Algorithm 1 DSMC algorithm for Boltzmann equation

1:Given N particles $({\mathbf{x}}_n^0,c_n^0)$, with $n=1,\dots ,N$ computed from the initial distribution $f_0(\mathbf{x},c)$; 2:for$t=1$to $N_t$ do 3:      set $n_p=\mathrm{Sround}(N/2)$; 4:     sample $n_p$ pairs $(i,j)$ uniformly without repetition among all possible pairs of particles at time step t; 5:      for each pair $(i,j)$, sample $\eta$,${\eta }_{*}$ 6:       for each pair $(i,j)$, compute the data change $$\begin{array}{cc}\hfill \Delta {\mathbf{x}}_i^t=& ϵP_{{\Delta }_1,{\Delta }_2}({\mathbf{x}}_i^t,{\mathbf{x}}_j^t,c_i^0,c_j^0)({\mathbf{x}}_j^t-{\mathbf{x}}_i^t)+\sqrt{2{\sigma }^{2}D\left(c_i^0\right)}\mathit{\eta }\hfill \\ \hfill \Delta {\mathbf{x}}_j^t=& ϵP_{{\Delta }_1,{\Delta }_2}({\mathbf{x}}_j^t,{\mathbf{x}}_i^t,c_j^0,c_i^0)({\mathbf{x}}_i^t-{\mathbf{x}}_j^t)+\sqrt{2{\sigma }^{2}D\left(c_j^0\right)}{\mathit{\eta }}_{*}\hfill \end{array}$$ (21) compute $${\mathbf{x}}_{i,j}^{t+1}={\mathbf{x}}_{i,j}^t+\Delta {\mathbf{x}}_{i,j}^t$$ (22) 7:end for