{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:41:09Z","timestamp":1760143269829,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T00:00:00Z","timestamp":1705968000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Higher Education of the Russian Federation as part of World-class Research Center program: Advanced Digital Technologies","award":["075-15-2020-903"],"award-info":[{"award-number":["075-15-2020-903"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The opportunities provided by new information technologies, object-oriented programming tools, and modern operating systems for solving boundary value problems in CFD described by partial differential equations are discussed. An approach to organizing vectorized calculations and implementing finite-difference methods for solving boundary value problems in CFD is considered. Vectorization in CFD problems, eliminating nested loops, is ensured through the appropriate data organization and the use of vectorized operations with arrays. The implementation of numerical algorithms with vectorized mesh structures, including access to internal and boundary mesh cells, is discussed. Specific examples are reported and the implementation of the developed computational algorithms is discussed. Despite the fact that the capabilities of the developed algorithms are illustrated by solving benchmark CFD problems, they enable a relatively simple generalization to more complex problems described by three-dimensional equations.<\/jats:p>","DOI":"10.3390\/a17020050","type":"journal-article","created":{"date-parts":[[2024,1,23]],"date-time":"2024-01-23T07:22:32Z","timestamp":1705994552000},"page":"50","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Vectorized Numerical Algorithms to Solve Internal Problems of Computational Fluid Dynamics"],"prefix":"10.3390","volume":"17","author":[{"given":"Konstantin","family":"Volkov","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Kingston University, London SW15 3DW, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1134\/S207004822304018X","article-title":"Methods for improving and evaluating the performance of unstructured CFD algorithms","volume":"15","author":"Soukov","year":"2023","journal-title":"Math. Model. Comput. Simul."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.compfluid.2018.03.011","article-title":"Multilevel parallelization for simulating compressible turbulent flows on most kinds of hybrid supercomputers","volume":"173","author":"Gorobets","year":"2018","journal-title":"Comput. Fluids"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5374","DOI":"10.2514\/1.J059953","article-title":"Progress in computational fluid dynamics discretizations algorithms and solvers for aerodynamic flows","volume":"9","author":"Mavriplis","year":"2021","journal-title":"AIAA J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1186\/s40712-021-00134-6","article-title":"CFD simulations of gearboxes: Implementation of a mesh clustering algorithm for efficient simulations of complex system\u2019s architectures","volume":"16","author":"Mastrone","year":"2021","journal-title":"Int. J. Mech. Mater. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1023\/A:1019155918070","article-title":"Remarks around 50 lines of Matlab: Short finite element implementation","volume":"20","author":"Alberty","year":"1999","journal-title":"Numer. Algorithms"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1137\/S0036144503429121","article-title":"A simple mesh generation in MATLAB","volume":"42","author":"Persson","year":"2004","journal-title":"SIAM Rev."},{"key":"ref_7","first-page":"157","article-title":"Vectorized MATLAB codes for linear two-dimensional elasticity","volume":"15","author":"Koko","year":"2007","journal-title":"Sci. Program."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"460","DOI":"10.2478\/cmam-2011-0026","article-title":"Efficient implementation of adaptive P1-FEM in MATLAB","volume":"11","author":"Funken","year":"2011","journal-title":"Comput. Methods Appl. Math."},{"key":"ref_9","first-page":"7151","article-title":"Fast MATLAB assembly of FEM matrices in 2D and 3D: Nodal elements","volume":"219","author":"Rahman","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1650010","DOI":"10.1142\/S1793962316500100","article-title":"Fast MATLAB assembly of fem matrices in 2d and 3d using cell array approach","volume":"7","author":"Koko","year":"2016","journal-title":"Int. J. Model. Simul. Sci. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"C234","DOI":"10.1137\/17M1128745","article-title":"FELICITY: A MATLAB\/C++ toolbox for developing finite element methods and simulation modelling","volume":"40","author":"Walker","year":"2018","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_12","first-page":"595","article-title":"Efficient and flexible Matlab implementation of 2D and 3D elasto-plastic problems","volume":"355","author":"Cermak","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_13","first-page":"117","article-title":"A finite volume solution of unsteady incompressible Navier\u2014Stokes equations using MATLAB","volume":"1","author":"Tsega","year":"2019","journal-title":"Numer. Comput. Methods Sci. