Математическая физика и компьютерное моделирование2587-63252587-690210.15688/mpcm.jvolsu.2024.4.4OtherЭФФЕКТИВНОСТЬ ПАРАЛЛЕЛЬНЫХ ВЫЧИСЛЕНИЙ ГРАВИТАЦИОННЫХ СИЛ МЕТОДОМ TREECODE В МОДЕЛЯХ 𝑁-ТЕЛEFFICIENCY OF PARALLEL COMPUTATIONS OF GRAVITATIONAL FORCES BY TREECODE METHOD IN 𝑁-BODY MODELSКузьминНиколай МихайловичKuzminNikolaynikolay.kuzmin@volsu.ru0000-0003-4074-0970СиротинДанила СергеевичSirotinDanilad.sirotin@volsu.ru0000-0001-8956-570XХоперсковАлександр ВалентиновичKhoperskovAlexanderkhoperskov@volsu.ru0000-0003-0149-7947Volgograd State University (Volgograd, Russian Federation)Волгоградский государственный университет (Волгоград, Российская Федерация)2712202427439550309202417112024https://creativecommons.org/licenses/by/4.0/CC BY 4.0

Моделирование бесстолкновительных галактических систем основывается на модели 𝑁-тел, которая требует больших вычислительных ресурсов из-за дальнодействующего характера гравитационных сил. Наиболее распространенным методом вычисления гравитации является приближенный алгоритм TreeCode, обеспечивающий более быстрое вычисление силы по сравнению с прямым суммированием вкладов от всех частиц. Проведен анализ вычислительной эффективности для моделей с числом частиц в пределах 108. Мы рассмотрели несколько процессоров с различной архитектурой с целью определения производительности параллельных симуляций на основе стандарта OpenMP. Анализ использования дополнительных потоков (extra threads) дополнительно к физическим ядрам показывает рост производительности симуляций только при загрузке всех логических потоков. Это дает прирост параллельной эффективности (efficiency of parallel computing) в среднем на 20 %.

Modeling of collisionless galactic systems is based on the 𝑁-body model, which requires large computational resources due to the long-range nature of gravitational forces. The most common method for calculating gravity is the TreeCode algorithm, which provides a faster calculation of the force compared to the direct summation of contributions from all particles for 𝑁-body simulation. An analysis of the computational efficiency is performed for models with the number of particles up to 108. We considered several processors with different architectures in order to determine the performance of parallel simulations based on the OpenMP standard. An analysis of the use of extra threads in addition to physical cores shows an increase in simulation performance only when all logical threads are loaded, which doubles the total number of threads. This gives an increase in the efficiency of parallel computing by 20 percent on average.

параллельные вычислениягравитирующие системыOpenMPархитектура процессоровтехнология Hyper-Threadingparallel computinggravitational systemsOpenMPprocessor architectureHyper-Threading technologyThis work supported by the Russian Science Foundation (grant no. 23-71-00016, https://rscf.ru/project/23-71-00016/).Athanassoula E. 𝑁-body Simulations of Galaxies and Groups of Galaxies with the Marseille GRAPE Systems. Annals of the New York Academy of Sciences, 1998, vol. 867, no. 1, pp. 141-155. DOI: https://doi.org/10.1111/j.1749-6632.1998.tb11255.xBagla J.S. Cosmological 𝑁-body Simulation: Techniques, Scope and Status. Current Science, 2005, vol. 88, pp. 1088-1100. DOI: https://doi.org/10.48550/arXiv.astroph/0411043Barnes J., Hut P. A Hierarchical O(N log N) Force-Calculation Algorithm. Nature, 1986, vol. 324, no. 4, pp. 446-449. DOI: https://doi.org/10.1038/324446a0Bissekenov A., Kalambay M., Abdikamalov E., Pang X., Berczik P., Shukirgaliyev B. Cluster Membership Analysis with Supervised Learning