{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T16:54:41Z","timestamp":1781110481839,"version":"3.54.1"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,1,16]],"date-time":"2021-01-16T00:00:00Z","timestamp":1610755200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,16]],"date-time":"2021-01-16T00:00:00Z","timestamp":1610755200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11771265"],"award-info":[{"award-number":["11771265"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Natural Science Foundation of Guangdong Provinvial Department of Education","award":["2018KQNCX156"],"award-info":[{"award-number":["2018KQNCX156"]}]},{"name":"key research projects of general universities in Guangdong Province","award":["2019KZDXM034"],"award-info":[{"award-number":["2019KZDXM034"]}]},{"name":"basic research and applied basic research projects in Guangdong Province","award":["20202002110000018"],"award-info":[{"award-number":["20202002110000018"]}]},{"name":"Natural Science Foundation of Guangdong Provincial Department of Education","award":["2020KZDZX1147"],"award-info":[{"award-number":["2020KZDZX1147"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,3]]},"DOI":"10.1007\/s10915-020-01398-4","type":"journal-article","created":{"date-parts":[[2021,1,16]],"date-time":"2021-01-16T13:02:42Z","timestamp":1610802162000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Banded Preconditioners for Riesz Space Fractional Diffusion Equations"],"prefix":"10.1007","volume":"86","author":[{"given":"Zi-Hang","family":"She","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cheng-Xue","family":"Lao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4099-6706","authenticated-orcid":false,"given":"Fu-Rong","family":"Lin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,1,16]]},"reference":[{"key":"1398_CR1","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1023\/A:1016539022492","volume":"29","author":"OP Agrawal","year":"2002","unstructured":"Agrawal, O.P.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 29, 145\u2013155 (2002)","journal-title":"Nonlinear Dyn."},{"issue":"6","key":"1398_CR2","doi-asserted-by":"publisher","first-page":"1403","DOI":"10.1029\/2000WR900031","volume":"36","author":"DA Benson","year":"2000","unstructured":"Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36(6), 1403\u20131412 (2000)","journal-title":"Water Resour. Res."},{"issue":"6","key":"1398_CR3","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1029\/2000WR900032","volume":"36","author":"DA Benson","year":"2000","unstructured":"Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of L\u00e9vy motion. Water Resour. Res. 36(6), 1413\u20131423 (2000)","journal-title":"Water Resour. Res."},{"key":"1398_CR4","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/0370-1573(90)90099-N","volume":"195","author":"JP Bouchaud","year":"1990","unstructured":"Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195, 127\u2013293 (1990)","journal-title":"Phys. Rep."},{"issue":"12","key":"1398_CR5","doi-asserted-by":"publisher","first-page":"5096","DOI":"10.1063\/1.1416180","volume":"8","author":"BA Carreras","year":"2001","unstructured":"Carreras, B.A., Lynch, V.E., Zaslavsky, G.M.: Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas 8(12), 5096\u20135103 (2001)","journal-title":"Phys. Plasmas"},{"issue":"4","key":"1398_CR6","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.1016\/j.jcp.2011.11.008","volume":"231","author":"C \u00c7elik","year":"2012","unstructured":"\u00c7elik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231(4), 1743\u20131750 (2012)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"1398_CR7","doi-asserted-by":"publisher","first-page":"766","DOI":"10.1137\/0909051","volume":"9","author":"TF Chan","year":"1988","unstructured":"Chan, T.F.: An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Comput. 9(4), 766\u2013771 (1988)","journal-title":"SIAM J. Sci. Comput."},{"key":"1398_CR8","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.apnum.2013.03.006","volume":"70","author":"MH Chen","year":"2013","unstructured":"Chen, M.H., Deng, W.H., Wu, Y.J.: Superlinearly convergent algorithms for the two-dimensional space-time Caputo\u2013Riesz fractional diffusion equation. Appl. Numer. Math. 70, 22\u201341 (2013)","journal-title":"Appl. Numer. Math."