{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T23:43:44Z","timestamp":1773791024403,"version":"3.50.1"},"reference-count":42,"publisher":"Elsevier","isbn-type":[{"value":"9780120121038","type":"print"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1962]]},"DOI":"10.1016\/s0065-2458(08)60620-8","type":"book-chapter","created":{"date-parts":[[2008,5,30]],"date-time":"2008-05-30T04:21:49Z","timestamp":1212121309000},"page":"189-273","source":"Crossref","is-referenced-by-count":77,"title":["Alternating Direction Implicit Methods"],"prefix":"10.1016","author":[{"given":"Garrett","family":"Birkhoff","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richard S.","family":"Varga","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Young","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/S0065-2458(08)60620-8_bib1","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1090\/S0002-9947-1959-0105814-4","article-title":"Implicit alternating direction methods","volume":"92","author":"Birkhoff","year":"1959","journal-title":"Trans. AMS"},{"key":"10.1016\/S0065-2458(08)60620-8_bib2","first-page":"79","article-title":"Calculation of unsteady-state gas flow through porous media","volume":"198","author":"Bruce","year":"1953","journal-title":"Trans. AIMME"},{"key":"10.1016\/S0065-2458(08)60620-8_bib3","doi-asserted-by":"crossref","first-page":"198","DOI":"10.2307\/2002021","article-title":"An alternating direction scheme for the biharmonic difference equations","volume":"12","author":"Conte","year":"1958","journal-title":"Math. Tables Aid Computing."},{"key":"10.1016\/S0065-2458(08)60620-8_bib4","unstructured":"de Boor C. M.J.R. Rice, Tchebycheff approximation by \u03b1II[(x \u2212rj)(x +rj)] and application to ADI iteration. To appear in J. Soc. Ind. AppL. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib5","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1090\/S0002-9939-1957-0090876-7","article-title":"A note on the alternating direction implcitcit method for the numerical solution of heat flow problems","volume":"8","author":"Douglas","year":"1957","journal-title":"Proc. AMS"},{"key":"10.1016\/S0065-2458(08)60620-8_bib6","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1137\/0103004","article-title":"On the numerical integration of \u03c32u\/\u03c3x2 + \u03c32u\/\u03c3y2 = \u03c3u\/\u03c3t by implicit methods","volume":"3","author":"Douglas","year":"1955","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib7","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/BF01386006","article-title":"Alternating direction iteration for mildly nonlinear elliptic differential equations","volume":"3","author":"Douglas","year":"1961","journal-title":"Numerical. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib8","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1090\/S0002-9947-1956-0084194-4","article-title":"On the numerical solution of heat conduction problems in two and three space variables","volume":"82","author":"Douglas","year":"1956","journal-title":"Trans. AMS"},{"key":"10.1016\/S0065-2458(08)60620-8_bib9","series-title":"Finite-difference Methods for Partial Differential Equations","author":"Forsythe","year":"1960"},{"key":"10.1016\/S0065-2458(08)60620-8_bib10","series-title":"Finite Differences.","author":"Fort","year":"1948"},{"key":"10.1016\/S0065-2458(08)60620-8_bib11","doi-asserted-by":"crossref","first-page":"65","DOI":"10.2307\/2002770","article-title":"Convergence rates of iterative treatments of partial differential equations","volume":"4","author":"Frankel","year":"1950","journal-title":"Math. Tables Aid Comput."},{"key":"10.1016\/S0065-2458(08)60620-8_bib12","first-page":"445","article-title":"Sur les matrices completement non-negatives et oscillatoires","volume":"4","author":"Gantmakher","year":"1937","journal-title":"Compositio Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib13_1","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01386013","article-title":"Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods, I","volume":"3","author":"Golub","year":"1961","journal-title":"Numer. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib13_2","first-page":"157","volume":"3","year":"1962","journal-title":"Numer. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib14","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1137\/0108009","article-title":"Simultaneous successive and alternating direction schemes","volume":"8","author":"Heller","year":"1960","journal-title":"J. Soc. Ind. and Appl. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib15","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1145\/320932.320933","article-title":"The approximate solution of matrix problems","volume":"5","author":"Householder","year":"1958","journal-title":"J. Assoc. Computing Machinery"},{"key":"10.1016\/S0065-2458(08)60620-8_bib16","first-page":"68","article-title":"Les m\u00e9thodes de solution approch\u00e9e des probl\u00e8mes de la physique math\u00e9matique","author":"Kryloff","year":"1931","journal-title":"M\u00e9m. Sci. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib17","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1007\/BF01386038","article-title":"Alternating direction and semi-explicit difference methods for parabolic partial differential equations","volume":"3","author":"Lees","year":"1962","journal-title":"Numer. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib18","first-page":"140","article-title":"On the linear iterative procedures for symmetric matrices","volume":"14","author":"Ostrowski","year":"1954","journal-title":"Rend. mat. appl. [5]"},{"key":"10.1016\/S0065-2458(08)60620-8_bib19","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/BF01386031","article-title":"\u201cMulti-line\u201d iterative methods for elliptic difference equations and fundamental frequencies","volume":"3","author":"appl","year":"1961","journal-title":"Numer. