# Simulation of Parameters of Plasma Dynamics of a Magneto Plasma Compressor
## 2 Model of the MPC Discharge and Numerical Solution Method
In order to solve the radiation transfer equations, the conditions for the absence of radiation incident from the outside were set at the boundaries of the computational domain, and a condition for the symmetry of radiation fluxes was set on the axis of symmetry. In order to determine the magnetic field strength \(\mathrm{B}_{\varphi}(\mathrm{r},\ \mathrm{z},\ \mathrm{t})\) at the left boundary of the calculated area \(\mathrm{I}\), the following relations were used: \(\mathrm{B}_{\varphi}=\frac{2\mathrm{I}}{\mathrm{cr}_{1}^{2}}\,\ \mathrm{r}\leq \mathrm{r}_{1}\mathrm{B}_{\varphi}=\frac{2\mathrm{I}}{\mathrm{cr}},\mathrm{r} \in[\mathrm{r}_{1},\ \mathrm{r}_{2}]\); \(\mathrm{B}_{\varphi}=\frac{2\mathrm{I}}{\mathrm{cr}}\frac{\mathrm{r}_{3}^{2} -\mathrm{r}^{2}}{\mathrm{r}_{1}^{2}-\mathrm{r}^{2}}\,\ \mathrm{r}\in[\mathrm{r}_{2},\ \mathrm{r}_{3}]\); \(\mathrm{B}_{\varphi}=0\,\ \mathrm{r}\in[\mathrm{r}_{3},\ \mathrm{r}_{\mathrm{k}}]\). The \"hyperbolic\" (\"convective\") part of the equations is:  
\[\frac{\partial\overset{\rightarrow}{\mathbf{U}}_{i}}{\partial t}+\frac{\overset{ \rightarrow}{\mathbf{F}}\left(\overset{\rightarrow}{\mathbf{U}}_{i+1/2}\right) -\overset{\rightarrow}{\mathbf{F}}\left(\overset{\rightarrow}{\mathbf{U}}_{i- 1/2}\right)}{\Delta_{\xi_{i}}}=\overset{\rightarrow}{\mathbf{F}}_{\text{r}}, \ \Delta_{\xi}=[\xi_{i-1/2}-\xi_{i},\xi_{i+1/2}-\xi_{i}]\]  
The reconstructed function \(\mathrm{Y}(\xi)\), \(\xi\in[-\Delta_{\xi}/2,\Delta_{\xi}/2]\), is:  
\[\mathrm{Y}(\xi)=\mathrm{F}_{\text{i}}^{n}(\xi)=\mathrm{R}(\xi)+\] \[+\mathrm{a}_{i}[\xi-\xi_{i}]^{3}+\mathrm{b}_{i}[\xi-\xi_{i}]^{4}+ \mathrm{c}_{i}[\xi-\xi_{i}]^{5}+\mathrm{d}_{i}[\xi-\xi_{i}]^{6}+\mathrm{e}_{ i}[\xi-\xi_{i}]^{7}+\mathrm{g}_{i}[\xi-\xi_{i}]^{8}+\mathrm{h}_{i}[\xi-\xi_{i}]^{9}\]  
A part of the \"reconstructed\" function at time \(\mathrm{t}^{n}\) is determined by an expression of the form:  
\[\mathrm{R}(\xi)=\] \[=\left\{ \begin{array}{cc}\mathrm{R}_{\text{T}}=\mathrm{Y}_{i}+ \phi(\mathrm{Y}_{i})\left(\frac{\partial\mathrm{Y}}{\partial\xi}\right)_{\text {i}}[\xi-\xi_{i}]+\frac{\phi(\mathrm{Y}_{i})}{2!}\left(\frac{\partial^{2} \mathrm{Y}}{\partial\xi^{2}}\right)_{\text{i}}[\xi-\xi_{i}]^{2},&\ell_{i}\approx 0,&\lambda_{\text{UB}}\xi\in\left[-\frac{\Delta_{i}}{2},\frac{\Delta_{i}}{2} \right],\\ \mathrm{R}_{\text{L-B}}=\mathrm{Y}_{i}+\phi(\mathrm{Y}_{i})\left[\mathrm{f}( \xi-\xi_{i})\mathrm{p}_{1}+\frac{\hat{\rho}(\xi-\xi_{i})}{2!}\mathrm{p}_{2} \right],&\ell_{i}\approx 1,&\lambda_{\text{UB}}\xi\in\left[-\frac{\Delta_{i}}{2}, \frac{\Delta_{i}}{2}\right],\end{array} \right.\]  
These are the first coefficients of the decomposition of the function \(\mathrm{Y}(\xi)\) into a truncated Burman-Lagrange series [32]. Obviously, the accuracy of the approximation (using the main part \(\mathrm{R}(\xi)\)) of the \"reconstructed\" function \(\mathrm{Y}(\xi)\) depends on the order of accuracy (approximation error) of the recovery of the values of the derivatives \(\left(\partial\mathrm{Y}/\partial\xi\right)_{\text{i}}\) and \(\left(\partial^{2}\mathrm{Y}/\partial\xi^{2}\right)_{\text{i}}\).