{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T10:42:52Z","timestamp":1775558572840,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,8,8]],"date-time":"2016-08-08T00:00:00Z","timestamp":1470614400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2017,5]]},"DOI":"10.1007\/s00211-016-0834-x","type":"journal-article","created":{"date-parts":[[2016,8,8]],"date-time":"2016-08-08T06:08:58Z","timestamp":1470636538000},"page":"75-99","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["Legendre spectral collocation in space and time for PDEs"],"prefix":"10.1007","volume":"136","author":[{"given":"S. H.","family":"Lui","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,8,8]]},"reference":[{"key":"834_CR1","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1145\/361573.361582","volume":"15","author":"RH Bartels","year":"1972","unstructured":"Bartels, R.H., Stewart, G.W.: A solution of the equation $$AX+XB=C$$ A X + X B = C . Commun. ACM 15, 820\u2013826 (1972)","journal-title":"Commun. ACM"},{"key":"834_CR2","series-title":"pp","first-page":"209","volume-title":"Handbook of Numerical Analysis","author":"C Bernardi","year":"1997","unstructured":"Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. pp, vol. 5, pp. 209\u2013485. North-Holland, Amsterdam (1997)"},{"key":"834_CR3","unstructured":"Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd Rev. Ed. Dover, Mineola (2001)"},{"key":"834_CR4","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1016\/j.cnsns.2014.05.030","volume":"20","author":"L Brugnano","year":"2015","unstructured":"Brugnano, L., Iavernaro, F., Trigiante, D.: Analysis of Hamiltonian boundary value methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynoial Hamiltonian systems. Commum. Nonlinear Sci. Numer. Simul. 20, 650\u2013667 (2015)","journal-title":"Commum. Nonlinear Sci. Numer. Simul."},{"key":"834_CR5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral Methods-Fundamentals in Single Domains","author":"C Canuto","year":"2006","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods-Fundamentals in Single Domains. Springer, New York (2006)"},{"key":"834_CR6","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1006\/jcph.1996.0234","volume":"129","author":"MH Carpenter","year":"1996","unstructured":"Carpenter, M.H., Gottlieb, D.: Spectral methods on arbitrary grids. J. Comput. Phys. 129, 74\u201386 (1996)","journal-title":"J. Comput. Phys."},{"key":"834_CR7","doi-asserted-by":"crossref","first-page":"C233","DOI":"10.1137\/110843484","volume":"34","author":"AJ Christlieb","year":"2012","unstructured":"Christlieb, A.J., Haynes, R.D., Ong, B.W.: A parallel space-time algorithm. SIAM J. Sci. Stat. Comput. 34, C233\u2013C248 (2012)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"834_CR8","doi-asserted-by":"crossref","first-page":"A52","DOI":"10.1137\/110861002","volume":"35","author":"X Dai","year":"2013","unstructured":"Dai, X., Maday, Y.: Stable parareal in time method for first- and second-order hyperbolic systems. SIAM J. Sci. Comput. 35, A52\u2013A78 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"834_CR9","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1023\/A:1022338906936","volume":"40","author":"A Dutt","year":"2000","unstructured":"Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40, 241\u2013266 (2000)","journal-title":"BIT"},{"key":"834_CR10","doi-asserted-by":"crossref","unstructured":"Falgout, R.D., Friedhoff, S., Kolev, Tz.V., Maclachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36, C625\u2013C661 (2014)","DOI":"10.1137\/130944230"},{"key":"834_CR11","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511626357","volume-title":"A Practical Guide to Pseudospectral Methods","author":"B Fornberg","year":"1996","unstructured":"Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)"},{"key":"834_CR12","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-59185-3","volume-title":"Spectral Elements for Transport-Dominated Equations","author":"D Funaro","year":"1997","unstructured":"Funaro, D.: Spectral Elements for Transport-Dominated Equations. Springer, Berlin (1997)"},{"key":"834_CR13","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1137\/05064607X","volume":"29","author":"MJ Gander","year":"2007","unstructured":"Gander, M.J., Vandewalle, S.: Analysis of the parareal time-parallel time-integration method. SIAM J. Sci. Comput. 