## 2 Theoretical Approach
### Numerical Calculation Approach  
Equations (1)-(3) constitute a system of first-order quasilinear partial differential equations. The solution methods have the finite difference method, the finite volume method, and the method of characteristics (MOC). The MOC is a widely used approach in the study of flow in pipes and the propagation of transience [28]. The MOC has good computational accuracy, and introducing Riemann variables makes the computation process simple and convenient. The MOC has numerous applications for pressure waves of high-speed railway tunnels [4; 41]. According to the theory of characteristics, the system of partial differential equations is transformed into characteristic line equations for each characteristic line. The dimensionless form of the characteristic line equation is as follows [4]:  
1. \(\lambda\) characteristic line Characteristic direction: \[\frac{dX}{dZ}=U+A\] (6) Characteristic equation: \[d\lambda=\delta\lambda_{A_{\Delta}}+\delta\lambda_{f}+\delta\lambda_{h}\] (7)  
Figure 4: The one-dimensional and three-dimensional characteristics of the initial compression waves. \(\beta\) characteristic line \[\frac{dX}{dZ}=U-A\] (8) Characteristic equation: \[d\beta=\delta\beta_{A_{\rm A}}+\delta\beta_{f}+\delta\beta_{h}\] (9) \(A_{\rm A}\) characteristic line \[\frac{dX}{dZ}=U\] (10) Characteristic equation: \[dA_{\rm A}=\delta A_{\rm A_{f}}+\delta A_{\rm A_{h}}\] (11)  
where the subscripts \(A_{\rm A}\), \(f\), and \(h\) denote the effect of entropy, friction, and heat transfer, respectively. The specific expressions are given below as follows:  
\[\delta\lambda_{A_{\rm A}}=\delta\beta_{A_{\rm A}}=\frac{\lambda+\beta}{2} \frac{dA_{\rm A}}{A_{\rm A}} \tag{12}\]  
\[\delta\lambda_{f}=\frac{1-\kappa}{2}\bigg{[}1-2\frac{\lambda-\beta}{\lambda+ \beta}\bigg{]}\frac{l_{\rm R}}{a_{\rm R}^{2}}G(u)dZ \tag{13}\]  
\[\delta\beta_{f}=\frac{\kappa-1}{2}\bigg{[}1+2\frac{\lambda-\beta}{\lambda+ \beta}\bigg{]}\frac{l_{\rm R}}{a_{\rm R}^{2}}G(u)dZ \tag{14}\]  
\[d\lambda=\delta\lambda_{A_{\Delta}}+\delta\lambda_{f}+\delta\lambda_{h}\]0  
\[d\lambda=\delta\lambda_{A_{\Delta}}+\delta\lambda_{f}+\delta\lambda_{h}\]1  
\[d\lambda=\delta\lambda_{A_{\Delta}}+\delta\lambda_{f}+\delta\lambda_{h}\]2  
where \(\lambda\), \(\beta\), and \(A_{\rm A}\) are generalized Riemann variables. The MOC equations, \(A\), \(U\), \(X\), and \(Z\) are the dimensionless speed of sound, the dimensionless velocity of airflow, dimensionless length, and dimensionless time, respectively, defined as:  
\[d\lambda=\delta\lambda_{A_{\Delta}}+\delta\lambda_{f}+\delta\lambda_{h}\]3  
where \(a_{\rm R}\) is the reference speed of sound and \(l_{\rm R}\) is the reference length. In the numerical computation process using MOC, the computational grid is first determined, and then linear interpolation is used to solve the flow parameters at the grid points. Since the characteristic line is a curve with a changing slope, it will bring large errors to the numerical calculation of the pressure gradient of the compression waves in the process of propagation evolution due to the accuracy of linear interpolation, so in order to improve the accuracy of the MOC with regard to the linear interpolation of the mesh, it is necessary to improve the MOC [42]. This paper used the modified iterative algorithm as an improved MOC. The determination of the interpolation points in the improved MOC is illustrated in \[d\lambda=\delta\lambda_{A_{\Delta}}+\delta\lambda_{f}+\delta\lambda_{h}\]4 To optimize the accuracy of interpolation, it is necessary to correct the position \(\delta X\) of the interpolation points. Based on the previously determined \(\lambda\), \(\beta\), and \(A_{\rm A}\) values at point \(R\), we recalculated the positions \(\delta X_{\rm Q}\), \(\delta X_{\rm W}\), and \(\delta X_{\rm E}\) of points Q, W, and E, respectively, using the following equations:  
\[\left[ \begin{array}{c}\delta X_{\rm Q}\\ \delta X_{\rm W}\\ \delta X_{\rm E}\end{array} \right]=\frac{1}{2}\left[