{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T17:59:51Z","timestamp":1776362391845,"version":"3.51.2"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T00:00:00Z","timestamp":1675209600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T00:00:00Z","timestamp":1675209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Postgraduate Scientific Research Innovation Project of Hunan Province","award":["CX20210607"],"award-info":[{"award-number":["CX20210607"]}]},{"name":"Postgraduate Scientific Research Innovation Project of Xiangtan University","award":["XDCX2021B110"],"award-info":[{"award-number":["XDCX2021B110"]}]},{"name":"Excellent Youth Foundation of SINOPEC","award":["P20009"],"award-info":[{"award-number":["P20009"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971472"],"award-info":[{"award-number":["11971472"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971414"],"award-info":[{"award-number":["11971414"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["CCF Trans. HPC"],"published-print":{"date-parts":[[2023,6]]},"DOI":"10.1007\/s42514-023-00136-0","type":"journal-article","created":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T03:52:43Z","timestamp":1675223563000},"page":"144-159","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An improved multistage preconditioner on GPUs for compositional reservoir simulation"],"prefix":"10.1007","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3850-4610","authenticated-orcid":false,"given":"Li","family":"Zhao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shizhe","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chen-Song","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunsheng","family":"Feng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shi","family":"Shu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,2,1]]},"reference":[{"key":"136_CR1","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1023\/A:1021291114475","volume":"6","author":"I Aavatsmark","year":"2002","unstructured":"Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405\u2013432 (2002). https:\/\/doi.org\/10.1023\/A:1021291114475","journal-title":"Comput. Geosci."},{"issue":"5","key":"136_CR2","doi-asserted-by":"publisher","first-page":"1329","DOI":"10.1002\/num.20320","volume":"24","author":"I Aavatsmark","year":"2008","unstructured":"Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ. 24(5), 1329\u20131360 (2008). https:\/\/doi.org\/10.1002\/num.20320","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"136_CR3","doi-asserted-by":"publisher","unstructured":"Al-Shaalan, T.M., Klie, H.M., Dogru, A.H., Wheeler, M.F.: Studies of robust two stage preconditioners for the solution of fully implicit multiphase flow problems. In: SPE Reservoir Simulation Symposium (2009). https:\/\/doi.org\/10.2118\/118722-MS","DOI":"10.2118\/118722-MS"},{"key":"136_CR4","first-page":"687","volume-title":"Petroleum reservoir simulation","author":"K Aziz","year":"1979","unstructured":"Aziz, K., Settari, A.: Petroleum reservoir simulation, p. 687. Applied Science Publishers, London (1979)"},{"key":"136_CR5","first-page":"257","volume-title":"Sparsity and its applications","author":"A Brandt","year":"1984","unstructured":"Brandt, A., Mccormick, S.F., Ruge, J.W.: Algebraic multigrid (AMG) for sparse matrix equations. In: Sparsity and its applications, pp. 257\u2013284. Cambridge University Press, Cambridge (1984)"},{"key":"136_CR6","doi-asserted-by":"publisher","unstructured":"Cao, H., Tchelepi, H.A., Wallis, J.R., Yardumian, H.E.: Parallel scalable unstructured CPR-Type linear solver for reservoir simulation. In: SPE Annual Technical Conference and Exhibition, vol. SPE-96809 (2005). https:\/\/doi.org\/10.2118\/96809-MS","DOI":"10.2118\/96809-MS"},{"key":"136_CR7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718942","volume-title":"Computational methods for multiphase flows in porous media","author":"ZX Chen","year":"2006","unstructured":"Chen, Z.X., Huan, G.R., Ma, Y.L.: Computational methods for multiphase flows in porous media. Society for Industrial and Applied Mathematics, New York (2006). https:\/\/doi.org\/10.1137\/1.9780898718942"},{"key":"136_CR8","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/978-3-319-05789-7_16","volume-title":"Domain decomposition methods in science and engineering XXI","author":"Z Chen","year":"2014","unstructured":"Chen, Z., Liu, H., Yu, S., Hsieh, B., Shao, L.: GPU-based parallel reservoir simulators. In: Domain decomposition methods in science and engineering XXI, pp. 199\u2013206. Springer, Cham (2014)"},{"issue":"04","key":"136_CR9","doi-asserted-by":"publisher","first-page":"308","DOI":"10.2118\/72469-PA","volume":"4","author":"MA Christie","year":"2001","unstructured":"Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4(04), 308\u2013317 (2001). https:\/\/doi.org\/10.2118\/72469-PA","journal-title":"SPE Reserv. Eval. Eng."},{"issue":"05","key":"136_CR10","doi-asserted-by":"publisher","first-page":"363","DOI":"10.2118\/8284-PA","volume":"20","author":"KH Coats","year":"1980","unstructured":"Coats, K.H.: An equation of state compositional model. Soc. Petrol. Eng. J. 20(05), 363\u2013376 (1980)","journal-title":"Soc. Petrol. Eng. J."},{"issue":"02","key":"136_CR11","doi-asserted-by":"publisher","first-page":"259","DOI":"10.2118\/15133-PA","volume":"7","author":"DA Collins","year":"1992","unstructured":"Collins, D.A., Nghiem, L.X., Li, Y.K., Grabonstotter, J.E.: An efficient approach to adaptive-implicit compositional simulation with an equation of state. SPE Reserv. Eng. 7(02), 259\u2013264 (1992). https:\/\/doi.org\/10.2118\/15133-PA","journal-title":"SPE Reserv. Eng."},{"issue":"1","key":"136_CR12","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1007\/BF01448839","volume":"100","author":"R Courant","year":"1928","unstructured":"Courant, R., Friedrichs, K., Lewy, H.: \u00dcber die partiellen differenzengleichungen der mathematischen physik. Mathematische Annalen 100(1), 32\u201374 (1928). https:\/\/doi.org\/10.1007\/BF01448839","journal-title":"Mathematische Annalen"},{"issue":"01","key":"136_CR13","doi-asserted-by":"publisher","first-page":"597","DOI":"10.2118\/203929-PA","volume":"27","author":"K Esler","year":"2022","unstructured":"Esler, K., Gandham, R., Patacchini, L., Garipov, T., Samardzic, A., Panfili, P., Caresani, F., Pizzolato, A., Cominelli, A.: A Graphics Processing Unit-based, industrial grade compositional reservoir simulator. SPE J. 27(01), 597\u2013612 (2022). https:\/\/doi.org\/10.2118\/203929-PA","journal-title":"SPE J."},{"issue":"6","key":"136_CR14","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1109\/MCSE.2006.105","volume":"8","author":"RD Falgout","year":"2006","unstructured":"Falgout, R.D.: An introduction to algebraic multigrid computing. Comput. Sci. Eng. 8(6), 24\u201333 (2006). https:\/\/doi.org\/10.1109\/MCSE.2006.105","journal-title":"Comput. Sci. Eng."},{"key":"136_CR15","doi-asserted-by":"crossref","unstructured":"Feng, C.S., Shu, S., Xu, J.C., Zhang, C.S.: A multi-stage preconditioner for the black oil model and its OpenMP implementation. In: Lecture Notes in Computational Science and Engineering 98, 141\u2013153 (2014)","DOI":"10.1007\/978-3-319-05789-7_11"},{"issue":"04","key":"136_CR16","doi-asserted-by":"publisher","first-page":"211","DOI":"10.2118\/6891-PA","volume":"19","author":"LT Fussell","year":"1979","unstructured":"Fussell, L.T., Fussell, D.D.: An iterative technique for compositional reservoir models. Soc. Petrol. Eng. J. 19(04), 211\u2013220 (1979)","journal-title":"Soc. Petrol. Eng. J."},{"issue":"12","key":"136_CR17","doi-asserted-by":"publisher","first-page":"2737","DOI":"10.1007\/s11425-013-4737-3","volume":"56","author":"XZ Hu","year":"2013","unstructured":"Hu, X.Z., Xu, J.C., Zhang, C.S.: Application of auxiliary space preconditioning in field-scale reservoir simulation. Sci. China Math. 56(12), 2737\u20132751 (2013). https:\/\/doi.org\/10.1007\/s11425-013-4737-3","journal-title":"Sci. China Math."