29, 556\u2013578 (2007)","journal-title":"SIAM J. Sci. Comput."},{"key":"834_CR14","doi-asserted-by":"crossref","unstructured":"Golub, G.H., Nash, S., Van Loan, C.F.: Hessenberg\u2013Schur method for the problem $$ax+xb=c$$ a x + x b = c . IEEE Trans. Autom. Control, AC-24, pp. 909\u2013913 (1979)","DOI":"10.1109\/TAC.1979.1102170"},{"key":"834_CR15","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970425","volume-title":"Numerical Analysis of Spectral Methods","author":"D Gottlieb","year":"1977","unstructured":"Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods. SIAM, Philadelphia (1977)"},{"key":"834_CR16","doi-asserted-by":"crossref","DOI":"10.1142\/3662","volume-title":"Spectral Methods and Their Applications","author":"BY Guo","year":"1998","unstructured":"Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)"},{"key":"834_CR17","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s10444-008-9067-6","volume":"30","author":"B-Y Guo","year":"2009","unstructured":"Guo, B.-Y., Wang, Z.-Q.: Legendre-Gauss collocation methods for ordinary differential equations. Adv. Comput. Math. 30, 249\u2013280 (2009)","journal-title":"Adv. Comput. Math."},{"key":"834_CR18","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511618352","volume-title":"Spectral Methods for Time-Dependent Problems","author":"J Hesthaven","year":"2007","unstructured":"Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Philadelphia (2007)"},{"key":"834_CR19","doi-asserted-by":"crossref","first-page":"848","DOI":"10.1137\/0916050","volume":"16","author":"G Horton","year":"1995","unstructured":"Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16, 848\u2013864 (1995)","journal-title":"SIAM J. Sci. Comput."},{"key":"834_CR20","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/s11075-015-0002-x","volume":"71","author":"W Liu","year":"2016","unstructured":"Liu, W., Sun, J., Wu, B.: Galerkin\u2013chebyshev spectral method and block boundary value methods for two-dimensional semilinear parabolic equations. Numer. Algorithms 71, 437\u2013455 (2016)","journal-title":"Numer. Algorithms"},{"key":"834_CR21","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1002\/num.21910","volume":"31","author":"W Liu","year":"2015","unstructured":"Liu, W., Wu, B., Sun, J.: Space-time spectral collocation method for the one-dimensional Sine\u2013Gordon equation. Numer. Methods PDEs 31, 670\u2013690 (2015)","journal-title":"Numer. Methods PDEs"},{"key":"834_CR22","volume-title":"Numerical Analysis of Partial Differential Equations","author":"SH Lui","year":"2011","unstructured":"Lui, S.H.: Numerical Analysis of Partial Differential Equations. Wiley, Hoboken (2011)"},{"key":"834_CR23","unstructured":"McDonald, E.G., Wathen, A.J.: A simple proposal for parallel computation over time of an evolutionary process with implicit time stepping. Technical report, The Mathematical Institute, University of Oxford Technical Report, vol. 1860 (2014)"},{"key":"834_CR24","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral Methods","author":"J Shen","year":"2011","unstructured":"Shen, J., Tang, T., Wang, L.-L.: Spectral Methods. Springer, Berlin (2011)"},{"key":"834_CR25","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1016\/j.apnum.2006.07.012","volume":"57","author":"J Shen","year":"2007","unstructured":"Shen, J., Wang, L.-L.: Fourierization of the Legendre\u2013Galerkin method and a new space-time spectral method. Appl. Numer. Math. 57, 710\u2013720 (2007)","journal-title":"Appl. Numer. Math."},{"key":"834_CR26","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1137\/0723002","volume":"23","author":"H Tal-Ezer","year":"1986","unstructured":"Tal-Ezer, H.: Spectral methods in time for hyperbolic equations. SIAM J. Numer. Anal. 23, 11\u201326 (1986)","journal-title":"SIAM J. Numer. Anal."