Parallel Numerical Algorithm for Solving Advection Equation for Coagulating Particles

Sergey Alexandrovich Matveev; Rishat R. Zagidullin; Alexander P. Smirnov; Eugene E. Tyrtyshnikov

doi:10.14529/jsfi180204

Authors

Sergey Alexandrovich Matveev Skolkovo Institute of Science and Technology and Marchuk Institute of Numerical Mathematics RAS
Rishat R. Zagidullin faculty of Computational Mathematics and Cybernetics of Lomonosov Moscow State University
Alexander P. Smirnov faculty of Computational Mathematics and Cybernetics of Lomonosov Moscow State University and Marchuk Institute of Numerical Mathematics RAS
Eugene E. Tyrtyshnikov Marchuk Institute of Numerical Mathematics RAS and faculty of Computational Mathematics and Cybernetics of Lomonosov Moscow State University

DOI:

https://doi.org/10.14529/jsfi180204

Abstract

In this work we present a parallel implementation of numerical algorithm solving the Cauchy problem for equation of advection of coagulating particles. This equation describes time-evolution of the concentration f(x, v, t) of particles of size v at the point x at the time-moment t. Our numerical algorithm is based on use of total variation diminishing (TVD) scheme and perfectly matching layers (PML) for approximation of advection operator along spatial coordinate x and utilization of the fast numerical method for evaluation of coagulation integrals exploiting low-rank decomposition of coagulation kernel coefficients and fast FFT-based implementation of convolution operation along particle size coordinate v. In our work we exploit one-dimensional domain decomposition approach along spatial coordinate x because it allows to avoid use of parallel FFT implementations which are very expensive in terms of data exchanges and have poor parallel scalability. Moreover, locality of finite-difference operator from TVD-scheme along x coordinate allows to obtain good scalability even for computing clusters with slow network interconnect due to modest volumes of data necessary for synchronization exchanges between times integration steps.

References

Aloyan, A.: Dynamics and kinematics of gas impurities and aerosols in the atmosphere. A Textbook (2002)

Ball, R., Connaughton, C., Jones, P., Rajesh, R., Zaboronski, O.: Collective oscillations in irreversible coagulation driven by monomer inputs and large-cluster outputs. Physical review letters 109(16), 168304 (2012), DOI: 10.1103/PhysRevLett.109.168304

Ball, R., Connaughton, C., Stein, T.H., Zaboronski, O.: Instantaneous gelation in

Smoluchowskis coagulation equation revisited. Physical Review E 84(1), 011111 (2011), DOI: 10.1103/PhysRevE.84.011111

Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 114(2), 185–200 (1994), DOI: 10.1006/jcph.1994.1159

Brilliantov, N., Krapivsky, P., Bodrova, A., Spahn, F., Hayakawa, H., Stadnichuk, V.,

Schmidt, J.: Size distribution of particles in saturns rings from aggregation and fragmentation. Proceedings of the National Academy of Sciences 112(31), 9536–9541 (2015), DOI: 10.1073/pnas.1503957112

Brilliantov, N., Bodrova, A., Krapivsky, P.: A model of ballistic aggregation and fragmentation. Journal of Statistical Mechanics: Theory and Experiment 2009(06), P06011 (2009), DOI: 10.1088/1742-5468/2009/06/P06011

Chaudhury, A., Oseledets, I., Ramachandran, R.: A computationally efficient technique for the solution of multi-dimensional PBMs of granulation via tensor decomposition. Computers & Chemical Engineering 61, 234–244 (2014), DOI: 10.1016/j.compchemeng.2013.10.020

Galkin, V.: Smoluchowski equation. Fizmatlit, Moscow (2001), (in Russian)

Gupta, A., Kumar, V.: The scalability of fft on parallel computers. Parallel and Distributed Systems, IEEE Transactions on 4(8), 922–932 (1993), DOI: 10.1109/71.238626

Leer, B.V.: Towards the ultimate conservative difference scheme. iv. a new approach to numerical convection. Journal of Computational Physics 23(3), 276–299 (1977), DOI: 10.1016/0021-9991(77)90095-X

Lyra, P.R.M., Morgan, K., Peraire, J., Peir, J.: TVD algorithms for the solution of the

compressible euler equations on unstructured meshes. International Journal for Numerical Methods in Fluids 19(9), 827–847, DOI: 10.1002/fld.1650190906

