**Acknowledgements**

**Figures 4** and **<sup>5</sup>** show the evolution of moments 1 and 3, respectively, obtained by the four splitting schemes and the analytical solutions (44d). While 3 directly represents the particle volume fraction (lacking a factor of /6), 1 do not have simple physical meaning, which may be used to define an average diameter through 1/0. For both 1 and 3, the characteristics of splitting errors are quite similar. Lie1 is found to overpredict, while Lie2 to underpredict. The splitting errors for the two Strang schemes are very small, almost indistinguishable from the analytical solutions. According to the error analyses (49c) and (51c), the splitting errors for 1 Strang schemes are zero. Close scrutiny shows in **Figure 4** that the numerical error is not actually zero, which actually arises from the ordinary differential equation (ODE) integration error, since a relatively large time step = 0.2 is used. For a more appropriate smaller time step, both the ODE integration error and splitting error are much smaller. Comparing **Figures 4** and **5**, the relatively errors are comparable for both 1 and 3 here. All the findings agree very

**Figure 5.** Evolution of moment 3 (volume fraction missing a factor /6) in case of constant nucleation and constant

The operator splitting Monte Carlo (OSMC) method has been developed recently [26] for solving the population balance equation for aerosol dynamics. Within the OSMC, nucleation and surface growth are handled with deterministic means, while coagulation is simulated with a stochastic method (the Marcus‐Lushnikov stochastic process). The deterministic and stochastic approaches in the algorithm are synthesized under the framework of operator splitting. The ultimate goal of the OSMC is to greatly improve the numerical efficiency and to

condensational growth in diameter (moment is defined with respect to particle diameter).

well with the analyses in Section 3.2.

26 Aerosols - Science and Case Studies

**5. Conclusions**

The financial support from the National Natural Science Foundation of China (No. 11402179 and 11572274) is greatly acknowledged. KZ would like to thank Dr. F. Bisetti at KAUST, Saudi Arabia, for his insightful discussion and valuable help. The financial support from the General Research Fund, Research Grants Council of the Hong Kong SAR, China (No. PolyU 152663/16E) is also greatly appreciated for allowing the authors' further development and extension of this OSCM foundation method.