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1134\/S0965542519010160","article-title":"Simulating flows of viscous incompressible fluid on graphics processors using the splitting scheme and multigrid method","volume":"59","author":"Volkov","year":"2019","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.cam.2006.11.021","article-title":"Vectorized adaptive quadrature in MATLAB","volume":"211","author":"Shampine","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1260\/174830109787913967","article-title":"A block SPP algorithm for multidimensional tridiagonal equations with optimal message vector length","volume":"3","author":"Guo","year":"2009","journal-title":"J. Algorithms Comput. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"14684","DOI":"10.1007\/s11227-023-05220-4","article-title":"SIMD vectorization for simultaneous solution of locally varying linear systems with multiple right-hand sides","volume":"79","author":"Kuhn","year":"2023","journal-title":"J. Supercomput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1311","DOI":"10.1007\/s11075-021-01154-1","article-title":"An efficient algorithm for solving piecewise-smooth dynamical systems","volume":"89","author":"Guglielmi","year":"2022","journal-title":"Numer. Algorithms"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"125603","DOI":"10.1016\/j.fuel.2022.125603","article-title":"PTFlash: A vectorized and parallel deep learning framework for two-phase flash calculation","volume":"331","author":"Qu","year":"2023","journal-title":"Fuel"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/s42774-019-0003-x","article-title":"Landmarks and new frontiers of computational fluid dynamics","volume":"1","author":"Shang","year":"2019","journal-title":"Adv. Aerodyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"051701","DOI":"10.1115\/1.4049731","article-title":"Support vector machines model of the nonlinear hydrodynamics of fixed cylinders","volume":"143","author":"Ma","year":"2021","journal-title":"J. Offshore Mech. Arct. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sofos, F., Stavrogiannis, C., Exarchou-Kouveli, K.K., Akabua, D., Charilas, G., and Karakasidis, T.E. (2022). Current trends in fluid research in the era of artificial intelligence: A review. Fluids, 7.","DOI":"10.3390\/fluids7030116"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Sharma, P., Chung, W.T., Akoush, B., and Ihme, M. (2023). A review of physics-informed machine learning in fluid mechanics. Energies, 16.","DOI":"10.3390\/en16052343"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"108847","DOI":"10.1016\/j.ijthermalsci.2023.108847","article-title":"Machine learning algorithms to study the relative roles of jet convection and natural convection in the presence of surface roughness elements on enhancement of jet impingement heat transfer","volume":"198","author":"Nagesha","year":"2024","journal-title":"Int. J. Therm. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1016\/j.actaastro.2022.09.044","article-title":"Simulation of high-temperature flowfield around hypersonic waverider using graphics processor units","volume":"204","author":"Dobrov","year":"2023","journal-title":"Acta Astronaut."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1006\/jcph.1997.5745","article-title":"Accurate monotonicity-preserving schemes with Runge\u2013Kutta time stepping","volume":"136","author":"Suresh","year":"1997","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1006\/jcph.2000.6443","article-title":"Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy","volume":"160","author":"Balsara","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0021-9991(78)90023-2","article-title":"A survey of several finite difference methods of systems of nonlinear hyperbolic conservation laws","volume":"27","author":"Sod","year":"1978","journal-title":"J. Comput. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1002\/cpa.3160070112","article-title":"Weak solutions of nonlinear hyperbolic equations and their numerical computation","volume":"7","author":"Lax","year":"1954","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.actaastro.2020.03.019","article-title":"Numerical simulation of aerodynamic problems based on adaptive mesh refinement method","volume":"172","author":"Struchkov","year":"2020","journal-title":"Acta Astronaut."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/2\/50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:47:41Z","timestamp":1760104061000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/2\/50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,23]]},"references-count":30,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["a17020050"],"URL":"https:\/\/doi.org\/10.3390\/a17020050","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2024,1,23]]}}}