**4. Numerical examples**

d

*M t Jt M* OP \*

*M t JGt Jd GM t M <sup>p</sup>* OP 2\* \* \* 1 0 1 <sup>1</sup> ( )= ( ) , <sup>2</sup>

*M t JG t Jd GM Gt Jd GM t M p p* OP 2 3 \* \* 2 \*2 \* \* 2 0 1 2 <sup>1</sup> ( )= ( ) ( 2 ) , <sup>4</sup>

*M t JG t Jd GM G t Jd GM Gt Jd GM t M p pp* OP 3 4 \* \* 23 \* 2 \* 2 \*3 \* \* 3 0 1 2 3 13 3 ( )= ( ) ( 3 ) ( 3 ) . 84 2



1

2(iv)

3(iv)

d

2\* 3 3 4

To summarize, there is no splitting error for the number density (0), which is solely deter‐ mined by nucleation. For 1 to 3, all Lie schemes are of first order (globally), and the leading errors are of opposite sign between Lie1 and Lie2. For Strang schemes, the splitting error for 1 is also zero, and the errors for 2 and 3 are of second order. In Lie1 and Strang1 scheme,

 d

2 3

The splitting error is found by comparing the splitting solution (51) with the exact solution (44)

3(iii)

1

nuc

and then Eq. (42) with half step again, the final solution at <sup>=</sup>

After sequentially solving Eq. (42) with half time step

<sup>2</sup>cond

(iv) Strang scheme 1

22 Aerosols - Science and Case Studies

+ 2\* 3 3 4

2cond (Denoted as Strang2)

 d

1 1 = . 2 4 *p t <sup>t</sup> Er JG d JG* (50d)

/2, then Eq. (41) with full time step,

is found as

0 0 ( )= , + (51a)

++ + (51b)

++ + + + (51c)

0(iv) *Er* = 0, (52a)

1(iv) *Er* = 0, (52b)

= , <sup>12</sup> *<sup>t</sup> Er JG* (52c)

1 1 = . 4 8 *p t <sup>t</sup> Er JG d JG* (52d)

+ + + + ++ + (51d)

### **4.1. Case of nucleation + coagulation**

Corresponding to the error analyses in Section 3.1, the case of constant nucleation and constant coagulation are simulated with the OSMC. The coagulation kernel and nucleation rate are set as unit 0 = 1, and = 1. Initially particles are mono dispersed with unit volume size, and the number density is 0 \*=1. Nucleated particles also have unit volume size.

**Figure 2** shows the number density (0) simulated by the four schemes, compared against the analytical solution (19). All the simulations use fixed time step = 0.1, with = 2000 Monte Carlo particles. The OSMC results are averaged over dozens of independent simulations to eliminate the randomness. Lie1 scheme is found to systematically underpredict the number density, while Lie2 to overpredict. Strang1 gives result negligibly higher than the analytical solution. Strang2 slightly underpredicts the true solution. All these findings agree extremely well with the analyses in Section 3.1. It is also found that Strang1 shows better precision than Strang2, that is because nucleation rate is higher than the coagulation rate ( =1> <sup>1</sup> <sup>2</sup>00 \* = <sup>1</sup> 2), and the higher rate process nucleation is split in Strang1, as explained in Section 3.1 also.

**Figure 3** compares 2 among the four splitting schemes and the analytical solution (39). Lie1 is found to overestimate 2 a little bit, while Lie2 to underestimate. The second‐order Strang

schemes give result almost indistinguishable from the analytical solution. Although second‐ order Strang schemes give better results than the Lie schemes, the discrepancy is not so significant as that for 0 in **Figure 2**.

**Figure 2.** Evolution of total particle number density 0 for constant kernel coagulation and constant rate nucleation ( = 0.1).

**Figure 3.** Evolution of moment 2 in case of constant kernel coagulation and constant rate nucleation (moment is defined with respect to the particle volume).

Both the error analyses and the numerical examples above show that the second‐order Strang schemes have much higher precision than the first‐order Lie schemes. Within the OSMC, coagulation is the most computationally intensive part, and nucleation only makes a tiny fraction of the total time cost. The Strang1 scheme has much higher precision than the

Lie schemes, and only needs to integrate nucleation twice often as in the Lie schemes, where the computational overhead is negligible. Strang2 needs nearly double computational cost as Strang1, and Strang2 outperforms Strang1 in precision only if 1 <sup>2</sup>00 \*<sup>2</sup> <sup>&</sup>gt; . The best per‐ formance of Strang2 over Strang1 is to nearly double the precision when 00 \*<sup>2</sup> ≫ . Hence, Strang1 is the optimal splitting scheme taking both the numerical precision and cost into account.

