{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T05:08:58Z","timestamp":1772860138989,"version":"3.50.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T00:00:00Z","timestamp":1720483200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T00:00:00Z","timestamp":1720483200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,5]]},"DOI":"10.1007\/s11075-024-01877-x","type":"journal-article","created":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T11:01:59Z","timestamp":1720522919000},"page":"251-284","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity"],"prefix":"10.1007","volume":"99","author":[{"given":"Anshima","family":"Singh","sequence":"first","affiliation":[]},{"given":"Sunil","family":"Kumar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,9]]},"reference":[{"key":"1877_CR1","doi-asserted-by":"crossref","unstructured":"Morton, K.W.: Revival: Numerical solution of convection-diffusion problems CRC Press, 2019. (1996)","DOI":"10.1201\/9780203711194"},{"issue":"3","key":"1877_CR2","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1080\/10407798808551388","volume":"13","author":"AD Brent","year":"1988","unstructured":"Brent, A.D., Voller, V.R., Reid, K.T.J.: Enthalpy-porosity technique for modeling convection-diffusion phase change: application to the melting of a pure metal. Numer. Heat Tran. Part A App. 13(3), 297\u2013318 (1988)","journal-title":"Numer. Heat Tran. Part A App."},{"issue":"3","key":"1877_CR3","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1080\/10407790.2013.797316","volume":"64","author":"Y Xue","year":"2013","unstructured":"Xue, Y., Yang, H.: Interval estimation of convection-diffusion heat transfer problems. Numer. Heat Tran. Part B Fundamentals 64(3), 263\u2013273 (2013)","journal-title":"Numer. Heat Tran. Part B Fundamentals"},{"issue":"1","key":"1877_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/1475-925X-9-34","volume":"9","author":"GT Carroll","year":"2010","unstructured":"Carroll, G.T., Devereux, P.D., Ku, D.N., McGloughlin, T.M., Walsh, M.T.: Experimental validation of convection-diffusion discretisation scheme employed for computational modelling of biological mass transport. BioMed. Eng. OnLine 9(1), 1\u201313 (2010)","journal-title":"BioMed. Eng. OnLine"},{"issue":"6","key":"1877_CR5","doi-asserted-by":"publisher","first-page":"1927","DOI":"10.1109\/TMC.2020.3034270","volume":"21","author":"X Chen","year":"2020","unstructured":"Chen, X., Xu, S., Liu, X., Xu, X., Noh, H.Y., Zhang, L., Zhang, P.: Adaptive hybrid model-enabled sensing system (HMSS) for mobile fine-grained air pollution estimation. IEEE Trans. Mob. Comput. 21(6), 1927\u20131944 (2020)","journal-title":"IEEE Trans. Mob. Comput."},{"issue":"2","key":"1877_CR6","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1016\/j.asej.2013.10.005","volume":"5","author":"G Hariharan","year":"2014","unstructured":"Hariharan, G.: An efficient wavelet based approximation method to water quality assessment model in a uniform channel. Ain Shams Eng. J. 5(2), 525\u2013532 (2014)","journal-title":"Ain Shams Eng. J."},{"issue":"5","key":"1877_CR7","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1080\/10255840412331303140","volume":"7","author":"D Hose","year":"2004","unstructured":"Hose, D., Narracott, A., Griffiths, B., Mahmood, S., Gunn, J., Sweeney, D., Lawford, P.: A thermal analogy for modelling drug elution from cardiovascular stents. Comput. Methods Biomech. Biomed. Eng. 7(5), 257\u2013264 (2004)","journal-title":"Comput. Methods Biomech. Biomed. Eng."},{"key":"1877_CR8","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1016\/j.addr.2015.04.017","volume":"92","author":"DF Katz","year":"2015","unstructured":"Katz, D.F., Yuan, A., Gao, Y.: Vaginal drug distribution modeling. Adv. Drug Deliv. Rev. 92, 2\u201313 (2015)","journal-title":"Adv. Drug Deliv. Rev."},{"issue":"4","key":"1877_CR9","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.89.043308","volume":"89","author":"L Li","year":"2014","unstructured":"Li, L., Chen, C., Mei, R., Klausner, J.F.: Conjugate heat and mass transfer in the lattice Boltzmann equation method. Phys. Rev. E 89(4), 043308 (2014)","journal-title":"Phys. Rev. E"},{"issue":"4","key":"1877_CR10","doi-asserted-by":"publisher","first-page":"528","DOI":"10.1080\/10408398.2016.1197881","volume":"58","author":"MIH Khan","year":"2018","unstructured":"Khan, M.I.H., Joardder, M.U.H., Kumar, C., Karim, M.A.: Multiphase porous media modelling: A novel approach to predicting food processing performance. Crit. Rev. Food Sci. Nutr. 58(4), 528\u2013546 (2018)","journal-title":"Crit. Rev. Food Sci. Nutr."},{"key":"1877_CR11","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.buildenv.2019.01.053","volume":"152","author":"Y Zhang","year":"2019","unstructured":"Zhang, Y., Jiang, J., Bai, Y., Liu, J., Shao, H., Wu, C., Guo, Z.: A fractional mass transfer model for simulating VOC emissions from porous, dry building material. Build. Environ. 