## 3 Mathematical Modeling  
It follows from what has been said that the mathematical model of the MPC discharge can be constructed on the basis of a two-dimensional (axis-symmetric), unsteady system of equations of viscous, one-temperature radiation, plasma dynamics. Electromagnetic processes are described by the system of Maxwell and Ohm equations in a plasma with finite conductivity. Radiation transfer is calculated within the multigroup diffusion approximation. The system is supplemented by equations, which describe the heating and evaporation (within the framework of the model with a Knudsen layer) of the surfaces of the structural elements of the electrode assembly. The system is closed by the equations of the electric circuit of the discharge. This system of equations takes into account the methods for calculating the equations of the state of matter, the absorption coefficients  
and the spatial \(\overline{\xi},\overline{\eta}\) and temporal \(\overline{t}\) variables, respectively, to the characteristic size \(\overline{t}\) and characteristic time \(t_{*}\). We introduce the following notation for the dimensionless variables:  
\[t=U_{*}\overline{t}/L_{*},\;\xi=\overline{\xi}/L_{*},\;\eta=\overline{\eta}/L_{ *},\;V_{\xi}=\overline{V}/V_{*},\;V_{\eta}=\overline{V}_{1}/V_{*},\;u= \overline{u}/V_{*},\;v=\overline{v}/V_{*},\]  
\[T=\overline{T}/T_{*},e=\overline{e}/e_{*},\;\rho=\overline{\rho}/\rho_{*},\;P= \overline{P}/P_{*},\;\mu_{\overline{\xi}}=\overline{\mu}_{\overline{\xi}}/\mu_ {*},\;\lambda_{\overline{\xi}}=\overline{\lambda}_{\overline{\xi}}/\lambda_{*},\]  
\[\gamma=C_{p*}/C_{v*},\;q=\overline{q}/q_{*},\omega=\overline{\omega}/\omega_{*}.\]  
To determine the spatial-temporal position of the contact boundary separating the plasma of the ablating material of the obstacle from the plasma of the environment, the method of fictitious impurity is used. For this, an additional equation is introduced into the system of the above equations \(\left(\rho_{g}\in[0,1]\right)\): \(\partial\rho_{g}/\partial t+\overset{\rightarrow}{V}\
abla\rho_{g}=0\). If \(\rho_{g}\) is known, one can determine the interface \(\Gamma(t)=\{(\xi,\eta):\;\rho_{g}(\xi,\eta,t)=0\}\), which divides the entire computational domain into two areas, each of which corresponds to the ablating material of the obstacle or the plasma of the environment, depending on the sign of the function. A complete description of the method for calculating the contact boundary and gasdynamic parameters near it is given in [15]. The research assumes a coaxial character of the MPC-discharge electrode system. In this case, in the axially symmetric discharge plasma formed above the surface of the end of the system, an electric field with an intensity \(\overset{\rightarrow}{E}(\xi,\eta,\;t)=(E_{\text{r}},\;E_{z},\;0)\) is created and a current flows with a density \(\overset{\rightarrow}{j}(\xi,\eta,\;t)=(j,\;j,\;0)\), providing energy release in the plasma with a specific (per unit volume) power \(\overset{\rightarrow}{j}\overset{\rightarrow}{E}\). The current flowing through the plasma is a source of its own magnetic field with an intensity \(\overset{\rightarrow}{B}\) having only an azimuthal component \(B_{\varphi}(k,\eta,\;t)\). Plasma dynamics processes occurring in the capillary discharge plume are considered under the assumption that the electromagnetic field has axial symmetry and one component \(B_{\varphi}\). \[