and N-Body Simulations. Astronomy & Astrophysics, 2024, vol. 689, article ID: A282. DOI: https://doi.org/10.1051/0004-6361/202449791Butenko M.A., Belikova I.V., Kuzmin N.M., Khokhlova S.S., Ivanchenko G.S., Ten A.V., Kudina I.G. Numerical Simulation of the Galaxies Outer Spiral Structure: the Influence of the Dark Halo Non-Axisymmetry on the Gaseous Disk Shape. Mathematical Physics and Computer Simulation, 2022, vol. 25 (3), pp. 73-83. DOI: https://doi.org/10.15688/mpcm.jvolsu.2022.3.5Ciambur B.C., Fragkoudi F., Khoperskov S., Di Matteo P., Combes F. Double X/Peanut Structures in Barred Galaxies − Insights from an 𝑁-Body Simulation. Monthly Notices of the Royal Astronomical Society, 2020, vol. 503, no. 2, pp. 2203-2214. DOI: https://doi.org/10.1093/mnras/staa3814Dubinski J., Kim J., Park C., Humble R. GOTPM: A Parallel Hybrid Particle-Mesh Treecode. New Astronomy, 2004, vol. 9, pp. 111-126. DOI: https://doi.org/10.1016/j.newast.2003.08.002Fattahi A., Navarro J.F., Sawala T., Frenk C.S., Oman K.A., Crain R.A., Furlong M., Schaller M., Schaye J., Theuns T., Jenkins A. The Apostle Project: Local Group Kinematic Mass Constraints and Simulation Candidate Selection. Monthly Notices of the Royal Astronomical Society, 2016, vol. 457, pp. 844-856. DOI: https://doi.org/10.1093/mnras/stv2970Fridman A.M., Khoperskov A.V. Physics of Galactic Disks. Cambridge International Science Publishing Ltd, 2012. 754 p.Grand R.J.J., Springel V., Gomez F.A., Marinacci F., Pakmor R., Campbell D.J.R., Jenkins A. Vertical Disc Heating in Milky Way-Sized Galaxies in a Cosmological Context. Monthly Notices of the Royal Astronomical Society, 2016, vol. 459, pp. 199-219.Hernandez-Aguayo C., Springel V., Pakmor R., Barrera M., Ferlito F., White S.D.M., Hernquis L., Hadzhiyska B., Delgado A.M., Kannan R., Bose S., Frenk Carlos The MillenniumTNG Project: High-Precision Predictions for Matter Clustering and Halo Statistics. Monthly Notices of the Royal Astronomical Society, 2023, vol. 524, pp. 2556-2578. DOI: https://doi.org/10.1093/mnras/stad1657Ishchenko M., Berczik P., Panamarev T., Kuvatova D., Kalambay M., Gluchshenko A., Veles O., Sobolenko M., Sobodar O., Omarov C. Dynamical Evolution of Milky Way Globular Clusters on the Cosmological Timescale - I. Mass Loss and Interaction with the Nuclear Star Cluster. Astronomy & Astrophysics, 2024, vol. 689, article ID: A178. DOI: https://doi.org/10.1051/0004-6361/202450399Ishchenko M., Berczik P., Sobolenko M. Milky Way Globular Clusters on Cosmological Timescales. IV. Guests in the Outer Solar System. Astronomy & Astrophysics, 2024, vol. 683, article ID: A146. DOI: https://doi.org/10.1051/0004-6361/202347990Just A., Piskunov A.E., Klos J.H., Kovaleva D.A., Polyachenko E.V. Global Survey of Star Clusters in the Milky Way − VII. Tidal Parameters and Mass Function. Astronomy & Astrophysics, 2023, vol. 672, article ID: A187. DOI: https://doi.org/10.1051/0004-6361/202244723Khoperskov A.V., Khrapov S.S., Sirotin D.S. Formation of Transitional cE/UCD Galaxies Through Massive Disc to Dwarf Galaxy Mergers. Galaxies, 2024, vol. 12, no. 1, article ID: 1. DOI: https://doi.org/10.3390/galaxies12010001Khrapov S., Khoperskov A. Study of the Effectiveness of Parallel Algorithms for Modeling the Dynamics of Collisionless Galactic Systems on GPUs. Supercomputing Frontiers and Innovations, 2024, vol. 11, no. 3, pp. 27-44. DOI: https://doi.org/10.14529/jsfi240302Khrapov S.S., Khoperskov A.V., Zaitseva N.A., Zasov A.V., Titov A.V. Formation of Spiral Dwarf Galaxies: Observational Data and Results of Numerical Simulation. Saint Petersburg State Polytechnical University Journal. Physics and Mathematics, 2023, vol. 16, no. 1.2, pp. 395-402. DOI: https://doi.org/10.18721/JPM.161.260Khrapov S.S., Khoperskov S.A., Khoperskov A.V. New features of parallel implementation of 𝑁-body problems on GPU. Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software. 2018, vol. 11, no. 1, pp. 124-136. DOI: https://doi.org/10.14529/mmp180111Kulikov I., Chernykh I., Protasov V. Mathematical Modeling of Formation, Evolution and Interaction of Galaxies in Cosmological Context. Journal of Physics: Conference Series, 2016, vol. 722, no. 1, article ID: 012023. DOI: https://dx.doi.org/10.1088/1742-6596/722/1/012023Kyziropoulos P.E., Filelis-Papadopoulos C.K., Gravvanis G.A. Parallel 𝑁-body Simulation Based on the PM and P3M Methods Using Multigrid Schemes in Conjunction with Generic Approximate Sparse Inverses. Mathematical Problems in Engineering, 2015, vol. 2015, article ID: 450980. DOI: https://doi.org/10.1155/2015/450980L’Huillier H.B., Park C., Kim J. GalaxyFlow: Upsampling Hydrodynamical Simulations for Realistic Mock Stellar Catalogues. New Astronomy, 2014, vol. 30, pp. 79-88. DOI: https://doi.org/10.1016/j.newast.2014.01.007Lim S.H., Raman K.A., Buckley M.R., Shih D. GalaxyFlow: Upsampling Hydrodynamical Simulations for Realistic Mock Stellar Catalogues. Monthly Notices of the Royal Astronomical Society, 2024, vol. 533, no. 1, pp. 143-164. DOI: https://doi.org/10.1093/mnras/stae1672Nipoti C., Cherchi G., Iorio G., Calura F. Effective 𝑁-Body Models of Composite Collisionless Stellar Systems. Monthly Notices of the Royal Astronomical Society, 2021, vol. 503, no. 3, pp. 4221-4230. DOI: https://doi.org/10.1093/mnras/stab763Pejch M.A., Morozov A.G., Khoperskov A.V. Modeling a Double-Hump Gas Rotation Curves in the Axisymmetric Gravitational Field of Galaxies. Mathematical Physics and Computer Simulation, 2023, vol. 3, pp. 91-104. DOI: https://doi.org/10.15688/mpcm.jvolsu.2023.3.7Pillepich A., Springel V., Nelson D., Genel S., Naiman J., Pakmor R., Hernquist L., Torrey P., Vogelsberger M., Weinberger R., Marinacci F. Simulating Galaxy Formation with the IllustrisTNG Model. Monthly Notices of the Royal Astronomical Society, 2018, vol. 473, pp. 4077-4106. DOI: https://doi.org/10.1093/mnras/stx2656Potter D., Stadel J., Teyssier R. PKDGRAV3: Beyond Trillion Particle Cosmological Simulations for the Next Era of Galaxy Surveys. Computational Astrophysics and Cosmology, 2017, vol. 4, article ID: 2. DOI: https://doi.org/10.1186/s40668-017-0021-1Ruan Ch.-Z., Hernandez-Aguayo C., Li B., Christian A., Carlton M.B., Klypin A., Prada F. Fast Full 𝑁-Body Simulations of Generic Modified Gravity: Conformal Coupling Models. Journal of Cosmology and Astroparticle Physics, 2022, vol. 2022, no. 5, article ID: 018. DOI: https://dx.doi.org/10.1088/1475-7516/2022/05/018Schaye J., Crain R.A., Bower R.G., Furlong F., Schaller M., Theuns T., Vecchia C.D., Frenk C.S., McCarthy I.G., Helly J.C., Jenkins A., Rosas-Guevara Y.M., White S.D.M., Baes M., Booth C.M., Camps P., Navarro J.F., Qu Y., Rahmati A., Sawala T., Thomas P.A., Trayford J. The EAGLE Project: Simulating the Evolution and Assembly of Galaxies and Their Environments. Monthly Notices of the Royal Astronomical Society, 2014, vol. 446, pp. 521-554. DOI: https://doi.org/10.1093/mnras/stu2058Schaye J., Kugel R., Schaller M., Helly J.C., Braspenning J. The FLAMINGO Project: Cosmological Hydrodynamical Simulations for Large-Scale Structure and Galaxy Cluster Surveys. Monthly Notices of the Royal Astronomical Society, 2023, vol. 526, pp. 4978-5020. DOI: https://doi.org/10.1093/mnras/stad2419Smirnov