},{"issue":"2","key":"1398_CR9","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1007\/s10915-016-0317-3","volume":"71","author":"H Ding","year":"2017","unstructured":"Ding, H., Li, C.: High-order numerical algorithms for Riesz derivatives via constructing new generating functions. J. Sci. Comput. 71(2), 759\u2013784 (2017)","journal-title":"J. Sci. Comput."},{"key":"1398_CR10","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/j.jcp.2015.11.061","volume":"307","author":"M Donatelli","year":"2016","unstructured":"Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262\u2013279 (2016)","journal-title":"J. Comput. Phys."},{"issue":"1\u20132","key":"1398_CR11","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/BF02935797","volume":"19","author":"FH Huang","year":"2005","unstructured":"Huang, F.H., Liu, F.W.: The space-time fractional diffusion equation with Caputo derivatives. J. Appl. Math. Comput. 19(1\u20132), 179 (2005)","journal-title":"J. Appl. Math. Comput."},{"key":"1398_CR12","volume-title":"Preconditioning Techniques for Toeplitz systems","author":"XQ Jin","year":"2010","unstructured":"Jin, X.Q.: Preconditioning Techniques for Toeplitz systems. Higher Education Press, Beijing (2010)"},{"key":"1398_CR13","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717907","volume-title":"Matrix Analysis for Scientists and Engineers","author":"AJ Laub","year":"2005","unstructured":"Laub, A.J.: Matrix Analysis for Scientists and Engineers. SIAM, Philadelphia (2005)"},{"key":"1398_CR14","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1016\/j.jcp.2013.02.025","volume":"242","author":"SL Lei","year":"2013","unstructured":"Lei, S.L., Sun, H.W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715\u2013725 (2013)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"1398_CR15","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1007\/s10444-015-9434-z","volume":"42","author":"C Li","year":"2016","unstructured":"Li, C., Deng, W.H.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42(3), 543\u2013572 (2016)","journal-title":"Adv. Comput. Math."},{"issue":"2","key":"1398_CR16","doi-asserted-by":"publisher","first-page":"282","DOI":"10.4208\/aamm.2015.m1069","volume":"9","author":"C Li","year":"2017","unstructured":"Li, C., Deng, W.H.: A new family of difference schemes for space fractional advection diffusion equation. Adv. Appl. Math. Mech. 9(2), 282\u2013306 (2017)","journal-title":"Adv. Appl. Math. Mech."},{"key":"1398_CR17","doi-asserted-by":"publisher","DOI":"10.1201\/b18503","volume-title":"Numerical Methods for Fractional Calculus","author":"CP Li","year":"2015","unstructured":"Li, C.P., Zeng, F.H.: Numerical Methods for Fractional Calculus. Chapman and Hall\/CRC, Cambridge (2015)"},{"key":"1398_CR18","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/j.cam.2019.06.008","volume":"363","author":"FR Lin","year":"2020","unstructured":"Lin, F.R., Liu, W.D.: The accuracy and stability of CN-WSGD schemes for space fractional diffusion equation. J. Comput. Appl. Math. 363, 77\u201391 (2020)","journal-title":"J. Comput. Appl. Math."},{"key":"1398_CR19","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.jcp.2013.07.040","volume":"256","author":"FR Lin","year":"2014","unstructured":"Lin, F.R., Yang, S.W., Jin, X.Q.: Preconditioned iterative methods for fractional diffusion equation. J. Comput. Phys. 256, 109\u2013117 (2014)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"1398_CR20","doi-asserted-by":"publisher","first-page":"1580","DOI":"10.1137\/17M1115447","volume":"38","author":"XL Lin","year":"2017","unstructured":"Lin, X.L., Ng, M.K., Sun, H.W.: A splitting preconditioner for Toeplitz-like linear systems arising from fractional diffusion equations. SIAM J. Matrix Anal. Appl. 38(4), 1580\u20131614 (2017)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"1398_CR21","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1007\/s10543-018-0699-8","volume":"58","author":"XL Lin","year":"2018","unstructured":"Lin, X.L., Ng, M.K., Sun, H.W.: Efficient preconditioner of one-sided space fractional diffusion equation. BIT Numer. Math. 58(3), 729\u2013748 (2018)","journal-title":"BIT Numer. Math."