## 2 Materials and Methods
### Study of the Approximation and Stability of a Family of Difference Schemes
\]  
It is clear that the terms containing \(\left(\frac{\partial p}{\partial t}\right)_{i}^{n+\frac{1}{2}}\)_and_\(\left(\frac{\partial Q}{\partial t}\right)_{i}^{n+\frac{1}{2}}\) cancel each other out and we obtain the following equations:\[\psi_{1}=0.5\tau(2\sigma-1)\bigg{(}\frac{\partial^{2}Q}{\partial x\partial t} \bigg{)}_{i}^{n+\frac{1}{2}}-\frac{\sigma h}{2}\cdot\bigg{(}\frac{\partial^{2}Q }{\partial x^{2}}\bigg{)}_{i}^{n+1}-\frac{(1-\sigma)h}{2}\cdot\bigg{(}\frac{ \partial^{2}Q}{\partial x^{2}}\bigg{)}_{i}^{n}+O\Big{(}\tau^{2}+h^{2}\Big{)}, \tag{70}\]  
\[\psi_{2}=0.5\tau(2\sigma-1)\bigg{(}\frac{\partial^{2}P}{\partial x\partial t} \bigg{)}_{i}^{n+\frac{1}{2}}-\frac{\sigma h}{2}\cdot\bigg{(}\frac{\partial^{2} P}{\partial x^{2}}\bigg{)}_{i}^{n+1}-\frac{(1-\sigma)h}{2}\cdot\bigg{(}\frac{ \partial^{2}P}{\partial x^{2}}\bigg{)}_{i}^{n}+ \tag{71}\]  
Let us expand the second derivatives in a Taylor series with respect to t in the vicinity of the point \(t=t_{n+\frac{1}{2}}\):  
\[ \begin{array}{c}\bigg{(}\frac{\partial^{2}Q}{\partial x^{2}}\bigg{)}_{i}^{n +1}=\bigg{(}\frac{\partial^{2}Q}{\partial x^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}+ O(\tau),\\ \bigg{(}\frac{\partial^{2}Q}{\partial x^{2}}\bigg{)}_{i}^{n}=\bigg{(}\frac{ \partial^{2}Q}{\partial x^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}+O(\tau),\\ \bigg{(}\frac{\partial^{2}P}{\partial x^{2}}\bigg{)}_{i}^{n+1}=\bigg{(}\frac{ \partial^{2}P}{\partial x^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}+O(\tau),\\ \bigg{(}\frac{\partial^{2}P}{\partial x^{2}}\bigg{)}_{i}^{n}=\bigg{(}\frac{ \partial^{2}P}{\partial x^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}+O(\tau).\end{array}  \tag{72}\]  
Suppose that \(Q_{i}^{n+1}=Q_{i}^{n+\frac{1}{2}}+O\big{(}\tau^{2}\big{)}\). Taking into account (72) from (70) and (71), we obtain the following equations:  
\[\psi_{1}=0.5\tau(2\sigma-1)\bigg{(}\frac{\partial^{2}Q}{\partial x\partial t} \bigg{)}_{i}^{n+\frac{1}{2}}-0.5h\cdot\bigg{(}\frac{\partial^{2}Q}{\partial x ^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}+O\Big{(}\tau^{2}+h^{2}\Big{)},\]  
\[\psi_{2}=0.5\tau(2\sigma-1)\bigg{(}\frac{\partial^{2}P}{\partial x\partial t} \bigg{)}_{i}^{n+\frac{1}{2}}-0.5h\cdot\bigg{(}\frac{\partial^{2}P}{\partial x ^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}+O\Big{(}\tau^{2}+h^{2}\Big{)}.\]  
From the system of Equations (34) and (35), we obtain the following equation:  
\[\frac{\partial^{2}Q}{\partial x^{2}}=-\frac{\partial^{2}P}{\partial t\partial x },\ \frac{\partial^{2}Q}{\partial x\partial t}=-\frac{\partial^{2}P}{\partial x^{2} }-2a\frac{\partial Q}{\partial x}.\]  
Consider the following equations:  
\[\psi_{1}+\psi_{2}=0.5\tau(2\sigma-1)\bigg{(}\frac{\partial^{2}Q}{\partial x \partial t}\bigg{)}_{i}^{n+\frac{1}{2}}-0.5h\cdot\bigg{(}\frac{\partial^{2}P}{ \partial x^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}+\]  
As well as the following equations:  
\[\bigg{(}\frac{\partial^{2}P}{\partial x^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}=- \bigg{(}\frac{\partial^{2}Q}{\partial x\partial t}\bigg{)}_{i}^{n+\frac{1}{2}} -2a\bigg{(}\frac{\partial Q}{\partial x}\bigg{)}_{i}^{n+\frac{1}{2}},\bigg{(} \frac{\partial^{2}Q}{\partial x^{2}}\bigg{)}_{i}^{n+\frac{1}{2}}=-\bigg{(} \frac{\partial^{2}P}{\partial x\partial t}\bigg{)}_{i}^{n+\frac{1}{2}}\]  
We obtain the following equations:  
\[\psi=\psi_{1}+\psi_{2}=0.5\tau(2\sigma\tau-\tau+h)\cdot\bigg{(}\frac{ \partial^{2}Q}{\partial x\partial t}\bigg{)}_{i}^{n+\frac{1}{2}}-ah\cdot \bigg{(}\frac{\partial Q}{\partial x}\bigg{)}_{i}^{n+\frac{1}{2}}+\]  
\[+0.5\tau(2\sigma\tau-\tau+h)\cdot\bigg{(}\frac{\partial^{2}P}{\partial x \partial t}\bigg{)}_{i}^{n+\frac{1}{2}}+O\big{(}\tau^{2}+h^{2}\big{)}.\]It can be seen that the scheme with weights has a second-order approximation. \[\psi=O\Big{(}\tau^{2}+h^{2}\Big{)},\]  
if the following equality holds:  
\[\sigma=\frac{1}{2}-\frac{h}{2\tau}=\sigma_{0}, \tag{73}\]  
and \(\sigma\
eq\sigma_{0}\)--the first order, \(\psi=O(\tau+h)\). Stability is based on the initial data.