{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T14:07:08Z","timestamp":1769868428552,"version":"3.49.0"},"reference-count":103,"publisher":"Association for Computing Machinery (ACM)","issue":"6","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62237001"],"award-info":[{"award-number":["62237001"]}],"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":["62122022"],"award-info":[{"award-number":["62122022"]}],"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":["62206064"],"award-info":[{"award-number":["62206064"]}],"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":["62206061"],"award-info":[{"award-number":["62206061"]}],"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":["62272298"],"award-info":[{"award-number":["62272298"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:p>This paper introduces a novel wavelet-based framework for simulating both single-phase (e.g., smoke) and two-phase (e.g., bubbly water) flows, featuring unified boundary condition handling for free surfaces and solid obstacles. In liquid simulations, conventional pressure projection methods enforce zero-pressure Dirichlet conditions at free surfaces by solving a simplified pressure Poisson equation. However, these approaches neglect air-phase incompressibility, leading to artificial bubble collapse. Stream function methods overcome this limitation by solving a density-variable vector potential Poisson equation, ensuring incompressibility in both simulated and unsimulated regions while maintaining divergence-free liquid phases independent of solver accuracy. Yet, they triple the linear system's dimensionality and exhibit poor convergence near solid boundaries. The fundamental limitation of both methods stems from their governing equations: singularities emerge as density approaches extreme values. The pressure Poisson equation becomes ill-conditioned when density nears zero (air phase), compromising air-phase incompressibility, while the vector potential equation degrades as density approaches infinity (solid phase), impeding solid-boundary convergence. To address these singularities, we first propose a novel decomposition where zero and infinite densities are well-defined. We then reformulate this decomposition as a fixed-point iteration using density-agnostic curl-free and divergence-free projections, eliminating the need for linear system solves. The error equation is derived, and a necessary and sufficient convergence condition is established. Building on this, we develop an iterative algorithm that efficiently solves the fixed-point problem through alternating wavelet-based non-orthogonal curl-free and divergence-free projections. Additionally, we investigate orthogonal curl-free projections (e.g., Fourier methods) and their complementary divergence-free counterparts, providing a comprehensive comparison between wavelet and Fourier approaches. Our method simultaneously computes pressure and stream functions, retaining the incompressibility benefits of stream function approaches while resolving their computational inefficiencies and solid-boundary convergence issues. Experiments demonstrate our framework's ability to efficiently simulate complex two-phase phenomena, such as the glugging effect during water pouring and multi-liquid-region interactions across zero-density air.<\/jats:p>","DOI":"10.1145\/3763364","type":"journal-article","created":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T17:15:39Z","timestamp":1764868539000},"page":"1-17","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Wavelet Fluids"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-9370-7334","authenticated-orcid":false,"given":"Luan","family":"Lyu","sequence":"first","affiliation":[{"name":"Guangdong University of Technology, Guangzhou, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3196-9880","authenticated-orcid":false,"given":"Xiaohua","family":"Ren","sequence":"additional","affiliation":[{"name":"WeChat Vision, Tencent Inc., Chengdu, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9921-2227","authenticated-orcid":false,"given":"Wei","family":"Cao","sequence":"additional","affiliation":[{"name":"Qingdao