ИСТИНА

In this work we present use of low-rank tensor decompositions for acceleration of evaluation of right-hand side of systems of kinetic equations with many-particle collisional terms. These equations can be interpreted as a generalization of classical Smoluchowski aggregation equations [1], allowing one to consider not only binary collisions of particles but also many particle collisions. Straight-forward evaluation of right-hand side for such system requires exponentially large quantity of numerical operations and we find such complexity too high for practical investigations. However, under assumptions that the kinetic coefficients can be represented with either canonical polyadic (CP) or tensor train decomposition (TT) with small ranks we can propose algorithms evaluating the right-hand side with much lower complexities. We check the accuracy of proposed approach for model Cauchy problem with constant kinetic coefficients and monodisperse initial conditions and obtain good agreement of numerical results with known explicit solution [2]. With use of our ideas we reach high level of accuracy of numerical solutions in really modest CPU-times [3]. We compare numerical solutions with different many-particle collision rates and obtain a significant influence of accounting triple collisional effects. [1] Smoluchowski M. Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Zeitschrift fur Physik 1916; 17: 557–585. [2] Krapivsky P L. Aggregation processes with n-particle elementary reactions. J of Phys A 1991; 24: 4697–4703. [3] Stefonishin D A, Matveev S A, Smirnov A P, Tyrtyshnikov E E. Tensor Decompositions for Solving the equations of Mathematical Models of Aggregation with Multiple Collisions of Particles. Numerical Methods and Programming 2018; 19: 390–404.