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib20","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1137\/0103003","article-title":"The numerical solution of parabolic and elliptic differential equations","volume":"3","author":"Peaceman","year":"1955","journal-title":"J. Soc. Ind. and Appl. Math."},{"issue":"88","key":"10.1016\/S0065-2458(08)60620-8_bib21_1","first-page":"2063","article-title":"Sur un proc\u00e9d\u00e9 convergent d'approximations successives pour d\u00e9terminer les polynomes d'approximation","volume":"19","author":"Remes","year":"1934","journal-title":"Compt. rend. acad. sci."},{"key":"10.1016\/S0065-2458(08)60620-8_bib21_2","first-page":"337","article-title":"Sur le calcul effectif des polynomes d'approximation de Tchebichef","volume":"199","year":"1934","journal-title":"Compt. rend. acad. sci."},{"key":"10.1016\/S0065-2458(08)60620-8_bib22","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1090\/S0002-9947-1961-0136913-8","article-title":"Tchebycheff approximations by functions unisolvent of variable degree","volume":"99","author":"Rice","year":"1961","journal-title":"Trans. AMS"},{"key":"10.1016\/S0065-2458(08)60620-8_bib23","doi-asserted-by":"crossref","first-page":"1326","DOI":"10.1063\/1.1699598","volume":"21","author":"Shortley","year":"1950","journal-title":"J. Appl. Phys."},{"key":"10.1016\/S0065-2458(08)60620-8_bib24","series-title":"On Numerical Approximations","first-page":"217","article-title":"Numerical methods of Tchebycheff approximation","author":"Stiefel","year":"1959"},{"key":"10.1016\/S0065-2458(08)60620-8_bib25","series-title":"Vector Spaces and Matrices","author":"Thrall","year":"1957"},{"key":"10.1016\/S0065-2458(08)60620-8_bib26","unstructured":"R.S. Varga, Overrelaxation applied to implicit alternating direction methods. Proc. Intern. Congr. on Information Processing, Paris, pp. 85\u201390, June (1958)"},{"key":"10.1016\/S0065-2458(08)60620-8_bib27","doi-asserted-by":"crossref","first-page":"617","DOI":"10.2140\/pjm.1959.9.617","article-title":"P-cyclic matrices: A generalization of the Young-Frankel successive overrelaxation scheme","volume":"9","author":"Varga","year":"1959","journal-title":"Pacific J. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib28","series-title":"\u201cMatrix Iterative Analysis\u201d","author":"Varga","year":"1962"},{"key":"10.1016\/S0065-2458(08)60620-8_bib29","series-title":"Boundary Problems in Differential Equations","article-title":"Factorization and normalized iterative methods","author":"Varga","year":"1960"},{"key":"10.1016\/S0065-2458(08)60620-8_bib30","doi-asserted-by":"crossref","first-page":"925","DOI":"10.2140\/pjm.1959.9.925","article-title":"Orderings of the successive overrelaxation scheme","volume":"9","author":"Varga","year":"1959","journal-title":"Pacific J. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib31","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1002\/sapm1961401220","article-title":"Higher order stable implicit methods for solving parabolic partial differential equations. J","volume":"40","author":"Varga","year":"1961","journal-title":"Math. and Phys."},{"key":"10.1016\/S0065-2458(08)60620-8_bib32","series-title":"CURE: a generalized two-space-dimension multigroup coding for the IBM 704","author":"Wachspress","year":"1957"},{"key":"10.1016\/S0065-2458(08)60620-8_bib33","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1137\/0108027","article-title":"An alternating-direction-implicit iteration technique","volume":"8","author":"Wachspress","year":"1960","journal-title":"J. Soc. Ind. and Appl. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib34","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1137\/0110025","article-title":"Optimum alternating-direction-implicit iteration parameters for a model problem","volume":"10","author":"Wachspress","year":"1962","journal-title":"J. Soc. Ind. and Appl. Math."},{"key":"10.1016\/S0065-2458(08)60620-8_bib35","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/BF01456804","article-title":"Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen","volume":"71","author":"Weyl","year":"1912","journal-title":"Math. Ann. (Leipzig)"},{"key":"10.1016\/S0065-2458(08)60620-8_bib36","unstructured":"D. Young, Iterative methods for solving partial difference equations of elliptic type. Ph.D. Thesis, Harvard (1950)"},{"key":"10.1016\/S0065-2458(08)60620-8_bib37","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1090\/S0002-9947-1954-0059635-7","article-title":"Iterative methods for solving partial difference equations of elliptic type","volume":"76","author":"Young","year":"1954","journal-title":"Trans. AMS"},{"key":"10.1016\/S0065-2458(08)60620-8_bib38","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1145\/320802.320803","article-title":"Ordvac solutions of the Dirichlet problem","volume":"2","author":"Young","year":"1955","journal-title":"J. Assoc. Computing Machinery"},{"key":"10.1016\/S0065-2458(08)60620-8_bib39","doi-asserted-by":"crossref","DOI":"10.1090\/psapm\/006\/0081539","article-title":"On the solution of linear systems by iteration","volume":"6","author":"Young","year":"1956","journal-title":"AMS Symposium on Numer. Anal."},{"key":"10.1016\/S0065-2458(08)60620-8_bib40","series-title":"Numerical Experiments Involving Boundary Problems in Differential Equations","author":"Young","year":"1960"}],"container-title":["Advances in Computers","Advances in Computers Volume 3"],"original-title":[],"language":"en","deposited":{"date-parts":[[2018,12,28]],"date-time":"2018-12-28T01:57:33Z","timestamp":1545962253000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0065245808606208"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1962]]},"ISBN":["9780120121038"],"references-count":42,"URL":"https:\/\/doi.org\/10.1016\/s0065-2458(08)60620-8","relation":{},"ISSN":["0065-2458"],"issn-type":[{"value":"0065-2458","type":"print"}],"subject":[],"published":{"date-parts":[[1962]]}}}