## 2 Results
### Balanced Up-Downwind Semi-Discretized Scheme
#### 2.1.2 Case for \(\rho\in(0,1]\)  
We need the following restrictions on mesh steps, in the case [18]  
\[\rho k_{\max}\leq\sigma h_{\min}\leq\sigma h_{\max}\leq\frac{1}{\rho}k_{\min}. \tag{26}\]  
Apparently, when \(\rho=1\), the above implies that a uniform spatial mesh with \(h=\sigma k\) must be used. Differing from (24), we consider a new, dynamically balanced cross-derivative approximation,  
\[u_{xy}(x_{m},y_{n},\tau)=\frac{1}{2}(\Delta_{x,-}\Delta_{y,-}+\Delta_{x,+} \Delta_{y,+})u_{m,n}(\tau)+\mathcal{O}(h_{\max}+k_{\max}). \tag{27}\]  
Computational stencils for (24) and (27) are shown in Figure 2.  
In this circumstance, we obtain the following new system  
\[w^{\prime}(\tau)=\tilde{A}w(\tau)+\tilde{f}(\tau), \tag{28}\]  
where \(w,\tilde{f}(\tau)\in\mathbb{R}^{MN}\) and \(\tilde{A}\in\mathbb{R}^{MN\times MN}\) is block tridiagonal; that is,  
\[\tilde{A} = \begin{bmatrix}\tilde{D}_{1}&\tilde{Q}_{1}&\cdots&\cdots&\cdots&0 \\ \tilde{P}_{2}&\tilde{D}_{2}&\tilde{Q}_{2}&\cdots&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\cdots&\vdots\\ \vdots&\cdots&\tilde{P}_{M-2}&\tilde{D}_{M-2}&\tilde{Q}_{M-2}&0\\ \cdots&\cdots&\cdots&\tilde{P}_{M-1}&\tilde{D}_{M-1}&\tilde{Q}_{M-1}\\ 0&\cdots&\cdots&\cdots&\tilde{P}_{M}&\tilde{D}_{M}\end{bmatrix} .\]  
Nontrivial entries of \(\tilde{P}_{m},\ \tilde{D}_{m},\) and \(\tilde{Q}_{m},\) within their respective ranges of \(m\), are given by  
Figure 2: Computational stencils of (24) (**left**) and (27) (**right**). \[\hat{p}_{n,n-1}^{(m)} = \frac{\rho\sigma y_{n}}{2h_{m}k_{n}},\ \ \ y_{n}>k_{1};\] \[\hat{p}_{n,n}^{(m)} = \left\{ \begin{aligned} &\frac{y_{1}}{h_{m}(h_{m}+h_{m+1})},\ \ \ y_{n}=k_{1},\\ &\frac{y_{n}}{h_{m}(h_{m}+h_{m+1})}-\frac{\rho\sigma y_{n}}{2h_{ m}k_{n}}+\frac{y_{n}-2r}{2h_{m}k_{n}},\ \ 2r<y_{n}\leq Y-k_{N+1};\\ &\hat{r}_{n,n-1}^{(m)}=\end{aligned} \right.\] \[\hat{r}_{n,n-1}^{(m)} = \left\{ \begin{aligned} &\frac{\sigma^{2}y_{n}}{k_{n}(k_{n}+k_{n+1})} -\frac{\rho\sigma y_{n}}{2h_{m}k_{n}},\ \ k_{1}<y_{n}\leq\eta,\\ &\frac{\sigma^{2}y_{n}}{k_{n}(k_{n}+k_{n+1})}-\frac{\rho\sigma y _{n}}{2h_{m}k_{n}}-\frac{\kappa(\eta-y_{n})}{k_{n}},\ \ \eta<y_{n}\leq Y-k_{N+1};\\ &\hat{r}_{n,n}^{(m)}=\end{aligned} \right.\] \[\hat{r}_{n,n}^{(m)} = \left\{ \begin{aligned} &\tilde{\alpha}_{m,1}+\frac{y_{1}-2r}{2h_{m+1} }-\frac{\kappa(\eta-y_{1})}{k_{n+1}},\ \ \ y_{1}=k_{1},\\ &\tilde{\beta}_{m,n}+\frac{y_{n}-2r}{2h_{m+1}}-\frac{\kappa(\eta -y_{n})}{k_{n+1}},\ \ k_{1}<y_{n}\leq 2r,\\ &\tilde{\beta}_{m,n}-\frac{y_{n}-2r}{2h_{m}}-\frac{\kappa(\eta -y_{n})}{k_{n+1}},\ \ \ 2r<y_{n}\leq\eta,\\ &\tilde{\beta}_{m,n}-\frac{y_{n}-2r}{2h_{m}}+\frac{\kappa(\eta -y_{n})}{k_{n}},\ \ \eta<y_{n}<Y-k_{N+1},\\ &\tilde{\gamma}_{m,N}-\frac{y_{N}-2r}{2h_{m}}+\frac{\kappa(\eta -y_{N})}{k_{N}},\ \ \ y_{N}=Y-k_{N+1};\\ \end{aligned} \right.\] \[\hat{r}_{n,n+1}^{(m)} = \left\{ \begin{aligned} &\frac{\sigma^{2}y_{n}}{k_{n+1}(k_{n}+k_{n+1})} -\frac{\rho\sigma y_{n}}{2h_{m+1}k_{n+1}}+\frac{\kappa(\eta-y_{n})}{k_{n+1}}, \ \ 0<y_{n}\leq\eta,\\ &\frac{\sigma^{2}y_{n}}{k_{n+1}(k_{n}+k_{n+1})}-\frac{\rho\sigma y _{n}}{2h_{m+1}k_{n+1}},\ \ \ \eta<y_{n}<Y-k_{N+1};\\ \end{aligned} \right.\] \[\hat{q}_{n,n}^{(m)} = \left\{ \begin{aligned} &\frac{y_{n}}{h_{m+1}(h_{m}+h_{m+1})} -\frac{\rho\sigma y_{n}}{2h_{m+1}k_{n+1}}-\frac{y_{n}-2r}{2h_{m+1}},\ \ 0<y_{n}\leq 2r,\\ &\frac{y_{n}}{h_{m+1}(h_{m}+h_{m+1})}-\frac{\rho\sigma y_{n}}{2h _{m+1}k_{n+1}},\ \ \ 2r<y_{n}<Y-k_{N+1},\\ &\frac{y_{N}}{h_{m+1}(h_{m}+h_{m+1})},\ \ \ y_{N}=Y-k_{N+1}; \\ \end{aligned}