Adaptive time-stepping for aggregation-shattering kinetics

TL;DR

This paper tackles the computational burden of simulating aggregation-fragmentation kinetics described by Smoluchowski-type equations, especially for large truncations and regimes with slow relaxation or rapid oscillations. It proposes and tests adaptive time-stepping schemes for explicit Runge-Kutta methods (RK2, RK4, RKF45) paired with low-rank kernel decompositions to reduce the cost of evaluating the nonlinear right-hand side from $O(M^2)$ to $O(M R \log M)$. Through two numerical experiments—complex stationary distributions with multiple sources and aggregation-shattering with dynamic oscillations—it shows that adaptive stepping yields substantial speedups (often dozens of times, and around 10–15× in oscillatory cases) while preserving accuracy, enabling reliable tracking of relaxation to stationary states and transient dynamics with modest resources. The results highlight the practical impact of dynamic time-stepping for large-scale kinetic systems and motivate exploring implicit methods for future extensions to spatially inhomogeneous coagulation problems.

Abstract

We propose an experimental study of adaptive time-stepping methods for efficient modeling of the aggregation-fragmentation kinetics. Precise modeling of this phenomena usually requires utilization of the large systems of nonlinear ordinary differential equations and intensive computations. We concentrate on performance of three explicit Runge-Kutta time-integration methods and provide simulations for two types of problems: finding of equilibrium solutions and simulations for kinetics with periodic solutions. The first class of problems may be analyzed through the relaxation of the solution to the stationary state after large time. In this case, the adaptive time-stepping may help to reach it using big steps reducing cost of the calculations without loss of accuracy. In the second case, the problem becomes numerically unstable at certain points of the phase space and may require tiny steps making the simulations very time-consuming. Adaptive criteria allows to increase the steps for most of points and speedup simulations significantly.

Figures (3)

• Figure 1: The problem with monodisperse conditions $M = 32768$ and a constant power of the source of monomers $P_1=1$ and several values of $P_{100}$ from $10^{-3}$ to $10^{-1}$. Left panel: the numerical stationary solutions oscillate for a wide range of the particle sizes in agreement with matveev2020oscillating and relax to the scaling $n_k \simeq k^{-3/2}$, $k \gg 1$. Center panel: dynamics of the concentrations $n_k(t)$ for multiple masses. Right panel: the adaptive time-step increase quite rapidly and oscillate around unity for a long segment of model time.
• Figure 2: Numerical experiments for the constant kernel with $\alpha=0$, $M=32768$ and $\lambda= 0.01$ and monodisperse initial conditions. Left panel: very slow relaxation of the $n_k$ for $k\gg 1$ to the stationary distribution $n_k \propto k^{-3/2}$, the scaling becomes stable only for $T=10^4$. The solutions become close to the basic asymptotic form only for the very large $T$. Right panel: the time steps grow up to very large values if the adaptive time-stepping rule is applied.
• Figure 3: Numerical experiments for the generalized Brownian kernel with $\alpha=0.95$, $M=32768$ and $\lambda=0.01$ and monodisperse initial conditions. Left panel: oscillations of the total density $N(t)$ in agreement with baseline paper matveev2017oscillations. Right panel: the adaptive time steps also oscillate with time and lead to speedup the computations.