Abstract
In this work we propose two parallel implementations of numerical method for the two-dimensional advection-coagulation equation: pure CPU and hybrid CPU/GPU. We approximate the advection component across the two dimensional space with use of unstructured grid and finite volume method with flux limiters. Smoluchowski coalescence operator corresponds to the coagulation process. We evaluate it within low complexity (\(O (N \log N)\)) via exploitation of the low-rank skeleton decomposition of coagulation kernel. We decompose spatial grid into the subdomains and solve the model equation in parallel using MPI. Even though we exploit the fast methods for evaluation of coalescence operator it is the most time-consuming part of numerical algorithm. Hence, we test performance of GPU accelerators for corresponding Smolushowski integrals. All in all, we evaluate the efficiency of incorporating MPI and Nvidia CuFFT library for speedup of calculations and obtain almost linear scalability of MPI implementation of our algorithm. We also find that hybrid exploitation of CPUs and GPUs leads to additional speedup of computations by 2–4 times.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Zagidullin, R.R., Smirnov, A.P., Matveev, S.A., Tyrtyshnikov, E.E.: An efficient numerical method for a mathematical model of a transport of coagulating particles. Moscow Univ. Comput. Math. Cybern. 41, 179–186 (2017)
Matveev, S.A., Zagidullin, R.R., Smirnov, A.P., Tyrtyshnikov, E.E.: Parallel numerical algorithm for solving advection equation for coagulating particles. Supercomput. Frontiers Innovations 5(2), 43–54 (2018)
Galkin, V.A.: Smoluchowski equation. Fizmatlit, Moscow, p. 336 (2001)
Darwish, M.S., Moukalled, F.: TVD schemes for unstructured grids. Int. J. Heat Mass Transf. 46, 599–611 (2003)
Denner, F., van Wachem, B.G.M.: TVD differencing on three-dimensional unstructured meshes with monotonicity-preserving correction of mesh skewness. J. Comput. Phys. 298, 466–479 (2015)
Syrakos, A., Varchanis, S., Dimakopoulos, Y., Goulas, A., Tsamopoulos, J.: A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods. Phys. Fluids 29, 127103 (2017)
Sozer, E., Brehm, C., Kiris, C.C.: Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers. In: Science and Technology Forum and Exposition: Conference Paper (2014)
Tyrtyshnikov, E.E.: Incomplete cross approximation in the mosaic-skeleton methods. Computing 64, 367–380 (2000)
Matveev, S.A.: A parallel implementation of a fast method for solving the smoluchowski-type kinetic equations of aggregation and fragmentation processes. Vychislitel’nye Metody i Programmirovanie 16, 360–368 (2015)
Steinbach, P., Werner, M.: gearshifft - the FFT Benchmark Suite for Heterogeneous Platforms. arXiv:1702.00629 (2017)
Sadovnichy, V., Tikhonravov, A., Voevodin, Vl., Opanasenko, V.: “Lomonosov”: supercomputing at Moscow State University. In: Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science), pp. 283–307. CRC Press, Boca Raton (2013)
Betelin, V.B., Galkin, V.A.: On the formation of structures in nonlinear problems of physical kinetics. Doklady Math. 99(1) (2019). Pleiades Publishing
Xu, Z., Zhao, H., Zheng, C.: Accelerating population balance-Monte Carlo simulation for coagulation dynamics from the Markov jump model, stochastic algorithm and GPU parallel computing. J. Comput. Phys. 281, 844–863 (2015)
Volochuk, V.M., Sedunov, Y.: Coagulation Processes in Dispersed Systems. Gidrometeoizd, Leningrad (1975)
Aloyan, A.E., Arutyunyan, V.O., Lushnikov, A.A., Zagaynov, V.A.: Transport of coagulating aerosol in the atmosphere. J. Aerosol Sci. 28(1), 67–85 (1997)
Hackbusch, W., John, V., Khachatryan, A., Suciu, C.: A numerical method for the simulation of an aggregation-driven population balance system. Int. J. Numer. Meth. Fluids 69(10), 1646–1660 (2012)
Boje, A., Akroyd, J., Kraft, M.: A hybrid particle-number and particle model for efficient solution of population balance equations. J. Comput. Phys. 389, 189–218 (2019)
Zacharov, I., et al.: ‘Zhores’-Petaflops supercomputer for data-driven modeling, machine learning and artificial intelligence installed in Skolkovo Institute of Science and Technology. arXiv preprint arXiv:1902.07490 (2019)
Acknowledgements
We would like to thank Dmitry Zheltkov for valuable consultations during preparation of the numerical experiments. The research is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University [11] and Zhores supercomputer of Skolkovo Institute of Science and Technology [18].
This article contains the results of the project performed in the framework of the implementation of the programs of the Central Competences of the National Technological Database “Center for Big Data Storage and Analysis” (project “Tensor methods for processing and analysis of Big Data”) of Lomonosov MSU with the Project Support Funding of the National Technological Reporting dated December 11, 2018 No. 13/1251/2018.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Zagidullin, R., Smirnov, A., Matveev, S., Tyrtyshnikov, E. (2019). Supercomputer Modelling of Spatially-heterogeneous Coagulation using MPI and CUDA. In: Voevodin, V., Sobolev, S. (eds) Supercomputing. RuSCDays 2019. Communications in Computer and Information Science, vol 1129. Springer, Cham. https://doi.org/10.1007/978-3-030-36592-9_33
Download citation
DOI: https://doi.org/10.1007/978-3-030-36592-9_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-36591-2
Online ISBN: 978-3-030-36592-9
eBook Packages: Computer ScienceComputer Science (R0)