# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 2 Numerical Method
### Gas-Kinetic BGK Scheme for the Time-Dependent Solution \(f\)
The coefficients \(a_{\alpha}\) and \(b_{\alpha}\) can be determined from the derivatives of Equation (8) in space, i.e.,  
\[ \begin{array}{ll}\int g_{0}a_{\alpha}\varphi_{\alpha}\varphi d\Xi=&\Big{(} \frac{\partial\mathbf{W}}{\partial x}\Big{)}_{i+1/2,j}\\ \int g_{0}b_{\alpha}\varphi_{\alpha}\varphi d\Xi=&\Big{(}\frac{\partial\mathbf{W} }{\partial y}\Big{)}_{i+1/2,j}\end{array} \,. \tag{17}\]  
The gradients of \(\mathbf{W}\) at the cell interface are calculated by applying the Green-Gaussian theorem to the auxiliary control volume; see Section 4.2 of [22] for the detailed implementation. As for the coefficients \(A_{\alpha}\), due to the fact that the non-equilibrium parts in \(f_{0}\) (terms involving \(\tau\)) have no direct contributions to the conservative variables, they can be determined from the following relation:  
\[\int g_{0}A_{\alpha}\varphi_{\alpha}\varphi d\Xi=-\int g_{0}(a_{\alpha}\varphi _{\alpha}\xi^{\prime}_{\ x}+b_{\alpha}\varphi_{\alpha}\xi^{\prime}_{\ y}) \varphi d\Xi+\omega\int s_{0}\varphi d\Xi, \tag{18}\]  
where the terms on the right side have been already calculated. For a smooth flow, the equilibrium state \(g\) around (\(\mathbf{x}_{i+1/2,j}=(0,0)\), \(t=0\)) is obtained from a second-order expansion in space and time:  
\[g(\mathbf{x}(t^{\prime}),t^{\prime},\xi,\zeta^{2})=g_{0}(1+At^{\prime}+ax+by)=g_{ 0}[1+At^{\prime}-a\xi^{\prime\prime}_{\ x}(t-t^{\prime})-b\xi^{\prime\prime }_{\ y}(t-t^{\prime})]. \tag{19}\]  
Like the equilibrium state \(g\), the source term \(s\) is assumed to have the form  
\[ \begin{array}{ll}s(\mathbf{x}(t^{\prime}),t^{\prime},\xi,\zeta^{2})&=s_{10}[1+A _{s1}t^{\prime}-a_{s1}\xi^{\prime}_{\ x}(t-t^{\prime})-b_{s1}\xi^{\prime}_{\ y}(t-t^{ \prime})]\\ &-s_{20}[1+A_{s2}t^{\prime}-a_{s2}\xi^{\prime}_{\ x}(t-t^{\prime})-b_{s2}\xi^ {\prime}_{\ y}(t-t^{\prime})]\end{array} \,, \tag{20}\]where \(s_{10}\) and \(s_{20}\) are defined at the cell interface and at time \(t=0\). They can be calculated using Equations (9) and (15), and we have \(s_{0}=s_{10}-s_{20}\). Then all the slopes in \(s\) are expanded as follows:  
\[ \begin{array}{ll}a_{s1}=a_{s1,A}\varphi_{a},&a_{s2}=a_{s2,A}\varphi_{a}\\ b_{s1}=b_{s1,A}\varphi_{a},&b_{s2}=b_{s2,A}\varphi_{a}\\ A_{s1}=A_{s1,A}\varphi_{a},&A_{s2}=A_{s2,A}\varphi_{a}\end{array} \quad. \tag{21}\]  
The unknown coefficients in the above relations are obtained by comparing the Taylor expansions of Equations (4) and (9), e.g., for \(a_{s1}\),  
\[ \begin{array}{ll}a_{s1,A}=a_{4}\\ a_{s1,2}=-0.5a_{3}\\ a_{s1,3}=0.5a_{2}\\ a_{s1,1}=a_{1}+a_{2}u+a_{3}v+0.5a_{4}(u^{2}+v^{2})-a_{s1,2}u_{s1}-a_{s1,3}v_{s 1}-0.5a_{s1,A}(u_{s1}^{2}+v_{s1}^{2})\end{array} \quad, \tag{22}\]  
and for \(a_{s2}\),  
\[ \begin{array}{ll}a_{s2,4}=a_{s1,A}\\ a_{s2,2}=-a_{s1,2}\\ a_{s2,3}=a_{s1,3}\\ a_{s2,1}=a_{s1,1}\end{array} \quad. \tag{23}\]  
Since \(a_{a}(\alpha=1\ \sim 4)\) have already been gained, \(a_{s1,A}\) and \(a_{s2,A}\) can be calculated with little computational effort. The other coefficients \(b_{s1,a}\), \(b_{s2,a}\), \(A_{s1,A}\) and \(A_{s2,A}\) have the same forms as Equations (22) and (23). The reason we assume that \(s\) is expanded in the form of Equation (20) can be explained below. From a mathematical point of view, it is constructed in such a way as to consider the time and space variations around the cell interface. Therefore, the constructed \(s\) at the cell interface has a second-order accuracy in both space and time, which is consistent with \(f\) and \(g\). Because \(s\) is entirely determined by the time-dependent variables \(\rho,p,u,v\) as shown in Equations (9) and (20), and there is a one-to-one correspondence between conservative variables and \(f\), Equation (20) is suitable for both stationary and unsteady solutions. From a physical point of view, we know that the key point of the BGK scheme is to accurately evaluate \(f\) at the cell interface.