Another general conclusion is that only 0 needs to be concerned when considering the operator splitting errors. Operator splitting generally produces less errors for higher order moments. The 2 splitting errors for the Strang schemes contain the factor of the critical volume of nucleated particles, which is generally very small. Taking a more practical numerical example of dibutyl phthalate (DBP) aerosol nucleation and coagulation in a turbulent mixing layer [26], the critical volume \* is of order 10−27m3, which renders very small splitting errors for high order moments. However, the splitting error for 0 may be still significant, and needs proper splitting scheme combined with small enough time step to satisfy the given error criteria.

**Figure 4.** Evolution of moment 1 in case of constant nucleation and constant condensational growth in diameter (moment is defined with respect to particle diameter).

#### **4.2. Case of nucleation + condensation**

schemes give result almost indistinguishable from the analytical solution. Although second‐ order Strang schemes give better results than the Lie schemes, the discrepancy is not so

**Figure 2.** Evolution of total particle number density 0 for constant kernel coagulation and constant rate nucleation

**Figure 3.** Evolution of moment 2 in case of constant kernel coagulation and constant rate nucleation (moment is

Both the error analyses and the numerical examples above show that the second‐order Strang schemes have much higher precision than the first‐order Lie schemes. Within the OSMC, coagulation is the most computationally intensive part, and nucleation only makes a tiny fraction of the total time cost. The Strang1 scheme has much higher precision than the

significant as that for 0 in **Figure 2**.

24 Aerosols - Science and Case Studies

defined with respect to the particle volume).

( = 0.1).

Corresponding to the error analysis in Section 3.2, the case of constant nucleation and constant condensational growth in diameter is simulated with the OSMC. Both the nucleation rate and growth rate are set to unit = 1 and = 1. Initially particles are monodispersed with unit volume size, and the number density is 0 \*=1. Nucleated particles also have unit volume size. The time step is = 0.2.

**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 well with the analyses in Section 3.2.

**Figure 5.** Evolution of moment 3 (volume fraction missing a factor /6) in case of constant nucleation and constant condensational growth in diameter (moment is defined with respect to particle diameter).

#### **5. Conclusions**

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 preserve the extraordinary flexibility and applicability of a stochastic method for aerosol dynamics simulation.

The OSMC has been validated through quite a few testing cases [26], and has also been used to simulate aerosol evolution under various conditions [32, 39]. However, the operator splitting errors in the OSMC have not been systematically investigated. Here, focused on two repre‐ sentative cases, i.e., constant nucleation and coagulation, and constant nucleation and con‐ densation, the splitting errors for four splitting schemes (two Lie schemes and two Strang schemes) are analyzed rigorously, combined with concrete numerical examples.

The number density, one of the most important statistics for aerosol particles, is found to be underestimated for the Lie1 scheme ( nuc coag), while overestimated for the Lie2 scheme (coag nuc). This is because coagulation rate would be overpredicted if nucleation is integrated before coagulation, hence the number density would be underpredicted as in Lie1, and vice versa. The second‐order Strang schemes exhibit much better precision that the first‐ order Lie schemes. If coagulation rate is higher than the nucleation rate, Strang2 ( 1 <sup>2</sup>coag nuc 1 2coag) has better precision than Strang1 (<sup>1</sup> 2 nuc coag 1 2 nuc). Other‐ wise, Strang1 has better precision. However, for the OSMC, Strang1 is always the most preferable splitting scheme after taking the numerical cost and precision into account.

The operator splitting errors for the case of nucleation and condensation show that splitting errors for different moments (except for 0) are comparable with each other. Splitting between nucleation and condensation has marginal impact on accurately predicting the moments. This is in great contrast to the prediction of 0, where splitting between nucleation and coagulation has significant impact.

To construct a good operator splitting scheme for the general case of nucleation, coagulation, and condensation the first priority is to consider the splitting error for 0. When a good scheme

for splitting between nucleation and coagulation is constructed, splitting among other aerosol dynamic processes can be constructed at more freedom. The overall splitting errors are pretty much determined by the splitting error for 0.

The analyses not only provide sound theoretical bases for selecting the most efficient operator splitting scheme for the usage of the OSMC, but also shed lights on how to adopt operator splitting in other PBE solving methods, i.e., direct discretization, method of moments etc.
