Adaptive time-stepping for the Super-Droplet Method Monte Carlo collision-coalescence scheme

We present an analysis of an adaptive time-stepping scheme for the Super-Droplet Method (SDM), a Monte Carlo algorithm for simulating particle coagulation. SDM represents cloud droplets as weighted superdroplets, enabling high-fidelity representations of microphysical processes such as collision-coalescence. However, the algorithm can undercount collisions when the expected number of events is not realizable given the superdroplet configuration, introducing a biased error referred here as the collision deficit. While SDM exhibits statistical spread inherent to Monte Carlo schemes, the deficit is a systematic underestimation of collision events. This error can be addressed with adaptive time-stepping, which dynamically adjusts simulation time steps to eliminate this deficit. We analyze the behavior of the deficit across a wide range of timesteps, superdroplet counts, and initialization strategies, and explore trade-offs between accuracy and efficiency. Using the classical Safranov-Golovin test case, we show that the deficit increases with timestep and superdroplet count, and that adaptive time-stepping effectively removes the associated error without significant cost. We test a smooth continuum of initial distributions with extrema representing two different initialization methods, and find that while the deficit is sensitive to the choice of attribute-space sampling strategies, adaptive time-stepping substantially reduces the difference, allowing for users to choose initialization methods optimized for other processes. We also propose a method of visualization, capturing both the attribute sampling, droplet interactions over multiple timesteps, and the deficit using network connectivity graphs. In 2-D flow-coupled simulations, we find the deficit can have a stronger effect on convergence than previously shown, with uncorrected deficit delaying the onset of precipitation.