We are presenting an analysis on an adaptive time-stepping scheme for the Super-Droplet Method (SDM) introduced in prior work to improve the efficiency and accuracy of probabilistic droplet coalescence in cloud microphysics. Superdroplets are computational particles that represent weighted ensembles of real cloud droplets, enabling high-fidelity representations of microphysical processes like collision–coalescence. SDM, first introduced by Shima et al. (2009), is a Monte Carlo algorithm that selects superdroplet pairs linearly within each timestep and simulation volume to test for collisions. The algorithm also includes logic that allows multiple collisions between the same candidate pair, accounting for the likelihood of repeated interactions within a single timestep.

Adaptive time-stepping dynamically adjusts simulation time steps to control the coalescence deficit: a quantifiable error that arises when the available droplet population within superdroplet candidate pairs is not large enough to undergo the expected collision events. While SDM exhibits inherent statistical spread due to its probabilistic nature, the deficit represents a systemic underestimation bias of collision events.

To validate the adaptive scheme, we implement it independently in two open-source models, PySDM and Droplets.jl. The results demonstrate consistency across platforms. Using the classical Golovin test case, we compare SDM and adaptive SDM scheme in terms of convergence, computational cost, and fidelity. We also draw conceptual parallels to the Weighted Flow Algorithm (WFA) Monte Carlo algorithm employed in PyPartMC, which employs analogous adaptive logic for eliminating the deficit within a different coalescence framework.

Our convergence analysis spans a range of timesteps, superdroplet counts, and initialization methods, examining how the deficit scales and exploring the trade-off between efficiency and error mitigation when adaptive time-stepping is implemented. We show that the substepping procedure maintains a computational cost similar to that of the original SDM using the “parent” timestep, while achieving accuracy comparable to smaller fixed timesteps. Overall, our results demonstrate that adaptive time stepping provides a more efficient path to reducing coalescence error than uniformly decreasing the timestep, with implications for large-scale cloud and precipitation modeling. While we focus on warm-phase microphysics, the results are also applicable to cold and mixed-phase systems, as well as other coagulation problems outside atmospheric science.

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