## 3 Numerical Examples
### Shock-Tube Problems  
This problem includes [PERSON]'s [48] and [PERSON]'s [49] problem, used to verify the shock wave capture capability. The initial condition for [PERSON]'s problem is:  
\[(\rho,u,p)=\left\{ \begin{array}{ll}(0.445,0.698,3.528),&\text{if}-0.5\leq x<0, \\ (0.5,0,0.5710),&\text{if}\ 0\leq x\leq 0.5,\end{array} \right. \tag{47}\]  
The computational domain spans from \([-0.5,0.5]\) in the x, and the uniform grid space is \(1/201\). Non-reflecting boundary conditions are applied, with the ultimate time set to \(t=0.14\). A fixed-time step size \(DT=1\times 10^{-4}\) is adopted. Exact representation of the theoretical solution. The initial condition for [PERSON]'s problem is  
\[(\rho,u,p)=\left\{ \begin{array}{ll}(1,0,1),&\text{if}\ 0\leq x<0.5,\\ (0.125,0.0,1),&\text{if}\ 0.5\leq x\leq 1.\end{array} \right. \tag{48}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Schemes** & \(\mathbf{h}\) & \(\mathbf{L_{1}}\) **Error** & \(\mathbf{L_{1}}\) **Order** & \(\mathbf{L_{2}}\) **Error** & \(\mathbf{L_{2}}\) **Order** & \(\mathbf{L_{\infty}}\) **Error** & \(\mathbf{L_{\infty}}\) **Order** \\ \hline \multirow{4}{*}{WENO7-LF} & \(1/5\) & \(1.46\times 10^{-3}\) & - & \(1.82\times 10^{-3}\) & - & \(2.99\times 10^{-3}\) & - \\  & \(1/10\) & \(3.34\times 10^{-5}\) & 5.450 & \(3.87\times 10^{-5}\) & 5.557 & \(7.42\times 10^{-5}\) & 5.332 \\  & \(1/20\) & \(2.27\times 10^{-7}\) & 7.202 & \(3.18\times 10^{-7}\) & 6.927 & \(8.15\times 10^{-7}\) & 6.508 \\  & \(1/40\) & \(7.65\times 10^{-10}\) & 8.209 & \(1.03\times 10^{-9}\) & 8.270 & \(2.60\times 10^{-9}\) & 8.294 \\ \hline \multirow{4}{*}{WENO7-RE} & \(1/5\) & \(1.37\times 10^{-3}\) & - & \(1.73\times 10^{-3}\) & - & \(2.82\times 10^{-3}\) & - \\  & \(1/10\) & \(3.15\times 10^{-5}\) & 5.441 & \(3.67\times 10^{-5}\) & 5.554 & \(7.07\times 10^{-5}\) & 5.320 \\ \cline{1-1}  & \(1/20\) & \(2.06\times 10^{-7}\) & 7.261 & \(2.89\times 10^{-7}\) & 6.993 & \(7.37\times 10^{-7}\) & 6.583 \\ \cline{1-1}  & \(1/40\) & \(6.81\times 10^{-10}\) & 8.239 & \(9.11\times 10^{-10}\) & 8.307 & \(2.26\times 10^{-9}\) & 8.348 \\ \hline \multirow{4}{*}{WENO7-LBFS} & \(1/5\) & \(1.21\times 10^{-3}\) & - & \(1.44\times 10^{-3}\) & - & \(2.41\times 10^{-3}\) & - \\  & \(1/10\) & \(2.64\times 10^{-5}\) & 5.525 & \(3.13\times 10^{-5}\) & 5.526 & \(6.12\times 10^{-5}\) & 5.300 \\ \cline{1-1}  & \(1/20\) & \(1.70\times 10^{-7}\) & 7.277 & \(2.37\times 10^{-7}\) & 7.044 & \(6.22\times 10^{-7}\) & 6.620 \\ \cline{1-1}  & \(1/40\) & \(5.43\times 10^{-10}\) & 8.289 & \(7.29\times 10^{-10}\) & 8.348 & \(1.89\times 10^{-9}\) & 8.361 \\ \hline \multirow{4}{*}{WENO7-URLBFS} & \(1/5\) & \(1.27\times 10^{-3}\) & - & \(1.53\times 10^{-3}\) & - & \(2.63\times 10^{-3}\) & - \\  & \(1/10\) & \(2.64\times 10^{-5}\) & 5.588 & \(3.13\times 10^{-5}\) & 5.610 & \(6.12\times 10^{-5}\) & 5.429 \\ \cline{1-1}  & \(1/20\) & \(1.70\times 10^{-7}\) & 7.277 & \(2.37\times 10^{-7}\) & 7.044 & \(6.22\times 10^{-7}\) & 6.620 \\ \cline{1-1}  & \(1/40\) & \(2.72\times 10^{-10}\) & 9.286 & \(3.64\times 10^{-10}\) & 9.349 & \(9.45\times 10^{-10}\) & 9.362 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Accuracy order and error of different flux solvers based on the WENO7 scheme at different grid resolutions. Figure 8: Advection of density perturbation problem. Figure **(a)** shows the accuracy of WENO7-LF, WENO7-ROE, WENO7-LBFS, and WENO7-URLBFS schemes; Figure **(b)** shows the CPU time and \(L_{2}\) error comparison of WENO7-LF, WENO7-ROE, WENO7-LBFS and WENO7-URLBFS schemes. The computational area spans from [0, 1] in the x, with the ultimate time set to \(t=0.25\), and other conditions are the same as the Lax problem. \"-NCST\" in Figure 9 indicates that the WENO reconstruction is carried out in physical space, while the others denote reconstruction in the eigenspace. Figures 9 and 10 compare the numerical and exact solutions of these two problems.