","journal-title":"Nonlinear Dyn."},{"issue":"2","key":"1676_CR11","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(2), 1057\u20131079 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"1676_CR12","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/j.aml.2019.04.030","volume":"96","author":"J Ren","year":"2019","unstructured":"Ren, J., Chen, H.: A numerical method for distributed order time fractional diffusion equation with weakly singular solutions. Appl. Math. Lett. 96, 159\u2013165 (2019)","journal-title":"Appl. Math. Lett."},{"key":"1676_CR13","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.jcp.2014.09.031","volume":"280","author":"AA Alikhanov","year":"2015","unstructured":"Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424\u2013438 (2015)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"1676_CR14","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s10915-022-01936-2","volume":"92","author":"N Kopteva","year":"2022","unstructured":"Kopteva, N., Stynes, M.: A posteriori error analysis for variable-coefficient multiterm time-fractional subdiffusion equations. J. Sci. Comput. 92(2), 73 (2022)","journal-title":"J. Sci. Comput."},{"key":"1676_CR15","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1016\/j.jcp.2014.09.033","volume":"280","author":"G Gao","year":"2015","unstructured":"Gao, G., Sun, H., Sun, Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J. Comput. Phys. 280, 510\u2013528 (2015)","journal-title":"J. Comput. Phys."},{"key":"1676_CR16","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s11075-018-0594-z","volume":"82","author":"S Zhai","year":"2019","unstructured":"Zhai, S., Weng, Z., Feng, X., Yuan, J.: Investigations on several high-order ADI methods for time-space fractional diffusion equation. Numer. Algorithms. 82, 69\u2013106 (2019)","journal-title":"Numer. Algorithms."},{"key":"1676_CR17","first-page":"124564","volume":"387","author":"L Wu","year":"2020","unstructured":"Wu, L., Zhai, S.: A new high order ADI numerical difference formula for time-fractional convection-diffusion equation. Appl. Math. Comput. 387, 124564 (2020)","journal-title":"Appl. Math. Comput."},{"issue":"6","key":"1676_CR18","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1080\/10407790.2014.985987","volume":"67","author":"S Zhai","year":"2015","unstructured":"Zhai, S., Wei, L., Huang, L., Feng, X.: An efficient algorithm with high accuracy for time-space fractional heat equations. Numer. Heat Tran. B-Fund. 67(6), 550\u2013562 (2015)","journal-title":"Numer. Heat Tran. B-Fund."},{"issue":"10","key":"1676_CR19","doi-asserted-by":"publisher","first-page":"2952","DOI":"10.1016\/j.camwa.2020.01.003","volume":"79","author":"R Du","year":"2020","unstructured":"Du, R., Alikhanov, A.A., Sun, Z.: Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations. Comput. Math. Appl. 79(10), 2952\u20132972 (2020)","journal-title":"Comput. Math. Appl."},{"issue":"6","key":"1676_CR20","doi-asserted-by":"publisher","first-page":"A2976","DOI":"10.1137\/130910865","volume":"35","author":"F Zeng","year":"2013","unstructured":"Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference\/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976\u2013A3000 (2013)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1676_CR21","doi-asserted-by":"publisher","first-page":"A55","DOI":"10.1137\/14096390X","volume":"37","author":"F Zeng","year":"2015","unstructured":"Zeng, F., Li, C., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37(1), A55\u2013A78 (2015)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1676_CR22","doi-asserted-by":"publisher","first-page":"1412241","DOI":"10.1080\/23311835.2017.1412241","volume":"4","author":"AT Balasim","year":"2017","unstructured":"Balasim, A.T., Ali, H.M.N.: New group iterative schemes in the numerical solution of the two-dimensional time fractional advection-diffusion equation. Cogent Math. 4(1), 1412241 (2017)","journal-title":"Cogent Math."},{"key":"1676_CR23","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.apnum.2021.05.025","volume":"168","author":"Y Wang","year":"2021","unstructured":"Wang, Y., Chen, H., Sun, T.: $$\\alpha $$-Robust $$H^{1}$$-norm convergence analysis of ADI scheme for two-dimensional time-fractional diffusion equation. Appl. Numer. Math. 168, 75\u201383 (2021)","journal-title":"Appl. Numer. Math."},{"key":"1676_CR24","first-page":"128192","volume":"457","author":"X Yang","year":"2023","unstructured":"Yang, X., Wu, L., Zhang, H.: A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl. Math. Comput. 457, 128192 (2023)","journal-title":"Appl. Math. Comput."