152, 182\u2013191 (2019)","journal-title":"Build. Environ."},{"key":"1877_CR12","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."},{"key":"1877_CR13","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.cam.2017.11.030","volume":"335","author":"J Zhang","year":"2018","unstructured":"Zhang, J., Zhang, X., Yang, B.: An approximation scheme for the time fractional convection-diffusion equation. Appl. Math. Comput. 335, 305\u2013312 (2018)","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"1877_CR14","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":"1877_CR15","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/j.jcp.2014.09.012","volume":"280","author":"M Cui","year":"2015","unstructured":"Cui, M.: Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients. J. Comput. Phys. 280, 143\u2013163 (2015)","journal-title":"J. Comput. Phys."},{"key":"1877_CR16","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/s12190-007-0013-4","volume":"26","author":"S Chen","year":"2008","unstructured":"Chen, S., Liu, F.: ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation. J. Appl. Math. Comput. 26, 295\u2013311 (2008)","journal-title":"J. Appl. Math. Comput."},{"issue":"3","key":"1877_CR17","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s10092-015-0150-3","volume":"53","author":"Y-M Wang","year":"2016","unstructured":"Wang, Y.-M., Wang, T.: Error analysis of a high-order compact ADI method for two-dimensional fractional convection-subdiffusion equations. Calcolo 53(3), 301\u2013330 (2016)","journal-title":"Calcolo"},{"key":"1877_CR18","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1016\/j.jcp.2017.03.061","volume":"340","author":"A Tayebi","year":"2017","unstructured":"Tayebi, A., Shekari, Y., Heydari, M.H.: A meshless method for solving two-dimensional variable-order time fractional advection\u2013diffusion equation. J. Comput. Phys. 340, 655\u2013669 (2017)","journal-title":"J. Comput. Phys."},{"key":"1877_CR19","doi-asserted-by":"crossref","unstructured":"Singh, A., Kumar, S., Vigo-Aguiar, J.: A fully discrete scheme based on cubic splines and its analysis for time-fractional reaction\u2013diffusion equations exhibiting weak initial singularity. J. Comput. Appl. Math. 115338 (2023)","DOI":"10.1016\/j.cam.2023.115338"},{"issue":"01","key":"1877_CR20","doi-asserted-by":"publisher","first-page":"1941006","DOI":"10.1142\/S179396231941006X","volume":"10","author":"H Chen","year":"2019","unstructured":"Chen, H., Hu, X., Ren, J., Sun, T., Tang, Y.: L1 scheme on graded mesh for the linearized time fractional KdV equation with initial singularity. Int. J. Model. Simul. Sci. Comput. 10(01), 1941006 (2019)","journal-title":"Int. J. Model. Simul. Sci. Comput."},{"issue":"3","key":"1877_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-021-01581-1","volume":"88","author":"M Chen","year":"2021","unstructured":"Chen, M., Jiang, S., Bu, W.: Two L1 schemes on graded meshes for fractional Feynman-Kac equation. J. Sci. Comput. 88(3), 1\u201324 (2021)","journal-title":"J. Sci. Comput."},{"issue":"8","key":"1877_CR22","first-page":"2874","volume":"75","author":"Z Cen","year":"2018","unstructured":"Cen, Z., Huang, J., Xu, A., Le, A.: Numerical approximation of a time-fractional Black\u2013Scholes equation. Comput. Math. App. 75(8), 2874\u20132887 (2018)","journal-title":"Comput. Math. App."},{"key":"1877_CR23","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1007\/s10915-018-0863-y","volume":"79","author":"H Chen","year":"2019","unstructured":"Chen, H., Stynes, M.: Error analysis of a second-order method on fitted meshes for a time-fractional diffusion problem. J. Sci. Comput. 79, 624\u2013647 (2019)","journal-title":"J. Sci. Comput."},{"key":"1877_CR24","first-page":"1","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. App. 126, 1\u201313 (2022)","journal-title":"Comput. Math. App."},{"key":"1877_CR25","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1016\/j.enganabound.2021.01.011","volume":"125","author":"H Qiao","year":"2021","unstructured":"Qiao, H., Cheng, A.: A fast finite difference\/rbf meshless approach for time fractional convection-diffusion equation with non-smooth solution. Eng. Anal. Boundary Elements 125, 280\u2013289 (2021)","journal-title":"Eng. Anal. Boundary Elements"},{"key":"1877_CR26","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."},{"issue":"1","key":"1877_CR27","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1190\/1.1442040","volume":"51","author":"MA Dablain","year":"1986","unstructured":"Dablain, M.A.: The application of high-order differencing to the scalar wave equation. Geophysics 51(1), 54\u201366 (1986)","journal-title":"Geophysics"},{"issue":"6","key":"1877_CR28","doi-asserted-by":"publisher","first-page":"4381","DOI":"10.1137\/080714907","volume":"47","author":"H-L Liao","year":"2010","unstructured":"Liao, H.