Parallel Numerical Algorithm for Solving Advection Equation for Coagulating Particles
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
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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)
Published
2018-06-29
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