ИСТИНА

In this work we present novel computational approach to numerical solution of the multicomponent Smoluchowski coagulation equation. New method is based on application of the low-rank approximations of both the coagulation kernel and the numerical solution in the tensor train(TT) format. Fast algorithms of linear algebra and the efficient implementations of TT-arithmetics allow us to reduce the complexity of well-known predictor-corrector scheme from O(N^{2d}) operations to just O(d^2 R^4 N log N)O(d^2R^4 N log ⁡N), where N is number of the grid nodes per each component axis, dd is number of coagulating species in the Smoluchowski coagulation equation and R is the maximal TT-rank of the used tensor trains. In contrast to method proposed by Chaudhury and Oseledets our scheme has the higher order of the accuracy. We test both the efficiency and the accuracy of novel methodology in comparison with the classical finite-difference predictor-corrector scheme and with Monte Carlo. The results of the numerical experiments allow us to claim that our method dramatically outperforms classical predictor-corrector scheme and also allows to reach the accuracy which seems to be unavailable for Monte Carlo solvers. Hence, we conclude that the finite-difference methods with the use of low-rank approximations can be successfully applied to complex problems of multicomponent coagulation processes.