A.A., Sotnikova N.Y., Koshkin A.A. Simulations of Slow Bars in Anisotropic Disk Systems. Astronomy Letters, 2017, vol. 43, pp. 61-74.Springel V., Yoshida N., White S.D.M. GADGET: A Code for Collisionless and Gasdynamical Cosmological Simulations. New Astronomy, 2001, vol. 6, pp. 79-117. DOI: https://doi.org/10.1016/S1384-1076(01)00042-2Tikhonenko I.S., Smirnov A.A., Sotnikova N.Ya. First Direct Identification of the Barlens Vertical Structure in Galaxy Models. Astronomy & Astrophysics, 2021, vol. 648, article ID: L4. DOI: https://doi.org/10.1051/0004-6361/202140703Titov A.V., Khoperskov A.V. Numerical Modeling of the Collisions of Spheroidal Galaxies: Mass Loss Efficiency by Baryon Components. Vestnik St. Petersburg University, Mathematics, 2022, vol. 55, no. 1, pp. 124-134. DOI: https://doi.org/10.1134/S1063454122010149Vogelsberger M., Marinacci F., Torrey P., Puchwein E. Cosmological Simulations of Galaxy Formation. Nature Reviews Physics, 2020, vol. 2, pp. 42-66. DOI: https://doi.org/10.1038/s42254-019-0127-2Walther J.H. An Influence Matrix Particle–Particle Particle-Mesh Algorithm with Exact Particle–Particle Correction. Journal of Computational Physics, 2003, vol. 184, pp. 670-678.Yokota R., Barba L.A. Treecode and Fast Multipole Method for 𝑁-Body Simulation with CUDA. GPU Computing Gems Emerald Edition. Boston, Morgan Kaufmann, 2011, pp. 113-132. DOI: https://doi.org/10.1016/B978-0-12-384988-5.00009-7Athanassoula E. 𝑁-body Simulations of Galaxies and Groups of Galaxies with the Marseille GRAPE Systems. Annals of the New York Academy of Sciences, 1998, vol. 867, no. 1, pp. 141-155. DOI: https://doi.org/10.1111/j.1749-6632.1998.tb11255.xBagla J.S. Cosmological 𝑁-body Simulation: Techniques, Scope and Status. Current Science, 2005, vol. 88, pp. 1088-1100. DOI: https://doi.org/10.48550/arXiv.astroph/0411043Barnes J., Hut P. A Hierarchical O(N log N) Force-Calculation Algorithm. Nature, 1986, vol. 324, no. 4, pp. 446-449. DOI: https://doi.org/10.1038/324446a0Bissekenov A., Kalambay M., Abdikamalov E., Pang X., Berczik P., Shukirgaliyev B. Cluster Membership Analysis with Supervised Learning and N-Body Simulations. Astronomy & Astrophysics, 2024, vol. 689, article ID: A282. DOI: https://doi.org/10.1051/0004-6361/202449791Butenko M.A., Belikova I.V., Kuzmin N.M., Khokhlova S.S., Ivanchenko G.S., Ten A.V., Kudina I.G. Numerical Simulation of the Galaxies Outer Spiral Structure: the Influence of the Dark Halo Non-Axisymmetry on the Gaseous Disk Shape. Mathematical Physics and Computer Simulation, 2022, vol. 25 (3), pp. 73-83. DOI: https://doi.org/10.15688/mpcm.jvolsu.2022.3.5Ciambur B.C., Fragkoudi F., Khoperskov S., Di Matteo P., Combes F. Double X/Peanut Structures in Barred Galaxies − Insights from an 𝑁-Body Simulation. Monthly Notices of the Royal Astronomical Society, 2020, vol. 503, no. 2, pp. 2203-2214. DOI: https://doi.org/10.1093/mnras/staa3814Dubinski J., Kim J., Park C., Humble R. GOTPM: A Parallel Hybrid Particle-Mesh Treecode. New Astronomy, 2004, vol. 9, pp. 111-126. DOI: https://doi.org/10.1016/j.newast.2003.08.002Fattahi A., Navarro J.F., Sawala T., Frenk C.S., Oman K.A., Crain R.A., Furlong M., Schaller M., Schaye J., Theuns T., Jenkins A. The Apostle Project: Local Group Kinematic Mass Constraints and Simulation Candidate Selection. Monthly Notices of the Royal Astronomical Society, 2016, vol. 457, pp. 844-856. DOI: https://doi.org/10.1093/mnras/stv2970Fridman A.M., Khoperskov A.V. Physics of Galactic Disks. Cambridge International Science Publishing Ltd, 2012. 