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.

Aggregation in non-uniform systems with advection and localized source

“…The detailed cross-validation of the accuracy and efficiency of the new method in comparison with coarse-graining and Monte Carlo approaches has been reported in our previous works [24,43,46]. Moreover, in the case of equation ( 2) and simple generalization of these equations, such fast algorithms can be utilized in parallel; a good parallel scaling may be achieved at modern computing clusters [47,48] supported by thousands of central processing unit (CPU)-cores and graphics processing unit (GPU) accelerators.…”

“…with K Br 1 = 2k B T/(3η). Both these kernels are homogeneous, K ai,a j = a λ K i, j , with λ = 4/3 for the ballistic kernel (46) and λ = 0 for the Brownian kernel (47). Thus, the ballistic kernel rapidly increases with masses of the aggregates, and this provides an extra justification for using it and ignoring the merging due to diffusion 7 Suppose the densities become stationary in the long-time limit.…”

“…7 For i = j however K ball i j = 0, that is, the diffusion mechanism may not be neglected. Hence, for i = j we use K Br i,i from equation (47), assuming that K Br 1 = 1.…”

“…Therefore the equations with the true ballistic kernel have been solved by traditional numerical methods. To get rid of zeros on the diagonal of the ballistic kernel (47), we add the diagonal elements of the Brownian kernel. In this way we take into account that particles of the same size, moving with the same sedimentation velocity, could still aggregate.…”