},{"issue":"12","key":"136_CR18","doi-asserted-by":"publisher","first-page":"213","DOI":"10.3390\/a11120213","volume":"11","author":"Z Kang","year":"2018","unstructured":"Kang, Z., Deng, Z., Han, W., Zhang, D.: Parallel reservoir simulation with OpenACC and domain decomposition. Algorithms 11(12), 213 (2018)","journal-title":"Algorithms"},{"key":"136_CR19","doi-asserted-by":"publisher","unstructured":"Killough, J., Kossack, C.: Fifth comparative solution project: evaluation of miscible flood simulators. In: SPE Symposium on Reservoir Simulation (1987). https:\/\/doi.org\/10.2118\/16000-MS","DOI":"10.2118\/16000-MS"},{"key":"136_CR20","first-page":"88","volume":"39","author":"RJ LeVeque","year":"2002","unstructured":"LeVeque, R.J.: Finite volume methods for hyperbolic problems. Meccanica 39, 88\u201389 (2002)","journal-title":"Meccanica"},{"issue":"2\u20133","key":"136_CR21","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s00791-016-0273-3","volume":"18","author":"Z Li","year":"2017","unstructured":"Li, Z., Wu, S.H., Zhang, C.S., Xu, J.C., Feng, C.S., Hu, X.Z.: Numerical studies of a class of linear solvers for fine-scale petroleum reservoir simulation. Comput. Vis. Sci. 18(2\u20133), 93\u2013102 (2017). https:\/\/doi.org\/10.1007\/s00791-016-0273-3","journal-title":"Comput. Vis. Sci."},{"key":"136_CR22","doi-asserted-by":"crossref","unstructured":"Manea, A.M., Almani, T.: A massively parallel algebraic multiscale solver for reservoir simulation on the GPU architecture. In: SPE Reservoir Simulation Conference (2019)","DOI":"10.3997\/2214-4609.201802253"},{"key":"136_CR23","doi-asserted-by":"publisher","unstructured":"Meyerink, J.A.: Iterative methods for the solution of linear equations based on incomplete block factorization of the matrix. In: SPE Reservoir Simulation Symposium, vol. SPE-12262 (1983). https:\/\/doi.org\/10.2118\/12262-MS","DOI":"10.2118\/12262-MS"},{"key":"136_CR24","doi-asserted-by":"publisher","unstructured":"Middya, U., Manea, A., Alhubail, M., Ferguson, T., Byer, T., Dogru, A.: A massively parallel reservoir simulator on the GPU architecture. In: SPE Reservoir Simulation Conference (2021). https:\/\/doi.org\/10.2118\/203918-MS","DOI":"10.2118\/203918-MS"},{"issue":"2","key":"136_CR25","doi-asserted-by":"publisher","first-page":"1079","DOI":"10.1137\/100818509","volume":"34","author":"A Napov","year":"2012","unstructured":"Napov, A., Notay, Y.: An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), 1079\u20131109 (2012). https:\/\/doi.org\/10.1137\/100818509","journal-title":"SIAM J. Sci. Comput."},{"key":"136_CR26","unstructured":"NVIDIA: CUDA C++ programming guide (2022). https:\/\/docs.nvidia.com\/cuda\/pdf\/CUDA_C_Programming_Guide.pdf"},{"issue":"01","key":"136_CR27","doi-asserted-by":"publisher","first-page":"13","DOI":"10.2118\/9723-PA","volume":"33","author":"AS Odeh","year":"1981","unstructured":"Odeh, A.S.: Comparison of solutions to a three-dimensional black-oil reservoir simulation problem. J Petrol Technol 33(01), 13\u201325 (1981). https:\/\/doi.org\/10.2118\/9723-PA","journal-title":"J Petrol Technol"},{"key":"136_CR28","unstructured":"OpenACC: OpenACC programming and best practices guide (2022). http:\/\/www.openacc.org\/"},{"key":"136_CR29","unstructured":"OpenCAEPoro: mlticomponent multiphase porous media flow simulator (2022). https:\/\/faspdevteam.github.io\/OpenCAEPoro"},{"key":"136_CR30","first-page":"190","volume-title":"Developments in petroleum science","author":"DW Peaceman","year":"1977","unstructured":"Peaceman, D.W.: Fundamentals of numerical reservoir simulation. In: Developments in petroleum science, p. 190. Elsevier, Amsterdam (1977)"},{"issue":"2","key":"136_CR31","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1002\/aic.690230202","volume":"23","author":"D Peng","year":"1977","unstructured":"Peng, D., Robinson, D.B.: A rigorous method for predicting the critical properties of multicomponent systems from an equation of state. AIChE J. 23(2), 137\u2013144 (1977). https:\/\/doi.org\/10.1002\/aic.690230202","journal-title":"AIChE J."