},{"issue":"1","key":"1398_CR22","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.cam.2004.01.033","volume":"172","author":"MM Meerschaert","year":"2004","unstructured":"Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65\u201377 (2004)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"1398_CR23","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1016\/j.apnum.2005.02.008","volume":"56","author":"MM Meerschaert","year":"2006","unstructured":"Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56(1), 80\u201390 (2006)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"1398_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0370-1573(00)00070-3","volume":"339","author":"R Metzler","year":"2000","unstructured":"Metzler, R., Klafter, J.: The random walk\u2019s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1\u201377 (2000)","journal-title":"Phys. Rep."},{"key":"1398_CR25","doi-asserted-by":"publisher","first-page":"992","DOI":"10.1016\/j.jcp.2017.08.064","volume":"350","author":"H Moghaderi","year":"2017","unstructured":"Moghaderi, H., Dehghan, M., Donatelli, M., Mazza, M.: Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations. J. Comput. Phys. 350, 992\u20131011 (2017)","journal-title":"J. Comput. Phys."},{"key":"1398_CR26","first-page":"173","volume":"6","author":"C Mortici","year":"2014","unstructured":"Mortici, C.: Methods and algorithms for approximating the Gamma function and related functions. A survey. Part I: Asymptotic series. Ann. Acad. Rom. Sci. Ser. Math. Appl. 6, 173 (2014)","journal-title":"Ann. Acad. Rom. Sci. Ser. Math. Appl."},{"issue":"6","key":"1398_CR27","doi-asserted-by":"publisher","first-page":"A2698","DOI":"10.1137\/130931795","volume":"36","author":"JY Pan","year":"2014","unstructured":"Pan, J.Y., Ke, R.H., Ng, M.K., Sun, H.W.: Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM J. Sci. Comput. 36(6), A2698\u2013A2719 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"1398_CR28","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1016\/j.jcp.2011.10.005","volume":"231","author":"HK Pang","year":"2012","unstructured":"Pang, H.K., Sun, H.W.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231, 693\u2013703 (2012)","journal-title":"J. Comput. Phys."},{"key":"1398_CR29","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003","volume-title":"Iterative Methods for Sparse Linear Systems","author":"Y Saad","year":"2003","unstructured":"Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)"},{"key":"1398_CR30","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.apnum.2014.11.007","volume":"90","author":"E Sousa","year":"2015","unstructured":"Sousa, E., Li, C.: A weighted finite difference method for the fractional diffusion equation based on the Riemann\u2013Liouville derivative. Appl. Numer. Math. 90, 22\u201337 (2015)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"1398_CR31","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.jcp.2005.08.008","volume":"213","author":"C Tadjeran","year":"2006","unstructured":"Tadjeran, C., Meershaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213(1), 205\u2013213 (2006)","journal-title":"J. Comput. Phys."},{"issue":"294","key":"1398_CR32","doi-asserted-by":"publisher","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","volume":"84","author":"W Tian","year":"2015","unstructured":"Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703\u20131727 (2015)","journal-title":"Math. Comput."},{"issue":"21","key":"1398_CR33","doi-asserted-by":"publisher","first-page":"8095","DOI":"10.1016\/j.jcp.2010.07.011","volume":"229","author":"H Wang","year":"2010","unstructured":"Wang, H., Wang, K., Sircar, T.: A direct $$\\cal{O}(N\\log ^{2}N)$$ finite difference method for fractional diffusion equations. J. Comput. Phys. 229(21), 8095\u20138104 (2010)","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01398-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-020-01398-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01398-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,26]],"date-time":"2021-02-26T08:29:50Z","timestamp":1614328190000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-020-01398-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,16]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["1398"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01398-4","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,16]]},"assertion":[{"value":"5 February 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 September 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 December 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"31"}}