{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T14:13:07Z","timestamp":1742911987550,"version":"3.40.3"},"publisher-location":"Cham","reference-count":28,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783319663197"},{"type":"electronic","value":"9783319663203"}],"license":[{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017]]},"DOI":"10.1007\/978-3-319-66320-3_25","type":"book-chapter","created":{"date-parts":[[2017,8,29]],"date-time":"2017-08-29T11:34:27Z","timestamp":1504006467000},"page":"346-361","source":"Crossref","is-referenced-by-count":2,"title":["The Method of Collocations and Least Residuals Combining the Integral Form of Collocation Equations and the Matching Differential Relations at the Solution of PDEs"],"prefix":"10.1007","author":[{"given":"Vasily P.","family":"Shapeev","sequence":"first","affiliation":[]},{"given":"Evgenii V.","family":"Vorozhtsov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,8,30]]},"reference":[{"key":"25_CR1","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1007\/978-3-319-02297-0_4","volume-title":"Computer Algebra in Scientific Computing","author":"P Amodio","year":"2013","unstructured":"Amodio, P., Blinkov, Y., Gerdt, V., La Scala, R.: On consistency of finite difference approximations to the Navier-Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 46\u201360. Springer, Cham (2013). doi: 10.1007\/978-3-319-02297-0_4"},{"key":"25_CR2","doi-asserted-by":"crossref","first-page":"161","DOI":"10.2516\/ogst:1999010","volume":"54","author":"O Bailly","year":"1999","unstructured":"Bailly, O., Buchou, C., Floch, A., Sainsaulieu, L.: Simulation of the intake and compression strokes of a motored 4-valve SI engine with a finite element code. Oil Gas Sci. Technol. 54, 161\u2013168 (1999)","journal-title":"Oil Gas Sci. Technol."},{"key":"25_CR3","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/S0045-7930(98)00002-4","volume":"27","author":"O Botella","year":"1998","unstructured":"Botella, O., Peyret, R.: Benchmark spectral results on the lid-driven cavity flow. Comput. Fluids 27, 421\u2013433 (1998)","journal-title":"Comput. Fluids"},{"key":"25_CR4","doi-asserted-by":"crossref","first-page":"4018","DOI":"10.1016\/j.jcp.2007.12.007","volume":"227","author":"PH Chiu","year":"2008","unstructured":"Chiu, P.H., Sheu, T.W.H., Lin, R.K.: An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier-Stokes equations. J. Comput. Phys. 227, 4018\u20134037 (2008)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"25_CR5","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0041-5553(64)90253-8","volume":"4","author":"RP Fedorenko","year":"1964","unstructured":"Fedorenko, R.P.: The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4(3), 227\u2013235 (1964)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"25_CR6","first-page":"740","volume":"10","author":"P Fritzson","year":"2008","unstructured":"Fritzson, P., Engelson, V., Sheshadri, K.: MathCode: a system for C++ or Fortran code generation from Mathematica. Math. J. 10, 740\u2013777 (2008)","journal-title":"Math. J."},{"key":"25_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1007\/3-540-15984-3_290","volume-title":"EUROCAL \u201985","author":"VG Ganzha","year":"1985","unstructured":"Ganzha, V.G., Mazurik, S.I., Shapeev, V.P.: Symbolic manipulations on a computer and their application to generation and investigation of difference schemes. In: Caviness, B.F. (ed.) EUROCAL 1985. LNCS, vol. 204, pp. 335\u2013347. Springer, Heidelberg (1985). doi: 10.1007\/3-540-15984-3_290"},{"key":"25_CR8","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1007\/978-3-642-04103-7_10","volume-title":"Computer Algebra in Scientific Computing","author":"VP Gerdt","year":"2009","unstructured":"Gerdt, V.P., Blinkov, Y.A.: Involution and difference schemes for the Navier\u2013Stokes Equations. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 94\u2013105. Springer, Heidelberg (2009). doi: 10.1007\/978-3-642-04103-7_10"},{"key":"25_CR9","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/0021-9991(82)90058-4","volume":"48","author":"U Ghia","year":"1982","unstructured":"Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387\u2013411 (1982)","journal-title":"J. Comput. Phys."},{"key":"25_CR10","doi-asserted-by":"crossref","first-page":"1670","DOI":"10.1134\/S0965542510100040","volume":"50","author":"VI Isaev","year":"2010","unstructured":"Isaev, V.I., Shapeev, V.P.: High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations. Comput. Math. Math. Phys. 50, 1670\u20131681 (2010)","journal-title":"Comput. Math. Math. Phys."},{"issue":"3\u20134","key":"25_CR11","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/0045-7825(94)90172-4","volume":"114","author":"B Jiang","year":"1994","unstructured":"Jiang, B., Lin, T.L., Povinelli, L.A.: Large-scale computation of incompressible viscous flow by least-squares finite element method. Comput. Meth. Appl. Mech. Eng. 114(3\u20134), 213\u2013231 (1994)","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"25_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03740-9","volume-title":"The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics","author":"BN Jiang","year":"1998","unstructured":"Jiang, B.N.: The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, Berlin (1998). doi: 10.1007\/978-3-662-03740-9"},{"key":"25_CR13","first-page":"491","volume":"4","author":"AN Krylov","year":"1931","unstructured":"Krylov, A.N.: On the numerical solution of the equation, which determines in technological questions the frequencies of small oscillations of material systems. Izv. AN SSSR, Otd. matem. i estestv. nauk 4, 491\u2013539 (1931). (in Russian)","journal-title":"Izv. AN SSSR, Otd. matem. i estestv. nauk"},{"issue":"4","key":"25_CR14","doi-asserted-by":"crossref","first-page":"498","DOI":"10.2514\/2.6795","volume":"17","author":"K Li","year":"2003","unstructured":"Li, K., Li, Q.: Three-dimensional gravity-jitter induced melt flow and solidification in magnetic fields. J. Thermophys. Heat Transf. 17(4), 498\u2013508 (2003)","journal-title":"J. Thermophys. Heat Transf."},{"issue":"4","key":"25_CR15","first-page":"116","volume":"18","author":"AV Plyasunova","year":"1987","unstructured":"Plyasunova, A.V., Sleptsov, A.G.: Collocation-grid method of solving the nonlinear parabolic equations on moving grids. Modelirovanie v mekhanike 18(4), 116\u2013137 (1987)","journal-title":"Modelirovanie v mekhanike"},{"key":"25_CR16","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/11555964_31","volume-title":"Computer Algebra in Scientific Computing","author":"L Semin","year":"2005","unstructured":"Semin, L., Shapeev, V.: Constructing the numerical method for Navier \u2014 Stokes equations using computer algebra system. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 367\u2013378. Springer, Heidelberg (2005). doi: 10.1007\/11555964_31"},{"issue":"2","key":"25_CR17","first-page":"90","volume":"1","author":"LG Semin","year":"1996","unstructured":"Semin, L.G., Sleptsov, A.G., Shapeev, V.P.: Collocation and least -squares method for Stokes equations. Comput. Technol. 1(2), 90\u201398 (1996). (in Russian)","journal-title":"Comput. Technol."},{"key":"25_CR18","doi-asserted-by":"crossref","first-page":"1874","DOI":"10.1137\/080719145","volume":"31","author":"AV Shapeev","year":"2009","unstructured":"Shapeev, A.V., Lin, P.: An asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows. SIAM J. Sci. Comput. 31, 1874\u20131900 (2009)","journal-title":"SIAM J. Sci. Comput."},{"key":"25_CR19","doi-asserted-by":"publisher","first-page":"01009","DOI":"10.1051\/epjconf\/201610801009","volume":"108","author":"V Shapeev","year":"2016","unstructured":"Shapeev, V.: Collocation and least residuals method and its applications. EPJ Web Conf. 108, 01009 (2016). doi: 10.1051\/epjconf\/201610801009","journal-title":"EPJ Web Conf."