},{"key":"834_CR27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/0726001","volume":"26","author":"H Tal-Ezer","year":"1989","unstructured":"Tal-Ezer, H.: Spectral methods in time for parabolic problems. SIAM J. Numer. Anal. 26, 1\u201311 (1989)","journal-title":"SIAM J. Numer. Anal."},{"key":"834_CR28","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1023\/A:1016273820035","volume":"17","author":"J-G Tang","year":"2002","unstructured":"Tang, J.-G., Ma, H.-P.: Single and multi-interval Legendre $$\\tau $$ \u03c4 -methods in time for parabolic equations. Adv. Comput. Math. 17, 349\u2013367 (2002)","journal-title":"Adv. Comput. Math."},{"key":"834_CR29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.apnum.2005.11.009","volume":"57","author":"J-G Tang","year":"2007","unstructured":"Tang, J.-G., Ma, H.-P.: A Legendre spectral method in time for first-order hyperbolic equations. Appl. Numer. Math. 57, 1\u201311 (2007)","journal-title":"Appl. Numer. Math."},{"key":"834_CR30","first-page":"779","volume":"5","author":"T Tang","year":"2009","unstructured":"Tang, T., Xu, X.: Accuracy enhancement using spectral postprocessing for differential equations and integral equations. Commun. Comput. Phys. 5, 779\u2013792 (2009)","journal-title":"Commun. Comput. Phys."},{"key":"834_CR31","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719598","volume-title":"Spectral Methods in Matlab","author":"LN Trefethen","year":"2000","unstructured":"Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)"},{"key":"834_CR32","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1137\/130922409","volume":"36","author":"L-L Wang","year":"2014","unstructured":"Wang, L.-L., Samson, M.D., Zhao, X.: A well-conditioned collocation method using a pseudospectral integration matrix. SIAM J. Sci. Comput. 36, 907\u2013929 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"834_CR33","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1007\/BF02025886","volume":"13","author":"S Wu","year":"1997","unstructured":"Wu, S., Liu, X.: Convergence of spectral method in time for Burgers\u2019 equation. Acta Math. Appl. Sin. 13, 314\u2013320 (1997)","journal-title":"Acta Math. Appl. Sin."},{"key":"834_CR34","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1090\/S0025-5718-2012-02645-7","volume":"82","author":"Z Xie","year":"2012","unstructured":"Xie, Z., Wang, L.-L., Zhao, X.: On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math. Comput. 82, 1017\u20131036 (2012)","journal-title":"Math. Comput."},{"key":"834_CR35","doi-asserted-by":"crossref","first-page":"1434","DOI":"10.1080\/00207160.2013.841901","volume":"91","author":"L Yi","year":"2014","unstructured":"Yi, L., Wang, Z.: Legendre\u2013Gauss-type spectral collocation algorithms for nonlinear ordinary\/partial differential equations. Int. J. Comput. Math. 91, 1434\u20131460 (2014)","journal-title":"Int. J. Comput. Math."},{"key":"834_CR36","first-page":"299","volume":"19","author":"L Yi","year":"2014","unstructured":"Yi, L., Wang, Z.: Legendre spectral collocation method for second-order nonlinear ordinary\/partial differential equations. Discrete Contin. Dyn. Syst. B 19, 299\u2013322 (2014)","journal-title":"Discrete Contin. Dyn. Syst. B"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-016-0834-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-016-0834-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-016-0834-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-016-0834-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,12]],"date-time":"2019-09-12T08:13:39Z","timestamp":1568276019000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-016-0834-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,8]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,5]]}},"alternative-id":["834"],"URL":"https:\/\/doi.org\/10.1007\/s00211-016-0834-x","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,8,8]]}}}