-L., Sun, Z.-Z., Shi, H.-S.: Error estimate of fourth-order compact scheme for linear Schr\u00f6dinger equations. SIAM J. Numer. Anal. 47(6), 4381\u20134401 (2010)","journal-title":"SIAM J. Numer. Anal."},{"issue":"20","key":"1877_CR29","doi-asserted-by":"publisher","first-page":"7792","DOI":"10.1016\/j.jcp.2009.07.021","volume":"228","author":"M Cui","year":"2009","unstructured":"Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228(20), 7792\u20137804 (2009)","journal-title":"J. Comput. Phys."},{"key":"1877_CR30","doi-asserted-by":"crossref","unstructured":"Gao, G.-h., Sun, Z.-z.: A compact finite difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230(3), 586\u2013595 (2011)","DOI":"10.1016\/j.jcp.2010.10.007"},{"key":"1877_CR31","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1016\/j.apnum.2018.03.005","volume":"129","author":"M Ran","year":"2018","unstructured":"Ran, M., Zhang, C.: New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order. Appl. Numer. Math. 129, 58\u201370 (2018)","journal-title":"Appl. Numer. Math."},{"issue":"8","key":"1877_CR32","doi-asserted-by":"publisher","first-page":"1638","DOI":"10.1080\/00207160.2021.1998467","volume":"99","author":"S Sumit","year":"2022","unstructured":"Sumit, S., Kumar, S., Kumar, M.: Optimal fourth-order parameter-uniform convergence of a non-monotone scheme on equidistributed meshes for singularly perturbed reaction\u2013diffusion problems. Int. J. Comput. Math. 99(8), 1638\u20131653 (2022)","journal-title":"Int. J. Comput. Math."},{"issue":"4","key":"1877_CR33","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1007\/s11786-019-00448-x","volume":"15","author":"LL Ferr\u00e1s","year":"2021","unstructured":"Ferr\u00e1s, L.L., Ford, N., Morgado, M.L., Rebelo, M.: High-order methods for systems of fractional ordinary differential equations and their application to time-fractional diffusion equations. Math. Comput. Sci. 15(4), 535\u2013551 (2021)","journal-title":"Math. Comput. Sci."},{"key":"1877_CR34","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1016\/j.matcom.2022.03.025","volume":"199","author":"SS Kumar","year":"2022","unstructured":"Kumar, S.S., Vigo-Aguiar, J.: A high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution. Math. Comput. Simul. 199, 287\u2013306 (2022)","journal-title":"Math. Comput. Simul."},{"issue":"16","key":"1877_CR35","doi-asserted-by":"publisher","first-page":"16521","DOI":"10.1002\/mma.9458","volume":"46","author":"A Singh","year":"2023","unstructured":"Singh, A., Kumar, S., Vigo-Aguiar, J.: High-order schemes and their error analysis for generalized variable coefficients fractional reaction-diffusion equations. Math. Methods Appl. Sci. 46(16), 16521\u201316541 (2023)","journal-title":"Math. Methods Appl. Sci."},{"key":"1877_CR36","doi-asserted-by":"crossref","unstructured":"Liao, H.-L., Sun, Z.-Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Diff. Equ. Int. J. 26(1), 37\u201360 (2010)","DOI":"10.1002\/num.20414"},{"key":"1877_CR37","doi-asserted-by":"crossref","unstructured":"Zhang, Y.-n., Sun, Z.-z.: Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 230(24), 8713\u20138728 (2011)","DOI":"10.1016\/j.jcp.2011.08.020"},{"issue":"2","key":"1877_CR38","doi-asserted-by":"publisher","first-page":"567","DOI":"10.4208\/cicp.OA-2020-0124","volume":"30","author":"H-L Liao","year":"2021","unstructured":"Liao, H.-L.: A Second-Order Scheme with Nonuniform Time Steps for a Linear Reaction-Subdiffusion Problem. Commun. Comput. Phys. 30(2), 567\u2013601 (2021)","journal-title":"Commun. Comput. Phys."},{"key":"1877_CR39","first-page":"155","volume":"99","author":"Y Wang","year":"2021","unstructured":"Wang, Y., Chen, H.: Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation. Comput. Math. App. 99, 155\u2013161 (2021)","journal-title":"Comput. Math. App."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01877-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01877-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01877-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T07:40:22Z","timestamp":1744616422000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01877-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,9]]},"references-count":39,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["1877"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01877-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,9]]},"assertion":[{"value":"20 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 June 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"Not applicable","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human and animal ethics"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"A preprint of this paper is available on arxiv with Ref. No. arXiv:2308.08971","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Preprint"}}]}}