},{"key":"25_CR20","unstructured":"Shapeev, V.P., Isaev, V.I., Idimeshev, S.V.: The collocations and least squares method: application to numerical solution of the Navier-Stokes equations. In: Eberhardsteiner, J., B\u00f6hm, H.J., Rammerstorfer, F.G. (eds.) CD-ROM Proceedings of the 6th ECCOMAS, September 2012. Vienna University of Technology (2012). ISBN: 978-3-9502481-9-7"},{"key":"25_CR21","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/978-3-319-02297-0_31","volume-title":"Computer Algebra in Scientific Computing","author":"VP Shapeev","year":"2013","unstructured":"Shapeev, V.P., Vorozhtsov, E.V.: CAS application to the construction of the collocations and least residuals method for the solution of 3D Navier\u2013Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 381\u2013392. Springer, Cham (2013). doi: 10.1007\/978-3-319-02297-0_31"},{"key":"25_CR22","first-page":"306","volume":"14","author":"VP Shapeev","year":"2013","unstructured":"Shapeev, V.P., Vorozhtsov, E.V., Isaev, V.I., Idimeshev, S.V.: The method of collocations and least residuals for three-dimensional Navier-Stokes equations. Vychislit. metody i programmirovanie 14, 306\u2013322 (2013). (in Russian)","journal-title":"Vychislit. metody i programmirovanie"},{"key":"25_CR23","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/978-3-642-32973-9_27","volume-title":"Computer Algebra in Scientific Computing","author":"VP Shapeev","year":"2012","unstructured":"Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numeric implementation of the method of collocations and least squares for 3D Navier\u2013Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 321\u2013333. Springer, Heidelberg (2012). doi: 10.1007\/978-3-642-32973-9_27"},{"key":"25_CR24","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/978-3-319-45641-6_30","volume-title":"Computer Algebra in Scientific Computing","author":"VP Shapeev","year":"2016","unstructured":"Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numerical optimization and realization of the method of collocations and least residuals for solving the Navier\u2013Stokes equations. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 473\u2013488. Springer, Cham (2016). doi: 10.1007\/978-3-319-45641-6_30"},{"key":"25_CR25","unstructured":"Valiullin, A.N., Ganzha, V.G., Meleshko, S.V., Murzin, F.A., Shapeev, V.P., Yanenko, N.N.: Application of Symbolic Manipulations on a Computer for Generation and Analysis of Difference Schemes. Prepr. Inst. Theor. Appl. Mech. Siberian Branch of the USSR Acad. Sci., Novosibirsk No. 7 (1981). (in Russian)"},{"key":"25_CR26","doi-asserted-by":"crossref","first-page":"80","DOI":"10.26089\/NumMet.v18r107","volume":"18","author":"EV Vorozhtsov","year":"2017","unstructured":"Vorozhtsov, E.V., Shapeev, V.P.: On combining the techniques for convergence acceleration of iteration processes during the numerical solution of Navier-Stokes equations. Vychislit. metody i programmirovanie 18, 80\u2013102 (2017). (in Russian)","journal-title":"Vychislit. metody i programmirovanie"},{"key":"25_CR27","volume-title":"An Introduction to Multigrid Methods","author":"P Wesseling","year":"1992","unstructured":"Wesseling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1992)"},{"key":"25_CR28","volume-title":"The Mathematica Book","author":"S Wolfram","year":"2003","unstructured":"Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media Inc., Champaign (2003)","edition":"5"}],"container-title":["Lecture Notes in Computer Science","Computer Algebra in Scientific Computing"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-66320-3_25","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T22:14:35Z","timestamp":1659392075000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-319-66320-3_25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319663197","9783319663203"],"references-count":28,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-66320-3_25","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2017]]}}}