},{"key":"136_CR32","unstructured":"Qiao, C.: General purpose compositional simulation for multiphase reactive flow with a fast linear solver. PhD thesis, The Pennsylvania State University (2015)"},{"key":"136_CR33","doi-asserted-by":"crossref","unstructured":"Quandalle, P., Savary, D.: An implicit in pressure and saturations approach to fully compositional simulation. In: SPE Symposium on Reservoir Simulation (1989)","DOI":"10.2118\/18423-MS"},{"key":"136_CR34","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, 2nd edn. Society for Industrial and Applied Mathematics, New York (2003). https:\/\/doi.org\/10.1137\/1.9780898718003","edition":"2"},{"key":"136_CR35","unstructured":"Schlumberger: ECLIPSE Technical Description Version 2017.2 (2017). https:\/\/www.software.slb.com\/products\/eclipse"},{"key":"136_CR36","doi-asserted-by":"publisher","unstructured":"St\u00fcben, K., Clees, T., Klie, H., Lu, B., Wheeler, M.F.: Algebraic multigrid methods (AMG) for the efficient solution of fully implicit formulations in reservoir simulation. In: SPE Reservoir Simulation Symposium (2007). https:\/\/doi.org\/10.2118\/105832-MS","DOI":"10.2118\/105832-MS"},{"key":"136_CR37","doi-asserted-by":"publisher","unstructured":"Sudan, H., Klie, H., Li, R., Saad, Y.: High performance manycore solvers for reservoir simulation. In: European Conference on the Mathematics of Oil Recovery (2010). https:\/\/doi.org\/10.3997\/2214-4609.20144961","DOI":"10.3997\/2214-4609.20144961"},{"key":"136_CR39","doi-asserted-by":"publisher","unstructured":"Wallis, J.R.: Incomplete gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: SPE Reservoir Simulation Symposium (1983). https:\/\/doi.org\/10.2118\/12265-MS","DOI":"10.2118\/12265-MS"},{"key":"136_CR38","doi-asserted-by":"publisher","unstructured":"Wallis, J.R., Kendall, R.P., Little, T.E.: Constrained residual acceleration of conjugate residual methods. In: SPE Reservoir Simulation Symposium (1985). https:\/\/doi.org\/10.2118\/13536-MS","DOI":"10.2118\/13536-MS"},{"issue":"4","key":"136_CR40","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1137\/1034116","volume":"34","author":"JC Xu","year":"1992","unstructured":"Xu, J.C.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581\u2013613 (1992)","journal-title":"SIAM Rev."},{"issue":"3","key":"136_CR41","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1007\/BF02238513","volume":"56","author":"JC Xu","year":"1996","unstructured":"Xu, J.C.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215\u2013235 (1996)","journal-title":"Computing"},{"key":"136_CR42","doi-asserted-by":"publisher","unstructured":"Yang, B., Liu, H., Chen, Z.X.: Accelerating linear solvers for reservoir simulation on GPU workstations. Society for Computer Simulation International 1, 1\u20138 (2016). https:\/\/doi.org\/10.22360\/SpringSim.2016.HPC.007","DOI":"10.22360\/SpringSim.2016.HPC.007"},{"issue":"1","key":"136_CR43","doi-asserted-by":"publisher","first-page":"1","DOI":"10.12288\/szjs.s2021-0813","volume":"43","author":"CS Zhang","year":"2022","unstructured":"Zhang, C.S.: Linear solvers for petroleum reservoir simulation. J. Numer. Methods Comput. Appl. 43(1), 1\u201326 (2022). https:\/\/doi.org\/10.12288\/szjs.s2021-0813","journal-title":"J. Numer. Methods Comput. Appl."},{"key":"136_CR44","doi-asserted-by":"publisher","DOI":"10.1016\/j.cageo.2022.105230","volume":"168","author":"L Zhao","year":"2022","unstructured":"Zhao, L., Feng, C.S., Zhang, C.S., Shu, S.: Parallel multi-stage preconditioners with adaptive setup for the black oil model. Comput. Geosci. 168, 105230 (2022). https:\/\/doi.org\/10.1016\/j.cageo.2022.105230","journal-title":"Comput. Geosci."}],"container-title":["CCF Transactions on High Performance Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42514-023-00136-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s42514-023-00136-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s42514-023-00136-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T09:13:03Z","timestamp":1687770783000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s42514-023-00136-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,1]]},"references-count":44,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["136"],"URL":"https:\/\/doi.org\/10.1007\/s42514-023-00136-0","relation":{},"ISSN":["2524-4922","2524-4930"],"issn-type":[{"value":"2524-4922","type":"print"},{"value":"2524-4930","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,1]]},"assertion":[{"value":"11 August 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}