A quasi-Monte Carlo scheme for Smoluchowski's coagulation equation

This paper analyzes a Monte Carlo algorithm for solving Smoluchowski’s coagulation equation. A finite number of particles approximates the initial mass distribution. Time is discretized and random numbers are used to move the particles according to the coagulation dynamics. Convergence is proved when quasi-random numbers are utilized and if the particles are relabeled according to mass in every time step. The results of some numerical experiments show that the error of the new algorithm is smaller than the error of a standard Monte Carlo algorithm using pseudo-random numbers without reordering the particles. Introduction Models of coalescence (i.e., coagulation, gelation, aggregation, agglomeration, accretion, etc.) mainly stem from the work of Smoluchowski on coagulation processes in colloids [15, 16]. Smoluchowski proposed the following infinite system of differential equations for the evolution of the number N0c(i, t) of clusters of mass i for i = 1, 2, 3 . . .: (1) ∂c ∂t (i, t) = 1 2 ∑ 1≤j t, the points xp with nb ≤ p < (n+ 1)b form a (t,m, s)-net in base b. The following result is shown in [12]. Lemma 1. Let X be a (t,m, s)-net in base b. For any elementary interval Q′ ⊂ Is−1 in base b and for any xs ∈ Ī, |Dbm(Q × [0, xs), X)| ≤ bt−m. The effectiveness of QMC methods has limitations. First, while they are valid for integration problems, they may not be directly applicable to simulations, due to the correlations between the points of a quasi-random sequence. This problem can be overcome by writing the desired result as an integral. This leads to a second limitation: the improved accuracy of QMC methods may be lost for problems in which the integrand is not smooth. It is necessary to take special measures to make optimal use of the greater uniformity associated with quasi-random sequences. This is achieved here through the additional step of reordering the particles at each time step. The aim of the paper is to construct and investigate a QMC method for Smoluchowski’s coagulation equation. In Section 1 we present a particle scheme using quasi-random numbers for the solution of the equation. In Section 2 we prove the convergence of the method, as the number of simulated particles increases. In Section 3 we carry out numerical experiments based on a comparison of the method with a standard MC scheme. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use A QUASI–MONTE CARLO SCHEME FOR SMOLUCHOWSKI’S EQUATION 1955 1. The algorithm We assume that the coagulation kernel K(i, j) is nonnegative and symmetric K(i, j) ≥ 0 and K(i, j) = K(j, i). Multiplying (1) by i and summing over all i, indicates that mass is conserved (2) d dt ∑