754 p.Grand R.J.J., Springel V., Gomez F.A., Marinacci F., Pakmor R., Campbell D.J.R., Jenkins A. Vertical Disc Heating in Milky Way-Sized Galaxies in a Cosmological Context. Monthly Notices of the Royal Astronomical Society, 2016, vol. 459, pp. 199-219.Hernandez-Aguayo C., Springel V., Pakmor R., Barrera M., Ferlito F., White S.D.M., Hernquis L., Hadzhiyska B., Delgado A.M., Kannan R., Bose S., Frenk Carlos The MillenniumTNG Project: High-Precision Predictions for Matter Clustering and Halo Statistics. Monthly Notices of the Royal Astronomical Society, 2023, vol. 524, pp. 2556-2578. DOI: https://doi.org/10.1093/mnras/stad1657Ishchenko M., Berczik P., Panamarev T., Kuvatova D., Kalambay M., Gluchshenko A., Veles O., Sobolenko M., Sobodar O., Omarov C. Dynamical Evolution of Milky Way Globular Clusters on the Cosmological Timescale - I. Mass Loss and Interaction with the Nuclear Star Cluster. Astronomy & Astrophysics, 2024, vol. 689, article ID: A178. DOI: https://doi.org/10.1051/0004-6361/202450399Ishchenko M., Berczik P., Sobolenko M. Milky Way Globular Clusters on Cosmological Timescales. IV. Guests in the Outer Solar System. Astronomy & Astrophysics, 2024, vol. 683, article ID: A146. DOI: https://doi.org/10.1051/0004-6361/202347990Just A., Piskunov A.E., Klos J.H., Kovaleva D.A., Polyachenko E.V. Global Survey of Star Clusters in the Milky Way − VII. Tidal Parameters and Mass Function. Astronomy & Astrophysics, 2023, vol. 672, article ID: A187. DOI: https://doi.org/10.1051/0004-6361/202244723Khoperskov A.V., Khrapov S.S., Sirotin D.S. Formation of Transitional cE/UCD Galaxies Through Massive Disc to Dwarf Galaxy Mergers. Galaxies, 2024, vol. 12, no. 1, article ID: 1. DOI: https://doi.org/10.3390/galaxies12010001Khrapov S., Khoperskov A. Study of the Effectiveness of Parallel Algorithms for Modeling the Dynamics of Collisionless Galactic Systems on GPUs. Supercomputing Frontiers and Innovations, 2024, vol. 11, no. 3, pp. 27-44. DOI: https://doi.org/10.14529/jsfi240302Khrapov S.S., Khoperskov A.V., Zaitseva N.A., Zasov A.V., Titov A.V. Formation of Spiral Dwarf Galaxies: Observational Data and Results of Numerical Simulation. Saint Petersburg State Polytechnical University Journal. Physics and Mathematics, 2023, vol. 16, no. 1.2, pp. 395-402. DOI: https://doi.org/10.18721/JPM.161.260Khrapov S.S., Khoperskov S.A., Khoperskov A.V. New features of parallel implementation of 𝑁-body problems on GPU. Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software. 2018, vol. 11, no. 1, pp. 124-136. DOI: https://doi.org/10.14529/mmp180111Kulikov I., Chernykh I., Protasov V. Mathematical Modeling of Formation, Evolution and Interaction of Galaxies in Cosmological Context. Journal of Physics: Conference Series, 2016, vol. 722, no. 1, article ID: 012023. DOI: https://dx.doi.org/10.1088/1742-6596/722/1/012023Kyziropoulos P.E., Filelis-Papadopoulos C.K., Gravvanis G.A. Parallel 𝑁-body Simulation Based on the PM and P3M Methods Using Multigrid Schemes in Conjunction with Generic Approximate Sparse Inverses. Mathematical Problems in Engineering, 2015, vol. 2015, article ID: 450980. DOI: https://doi.org/10.1155/2015/450980L’Huillier H.B., Park C., Kim J. GalaxyFlow: Upsampling Hydrodynamical Simulations for Realistic Mock Stellar Catalogues. New Astronomy, 2014, vol. 30, pp. 79-88. DOI: https://doi.org/10.1016/j.newast.2014.01.007Lim S.H., Raman K.A., Buckley M.R., Shih D. GalaxyFlow: Upsampling Hydrodynamical Simulations for Realistic Mock Stellar Catalogues. Monthly Notices of the Royal Astronomical Society, 2024, vol. 533, no. 1, pp. 143-164. DOI: https://doi.org/10.1093/mnras/stae1672Nipoti