University, Qingdao, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2551-2024","authenticated-orcid":false,"given":"Jian","family":"Zhu","sequence":"additional","affiliation":[{"name":"Guangdong University of Technology, Guangzhou, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-9713-1957","authenticated-orcid":false,"given":"Ziyang","family":"Ma","sequence":"additional","affiliation":[{"name":"WeChat Vision, Tencent Inc., Beijing, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2174-1428","authenticated-orcid":false,"given":"Enhua","family":"Wu","sequence":"additional","affiliation":[{"name":"SKLCS, Institute of Software, Chinese Academy of Sciences, Beijing, China"},{"name":"University of Macau, Macau, China"}]}],"member":"320","published-online":{"date-parts":[[2025,12,4]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2013.03.048"},{"key":"e_1_2_2_2_1","volume-title":"Algebraic Multigrid Solver","author":"X.","unstructured":"AmgX. 2025. Algebraic Multigrid Solver (AmgX) Library. https:\/\/github.com\/NVIDIA\/AMGX."},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/2766935"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073368.1073380"},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1275808.1276502"},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2018.2832080"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/2159516.2159522"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/3730854"},{"key":"e_1_2_2_9_1","volume-title":"Fluid Simulation for Computer Graphics","author":"Bridson R.","unstructured":"R. Bridson. 2018. Fluid Simulation for Computer Graphics. CRC Press."},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276435"},{"key":"e_1_2_2_11_1","volume-title":"Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"Brochu Tyson","year":"2012","unstructured":"Tyson Brochu, Todd Keeler, and Robert Bridson. 2012. Linear-time smoke animation with vortex sheet meshes. In Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation (Lausanne, Switzerland) (SCA '12). Eurographics Association, Goslar, DEU, 87\u201395."},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185559"},{"key":"e_1_2_2_13_1","doi-asserted-by":"publisher","DOI":"10.1145\/3550454.3555498"},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160450502"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073264"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/3478513.3480536"},{"key":"e_1_2_2_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201352"},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970104"},{"key":"e_1_2_2_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/2077341.2077351"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1080\/14685240500260547"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2008.06.001"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/1189762.1189766"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6236"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/3414685.3417842"},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.5555\/792756.792862"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1145\/3197517.3201309"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1145\/3386569.3392455"},{"key":"e_1_2_2_28_1","volume-title":"Proceedings of the 2004 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"Greenwood S. T.","unstructured":"S. T. Greenwood and D. H. House. 2004. Better with bubbles: enhancing the visual realism of simulated fluid. In Proceedings of the 2004 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation (Grenoble, France) (SCA '04). Eurographics Association, Goslar, DEU, 287\u2013296."},{"key":"e_1_2_2_29_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2016.2532335"},{"key":"e_1_2_2_30_1","volume-title":"Legendre Fluids: A Unified Framework for Analytic Reduced Space Modeling and Rendering of Participating Media. In 2007 ACM SIGGRAPH \/ Eurographics Symposium on Computer Animation.","author":"Gupta Mohit","unstructured":"Mohit Gupta and Srinivasa G. Narasimhan. 