Matveev, S.A., Krapivsky, P.L., Smirnov, A.P., Tyrtyshnikov, E.E., Brilliantov, N.V.: Oscillations in aggregation-shattering processes. Phys. Rev. Lett. 119, 260601 (Dec 2017), DOI: 10.1103/PhysRevLett.119.260601

Matveev, S., Smirnov, A., Tyrtyshnikov, E.: A fast numerical method for the cauchy problem for the Smoluchowski equation. Journal of Computational Physics 282, 23–32 (2015), DOI: 10.1016/j.jcp.2014.11.003

Matveev, S., Stadnichuk, V., Tyrtyshnikov, E., Smirnov, A., Ampilogova, N., Brilliantov, N.: Anderson acceleration method of finding steady-state particle size distribution for a wide class of aggregationfragmentation models. Computer Physics Communications 224, 154–163 (2018), DOI: 10.1016/j.cpc.2017.11.002

Matveev, S., Zheltkov, D., Tyrtyshnikov, E., Smirnov, A.: Tensor train versus Monte

Carlo for the multicomponent Smoluchowski coagulation equation. Journal of Computational Physics 316, 164–179 (2016), DOI: 10.1016/j.jcp.2016.04.025

Matveev, S.A.: A parallel implementation of a fast method for solving the smoluchowskitype kinetic equations of aggregation and fragmentation processes. Vychislitel’nye Metody i Programmirovanie 16(3), 360–368 (2015), (in Russian)

Mirzaev, I., Byrne, E.C., Bortz, D.M.: An inverse problem for a class of conditional

probability measure-dependent evolution equations. Inverse Problems 32(9), 095005 (2016), DOI: 10.1088/0266-5611/32/9/095005

Muller, H.: Zur allgemeinen theorie ser raschen koagulation. Fortschrittsberichte uber Kolloide und Polymere 27(6), 223–250 (1928), DOI: 10.1007/BF02558510

Okuzumi, S., Tanaka, H., Kobayashi, H., Wada, K.: Rapid coagulation of porous dust

aggregates outside the snow line: A pathway to successful icy planetesimal formation. The Astrophysical Journal 752(2), 106 (2012), DOI: 10.1088/0004-637X/752/2/106

Piskunov, V.: Analytical solutions for coagulation and condensation kinetics

of composite particles. Physica D: Nonlinear Phenomena 249, 38–45 (2013),

DOI: 10.1016/j.physd.2013.01.008

Rakhuba, M.V., Oseledets, I.V.: Fast multidimensional convolution in low-rank tensor formats via cross approximation. SIAM Journal on Scientific Computing 37(2), A565–A582 (2015), DOI: 10.1137/140958529

Sabelfeld, K.: A random walk on spheres based kinetic monte carlo method for simulation of the fluctuation-limited bimolecular reactions. Mathematics and Computers in Simulation (2016), DOI: 10.1016/j.matcom.2016.03.011

Smirnov, A., Matveev, S., Zheltkov, D., Tyrtyshnikov, E.: Fast and accurate finite-difference method solving multicomponent Smoluchowski coagulation equation with source and sink terms. Procedia Computer Science 80, 2141–2146 (2016), DOI: 10.1016/j.procs.2016.05.533

von Smoluchowski, M.: Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Zeitschrift fur Physik 17, 557–585 (1916)

Sorokin, A., Strizhov, V., Demin, M., Smirnov, A.: Monte-Carlo modeling of aerosol kinetics. Atomic Energy 117(4), 289–293 (2015), DOI: 10.1007/s10512-015-9923-7

Stadnichuk, V., Bodrova, A., Brilliantov, N.: Smoluchowski aggregation–fragmentation equations: Fast numerical method to find steady-state solutions. International Journal of Modern Physics B 29(29), 1550208 (2015), DOI: 10.1142/S0217979215502082

Tyrtyshnikov, E.E.: Incomplete cross approximation in the mosaic–skeleton method. Computing 64(4), 367–380 (2000), DOI: 10.1007/s006070070031

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 University Computational Mathematics and Cybernetics 41(4), 179–186 (Oct 2017), DOI: 10.3103/S0278641917040082

Zheltkov, D.A., Tyrtyshnikov, E.E.: A parallel implementation of the matrix cross approximation method. Vychislitel’nye Metody i Programmirovanie 16(3), 369–375 (2015), (in Russian)