C., Cherchi G., Iorio G., Calura F. Effective 𝑁-Body Models of Composite Collisionless Stellar Systems. Monthly Notices of the Royal Astronomical Society, 2021, vol. 503, no. 3, pp. 4221-4230. DOI: https://doi.org/10.1093/mnras/stab763Pejch M.A., Morozov A.G., Khoperskov A.V. Modeling a Double-Hump Gas Rotation Curves in the Axisymmetric Gravitational Field of Galaxies. Mathematical Physics and Computer Simulation, 2023, vol. 3, pp. 91-104. DOI: https://doi.org/10.15688/mpcm.jvolsu.2023.3.7Pillepich A., Springel V., Nelson D., Genel S., Naiman J., Pakmor R., Hernquist L., Torrey P., Vogelsberger M., Weinberger R., Marinacci F. Simulating Galaxy Formation with the IllustrisTNG Model. Monthly Notices of the Royal Astronomical Society, 2018, vol. 473, pp. 4077-4106. DOI: https://doi.org/10.1093/mnras/stx2656Potter D., Stadel J., Teyssier R. PKDGRAV3: Beyond Trillion Particle Cosmological Simulations for the Next Era of Galaxy Surveys. Computational Astrophysics and Cosmology, 2017, vol. 4, article ID: 2. DOI: https://doi.org/10.1186/s40668-017-0021-1Ruan Ch.-Z., Hernandez-Aguayo C., Li B., Christian A., Carlton M.B., Klypin A., Prada F. Fast Full 𝑁-Body Simulations of Generic Modified Gravity: Conformal Coupling Models. Journal of Cosmology and Astroparticle Physics, 2022, vol. 2022, no. 5, article ID: 018. DOI: https://dx.doi.org/10.1088/1475-7516/2022/05/018Schaye J., Crain R.A., Bower R.G., Furlong F., Schaller M., Theuns T., Vecchia C.D., Frenk C.S., McCarthy I.G., Helly J.C., Jenkins A., Rosas-Guevara Y.M., White S.D.M., Baes M., Booth C.M., Camps P., Navarro J.F., Qu Y., Rahmati A., Sawala T., Thomas P.A., Trayford J. The EAGLE Project: Simulating the Evolution and Assembly of Galaxies and Their Environments. Monthly Notices of the Royal Astronomical Society, 2014, vol. 446, pp. 521-554. DOI: https://doi.org/10.1093/mnras/stu2058Schaye J., Kugel R., Schaller M., Helly J.C., Braspenning J. The FLAMINGO Project: Cosmological Hydrodynamical Simulations for Large-Scale Structure and Galaxy Cluster Surveys. Monthly Notices of the Royal Astronomical Society, 2023, vol. 526, pp. 4978-5020. DOI: https://doi.org/10.1093/mnras/stad2419Smirnov A.A., Sotnikova N.Y., Koshkin A.A. Simulations of Slow Bars in Anisotropic Disk Systems. Astronomy Letters, 2017, vol. 43, pp. 61-74.Springel V., Yoshida N., White S.D.M. GADGET: A Code for Collisionless and Gasdynamical Cosmological Simulations. New Astronomy, 2001, vol. 6, pp. 79-117. DOI: https://doi.org/10.1016/S1384-1076(01)00042-2Tikhonenko I.S., Smirnov A.A., Sotnikova N.Ya. First Direct Identification of the Barlens Vertical Structure in Galaxy Models. Astronomy & Astrophysics, 2021, vol. 648, article ID: L4. DOI: https://doi.org/10.1051/0004-6361/202140703Titov A.V., Khoperskov A.V. Numerical Modeling of the Collisions of Spheroidal Galaxies: Mass Loss Efficiency by Baryon Components. Vestnik St. Petersburg University, Mathematics, 2022, vol. 55, no. 1, pp. 124-134. DOI: https://doi.org/10.1134/S1063454122010149Vogelsberger M., Marinacci F., Torrey P., Puchwein E. Cosmological Simulations of Galaxy Formation. Nature Reviews Physics, 2020, vol. 2, pp. 42-66. DOI: https://doi.org/10.1038/s42254-019-0127-2Walther J.H. An Influence Matrix Particle–Particle Particle-Mesh Algorithm with Exact Particle–Particle Correction. Journal of Computational Physics, 2003, vol. 184, pp. 670-678.Yokota R., Barba L.A. Treecode and Fast Multipole Method for 𝑁-Body Simulation with CUDA. GPU Computing Gems Emerald Edition. Boston, Morgan Kaufmann, 2011, pp. 113-132. DOI: https://doi.org/10.1016/B978-0-12-384988-5.00009-7