2007. Legendre Fluids: A Unified Framework for Analytic Reduced Space Modeling and Rendering of Participating Media. In 2007 ACM SIGGRAPH \/ Eurographics Symposium on Computer Animation."},{"key":"e_1_2_2_31_1","volume-title":"Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. The Physics of Fluids 8, 12 (12","author":"Harlow Francis H.","year":"1965","unstructured":"Francis H. Harlow and J. Eddie Welch. 1965. Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. The Physics of Fluids 8, 12 (12 1965), 2182\u20132189."},{"key":"e_1_2_2_32_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-27413-8_20"},{"key":"e_1_2_2_33_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2012.07.029"},{"key":"e_1_2_2_34_1","doi-asserted-by":"publisher","DOI":"10.1137\/140969208"},{"key":"e_1_2_2_35_1","volume-title":"Cartesian CFD Methods for Complex Applications","author":"Harouna Souleymane Kadri","unstructured":"Souleymane Kadri Harouna and Val\u00e9rie Perrier. 2021. No-Slip and Free-Slip Divergence-Free Wavelets for the Simulation of Incompressible Viscous Flows. In Cartesian CFD Methods for Complex Applications. Springer, 37\u201365."},{"key":"e_1_2_2_36_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2021.125479"},{"key":"e_1_2_2_37_1","volume-title":"Fabian L\u00f6schner, Jan Bender, Yin Yang, and Sheldon Andrews.","author":"Holz Daniel","year":"2025","unstructured":"Daniel Holz, Stefan Rhys Jeske, Fabian L\u00f6schner, Jan Bender, Yin Yang, and Sheldon Andrews. 2025. Multiphysics Simulation Methods in Computer Graphics. Computer Graphics Forum n\/a, n\/a (2025), e70082."},{"key":"e_1_2_2_38_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073283"},{"key":"e_1_2_2_39_1","doi-asserted-by":"publisher","DOI":"10.1145\/1399504.1360647"},{"key":"e_1_2_2_40_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00371-012-0697-9"},{"key":"e_1_2_2_41_1","volume-title":"Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"Jones Aaron Demby","year":"2016","unstructured":"Aaron Demby Jones, Pradeep Sen, and Theodore Kim. 2016. Compressing fluid sub-spaces. In Proceedings of the ACM SIGGRAPH\/Eurographics Symposium on Computer Animation (Zurich, Switzerland) (SCA '16). Eurographics Association, Goslar, DEU, 77\u201384."},{"key":"e_1_2_2_42_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1011178417620"},{"key":"e_1_2_2_43_1","unstructured":"Byungmoon Kim Yingjie Liu Ignacio Llamas Xiangmin Jiao and Jarek Rossignac. 2007."},{"key":"e_1_2_2_44_1","volume-title":"ACM SIGGRAPH 2007 Papers","author":"Simulation","unstructured":"Simulation of bubbles in foam with the volume control method. In ACM SIGGRAPH 2007 Papers (San Diego, California) (SIGGRAPH '07). Association for Computing Machinery, New York, NY, USA, 98\u2013es."},{"key":"e_1_2_2_45_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461987"},{"key":"e_1_2_2_46_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461987"},{"key":"e_1_2_2_47_1","doi-asserted-by":"publisher","DOI":"10.1145\/1399504.1360649"},{"key":"e_1_2_2_48_1","volume-title":"Analyses multi-r\u00e9solutions non orthogonales, commutation entre projecteurs et derivation et ondelettes vecteurs \u00e0 divergence nuIIe. Revista Matem\u00e1tica Iberoamericana 8, 2","author":"Lemarie-Rieusset Pierre Gilles","year":"1992","unstructured":"Pierre Gilles Lemarie-Rieusset. 1992. Analyses multi-r\u00e9solutions non orthogonales, commutation entre projecteurs et derivation et ondelettes vecteurs \u00e0 divergence nuIIe. Revista Matem\u00e1tica Iberoamericana 8, 2 (1992), 221\u2013237."},{"key":"e_1_2_2_49_1","doi-asserted-by":"publisher","DOI":"10.1145\/3592138"},{"key":"e_1_2_2_50_1","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530132"},{"key":"e_1_2_2_51_1","doi-asserted-by":"publisher","DOI":"10.1145\/3658178"},{"key":"e_1_2_2_52_1","doi-asserted-by":"publisher","DOI":"10.1145\/2816795.2818130"},{"key":"e_1_2_2_53_1","doi-asserted-by":"publisher","DOI":"10.1145\/1507149.1507165"},{"key":"e_1_2_2_54_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141960"},{"key":"e_1_2_2_55_1","volume-title":"Efficient odd-even multigrid for pointwise incompressible fluid simulation on GPU. The Visual Computer 40, 12 (01","author":"Lyu Luan","year":"2024","unstructured":"Luan Lyu, Wei Cao, Xiaohua Ren, Enhua Wu, and Zhi-Xin Yang. 2024a. Efficient odd-even multigrid for pointwise incompressible fluid simulation on GPU. The Visual Computer 40, 12 (01 Dec 2024), 8675\u20138691."