},{"issue":"6","key":"1676_CR25","doi-asserted-by":"publisher","first-page":"2599","DOI":"10.1137\/130934192","volume":"52","author":"F Zeng","year":"2014","unstructured":"Zeng, F., Liu, F., Li, C., Turner, I., Anh, V.: A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52(6), 2599\u20132622 (2014)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"1676_CR26","doi-asserted-by":"publisher","first-page":"A55","DOI":"10.1137\/14096390X","volume":"37","author":"F Zeng","year":"2015","unstructured":"Zeng, F., Li, C., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37(1), A55\u2013A78 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"1676_CR27","doi-asserted-by":"crossref","unstructured":"Zhou, Z., Zhang, H., Yang, X., Tang, J.: An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions. Int. J. Comput. Math. 1\u201318 (2023)","DOI":"10.1080\/00207160.2023.2212307"},{"issue":"6","key":"1676_CR28","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1007\/s40314-023-02373-z","volume":"42","author":"Q Tian","year":"2023","unstructured":"Tian, Q., Yang, X., Zhang, H., Xu, D.: An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties. Comput. Appl. Math. 42(6), 246 (2023)","journal-title":"Comput. Appl. Math."},{"key":"1676_CR29","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1016\/j.jcp.2014.03.020","volume":"269","author":"S Zhai","year":"2014","unstructured":"Zhai, S., Feng, X., He, Y.: An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation. J. Comput. Phys. 269, 138\u2013155 (2014)","journal-title":"J. Comput. Phys."},{"key":"1676_CR30","doi-asserted-by":"publisher","first-page":"440","DOI":"10.1016\/j.ijheatmasstransfer.2015.01.028","volume":"84","author":"S Zhai","year":"2015","unstructured":"Zhai, S., Weng, Z., Gui, D., Feng, X.: High-order compact operator splitting method for three-dimensional fractional equation with subdiffusion. Int. J. Heat Mass Tran. 84, 440\u2013447 (2015)","journal-title":"Int. J. Heat Mass Tran."},{"key":"1676_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.camwa.2022.09.006","volume":"126","author":"P Roul","year":"2022","unstructured":"Roul, P., Rohil, V.: A high-order numerical scheme based on graded mesh and its analysis for the two-dimensional time-fractional convection-diffusion equation. Comput. Math. Appl. 126, 1\u201313 (2022)","journal-title":"Comput. Math. Appl."},{"issue":"24","key":"1676_CR32","doi-asserted-by":"publisher","first-page":"8713","DOI":"10.1016\/j.jcp.2011.08.020","volume":"230","author":"Y Zhang","year":"2011","unstructured":"Zhang, Y., Sun, Z.: Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 230(24), 8713\u20138728 (2011)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"1676_CR33","doi-asserted-by":"publisher","first-page":"974","DOI":"10.1093\/imanum\/draa015","volume":"41","author":"H Chen","year":"2021","unstructured":"Chen, H., Stynes, M.: Blow-up of error estimates in time-fractional initial-boundary value problems. IMA J. Numer. Anal. 41(2), 974\u2013997 (2021)","journal-title":"IMA J. Numer. Anal."},{"issue":"1","key":"1676_CR34","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1137\/16M1175742","volume":"57","author":"H Liao","year":"2019","unstructured":"Liao, H., McLean, W., Zhang, J.: A discrete Gronwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57(1), 218\u2013237 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"1676_CR35","doi-asserted-by":"publisher","first-page":"1749","DOI":"10.1007\/s11075-020-01036-y","volume":"87","author":"C Huang","year":"2021","unstructured":"Huang, C., Stynes, M.: $$\\alpha $$-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation. Numer. Algorithms 87, 1749\u20131766 (2021)","journal-title":"Numer. Algorithms"},{"key":"1676_CR36","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/j.camwa.2023.01.028","volume":"135","author":"C Huang","year":"2023","unstructured":"Huang, C., An, N., Chen, H.: Optimal pointwise-in-time error analysis of a mixed finite element method for a multi-term time-fractional fourth-order equation. Comput. Math. Appl. 135, 149\u2013156 (2023)","journal-title":"Comput. Math. Appl."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01676-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01676-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01676-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T08:25:46Z","timestamp":1720772746000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01676-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,16]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,8]]}},"alternative-id":["1676"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01676-w","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,16]]},"assertion":[{"value":"15 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 October 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors consent to participate and consent to publish of this manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}