},{"key":"e_1_2_2_56_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11432-018-9518-3"},{"key":"e_1_2_2_57_1","volume-title":"Wavelet Potentials: An Efficient Potential Recovery Technique for Pointwise Incompressible Fluids. Computer Graphics Forum","author":"Lyu Luan","year":"2024","unstructured":"Luan Lyu, Xiaohua Ren, Wei Cao, Jian Zhu, Enhua Wu, and Zhi-Xin Yang. 2024b. Wavelet Potentials: An Efficient Potential Recovery Technique for Pointwise Incompressible Fluids. Computer Graphics Forum (2024)."},{"key":"e_1_2_2_58_1","volume-title":"Fast Wavelet-domain Smoke Guiding. In Pacific Graphics Conference Papers and Posters. The Eurographics Association.","author":"Lyu Luan","year":"2024","unstructured":"Luan Lyu, Xiaohua Ren, Enhua Wu, and Zhi-Xin Yang. 2024c. Fast Wavelet-domain Smoke Guiding. In Pacific Graphics Conference Papers and Posters. The Eurographics Association."},{"key":"e_1_2_2_59_1","volume-title":"A Wavelet Tour of Signal Processing","author":"Mallat Stphane","unstructured":"Stphane Mallat. 2008. A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way (3rd ed.). Academic Press, Inc., USA."},{"key":"e_1_2_2_60_1","volume-title":"Proceedings of the 2010 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"McAdams A.","unstructured":"A. McAdams, E. Sifakis, and J. Teran. 2010. A parallel multigrid Poisson solver for fluids simulation on large grids. In Proceedings of the 2010 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation (Madrid, Spain) (SCA '10). Eurographics Association, Goslar, DEU, 65\u201374."},{"key":"e_1_2_2_61_1","volume-title":"Proceedings of the 2006 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation","author":"Mihalef V.","unstructured":"V. Mihalef, B. Unlusu, D. Metaxas, M. Sussman, and M. Y. Hussaini. 2006. Physics based boiling simulation. In Proceedings of the 2006 ACM SIGGRAPH\/Eurographics Symposium on Computer Animation (Vienna, Austria) (SCA '06). Eurographics Association, Goslar, DEU, 317\u2013324."},{"key":"e_1_2_2_62_1","doi-asserted-by":"publisher","DOI":"10.5555\/1632592.1632595"},{"key":"e_1_2_2_63_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073368.1073402"},{"key":"e_1_2_2_64_1","doi-asserted-by":"publisher","DOI":"10.1137\/140980260"},{"key":"e_1_2_2_65_1","doi-asserted-by":"publisher","DOI":"10.1145\/3747872"},{"key":"e_1_2_2_66_1","doi-asserted-by":"publisher","DOI":"10.1145\/2485895.2485912"},{"key":"e_1_2_2_67_1","doi-asserted-by":"publisher","DOI":"10.1145\/2185520.2185608"},{"key":"e_1_2_2_68_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2021.3062643"},{"key":"e_1_2_2_69_1","doi-asserted-by":"publisher","DOI":"10.1145\/2645703"},{"key":"e_1_2_2_70_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11390-018-1829-0"},{"key":"e_1_2_2_71_1","volume-title":"Computer Graphics Forum","author":"Ren Xiaohua","unstructured":"Xiaohua Ren, Luan Lyu, Xiaowei He, Wei Cao, Zhixin Yang, Bin Sheng, Yanci Zhang, and Enhua Wu. 2018a. Biorthogonal wavelet surface reconstruction using partial integrations. In Computer Graphics Forum, Vol. 37. Wiley Online Library, 13\u201324."},{"key":"e_1_2_2_72_1","volume-title":"Computer Graphics Forum","author":"Ren Xiaohua","unstructured":"Xiaohua Ren, Luan Lyu, Xiaowei He, Yanci Zhang, and Enhua Wu. 2017. Efficient Gradient-Domain Compositing Using an Approximate Curl-free Wavelet Projection. In Computer Graphics Forum, Vol. 36. Wiley Online Library, 207\u2013215."},{"key":"e_1_2_2_73_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459846"},{"key":"e_1_2_2_74_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00371-015-1122-y"},{"key":"e_1_2_2_75_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073282"},{"key":"e_1_2_2_76_1","doi-asserted-by":"publisher","DOI":"10.1145\/1037957.1037962"},{"key":"e_1_2_2_77_1","doi-asserted-by":"publisher","DOI":"10.1145\/311535.311548"},{"key":"e_1_2_2_78_1","first-page":"2","article-title":"A simple fluid solver based on the FFT","volume":"6","author":"Stam Jos","year":"2002","unstructured":"Jos Stam. 2002. A simple fluid solver based on the FFT. J. Graph. Tools 6, 2 (Sept. 2002), 43\u201352.","journal-title":"J. Graph. Tools"},{"key":"e_1_2_2_79_1","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2462006"},{"key":"e_1_2_2_80_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.01.007"},{"key":"e_1_2_2_81_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-016-9325-7"},{"key":"e_1_2_2_82_1","doi-asserted-by":"publisher","DOI":"10.1145\/3388767.3407390"},{"key":"e_1_2_2_83_1","doi-asserted-by":"publisher","DOI":"10.1145\/3687963"},{"key":"e_1_2_2_84_1","unstructured":"Wim Sweldens and Peter Schr\u00f6der. 1996. Building your own wavelets at home. In Wavelets in Computer Graphics. ACM SIGGRAPH Course notes 15\u201387."},{"key":"e_1_2_2_85_1","doi-asserted-by":"publisher","DOI":"10.1145\/3072959.3073651"},{"key":"e_1_2_2_86_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141911.1141962"},{"key":"e_1_2_2_87_1","unstructured":"Karsten Urban. 1996. Using divergence free wavelets for the numerical solution of the Stokes problem. Citeseer."},{"key":"e_1_2_2_88_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01245-X"},{"key":"e_1_2_2_89_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2013.95"},{"key":"e_1_2_2_90_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2024.3404151"},{"key":"e_1_2_2_91_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.12577"},{"key":"e_1_2_2_92_1","doi-asserted-by":"publisher","DOI":"10.1145\/1778765.1778852"},{"key":"e_1_2_2_93_1","doi-asserted-by":"publisher","DOI":"10.1145\/1531326.1531345"},{"key":"e_1_2_2_94_1","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530059"},{"key":"e_1_2_2_95_1","volume-title":"A unified multi-scale method for simulating immersed bubbles. Computer Graphics Forum n\/a, n\/a","author":"Wretborn Joel","year":"2025","unstructured":"Joel Wretborn, Alexey Stomakhin, and Christopher Batty. 2025. A unified multi-scale method for simulating immersed bubbles. Computer Graphics Forum n\/a, n\/a (2025), e70033."},{"key":"e_1_2_2_96_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13350"},{"key":"e_1_2_2_97_1","doi-asserted-by":"publisher","DOI":"10.1145\/3618347"},{"key":"e_1_2_2_98_1","doi-asserted-by":"publisher","DOI":"10.1111\/cgf.13523"},{"key":"e_1_2_2_99_1","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2015.2446474"},{"key":"e_1_2_2_100_1","doi-asserted-by":"publisher","DOI":"10.1145\/2661229.2661261"},{"key":"e_1_2_2_101_1","doi-asserted-by":"publisher","DOI":"10.5555\/1218064.1218109"},{"key":"e_1_2_2_102_1","volume-title":"Using divergence free wavelets for the numerical solution of the 2-D stationary Navier-Stokes equations. Applied mathematics and computation 163, 2","author":"Zhou Xiaolin","year":"2005","unstructured":"Xiaolin Zhou and Yinnian He. 2005. Using divergence free wavelets for the numerical solution of the 2-D stationary Navier-Stokes equations. Applied mathematics and computation 163, 2 (2005), 593\u2013607."},{"key":"e_1_2_2_103_1","doi-asserted-by":"publisher","DOI":"10.1145\/1073204.1073298"}],"container-title":["ACM Transactions on Graphics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3763364","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T23:16:49Z","timestamp":1769815009000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3763364"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12]]},"references-count":103,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["10.1145\/3763364"],"URL":"https:\/\/doi.org\/10.1145\/3763364","relation":{},"ISSN":["0730-0301","1557-7368"],"issn-type":[{"value":"0730-0301","type":"print"},{"value":"1557-7368","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12]]},"assertion":[{"value":"2025-05-24","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-08-09","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-12-04","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}