116
A. G. Aksenov
equations for distribution functions f α (r, p, t) and classical single-component gas
dynamics. Taking into account possible large opacities for the fast particles, one
should consider the gas for the mixture without transfer to the separate description
of the gas dynamics of the matter and “massless” fast particles. The joined treatment
allows to integrate the gas dynamic transport with the maximal possible time steps
in the explicit scheme even at the optically thick cases.
Efficient Riemann problem solvers for such hydrodynamic equations are
constructed, see, e.g., for some special case of EOS [5–7] and general EOS [8]. Below
an original method based on the Riemann problem solver for the multi-temperature
nonequilibrium gas [9–11] is briefly described. This method operates with homogenous mixture of components of the matter on the fixed Eulerian grid to carry out
the deep phase of the development of the hydrodynamic instabilities. To improve
the spatial resolution, it uses the reconstruction of the contact discontinuities on
fixed grid as an original method [12]. The method was applied within the plasma
physics for the inertial heavy-ion fusion [1, 13] and is useful in astrophysical tasks
with hydrodynamic and the radiation transfer [14, 15]. In the local model for EOS
proposed, it is assumed that the entropy variations in neighboring mesh cells are small
at the evaluation of the dimensionless coefficients EOS from the pressure jump across
the discontinuity. In the case of an arbitrarily large pressure jump, the model yields
physically reasonable results. In real cases, the pressure jumps are not small on the
surfaces of nearest cells of the computational grid without viscosity.
The simplest multi-temperature shock wave (SW) structure in plasma was considered in [16]. This solution is a suitable test for the method. This work prevents
disadvantages of the strong SW test from the viewpoint of the physics in the last
publications [9–11]. To provide the correct simple physical description, one starts
from the heated ionized hydrogen plasma in the initial state. Thus, the temperature
after the strong SW becomes huge enough for the important role of the disregarded
radiation. Do the obtained mathematical results for strong SW contain physical applications? It is possible to give the answer on the base of qualitative estimates. Also by
means of introducing nonequilibrium radiation into the developed code, it is possible
to give the qualitative answer. EOS for the mixture of protons, electrons, and radiation is not ideal gas low in comparison with EOS for protons and electrons in fully
ionized hydrogen plasma. The introducing of radiation also illustrates an application
of the method to the general EOS. General EOS can contain domains with a negative
square of the sound speed c
2
≡ (dP/dρ) s at phase transitions. In these domains, the
gas enthalpy should be corrected to provide the nonnegative sound speed square.
The chapter is organized as follows. Section 9.2 provides a formulation of the
problem and the numerical method. Shock wave structure in hydrogen plasma is
discussed in Sect. 9.3. Discussion about radiation effects is given in Sect. 9.4.
Section 9.5 concludes the chapter.
A. G. Aksenov
equations for distribution functions f α (r, p, t) and classical single-component gas
dynamics. Taking into account possible large opacities for the fast particles, one
should consider the gas for the mixture without transfer to the separate description
of the gas dynamics of the matter and “massless” fast particles. The joined treatment
allows to integrate the gas dynamic transport with the maximal possible time steps
in the explicit scheme even at the optically thick cases.
Efficient Riemann problem solvers for such hydrodynamic equations are
constructed, see, e.g., for some special case of EOS [5–7] and general EOS [8]. Below
an original method based on the Riemann problem solver for the multi-temperature
nonequilibrium gas [9–11] is briefly described. This method operates with homogenous mixture of components of the matter on the fixed Eulerian grid to carry out
the deep phase of the development of the hydrodynamic instabilities. To improve
the spatial resolution, it uses the reconstruction of the contact discontinuities on
fixed grid as an original method [12]. The method was applied within the plasma
physics for the inertial heavy-ion fusion [1, 13] and is useful in astrophysical tasks
with hydrodynamic and the radiation transfer [14, 15]. In the local model for EOS
proposed, it is assumed that the entropy variations in neighboring mesh cells are small
at the evaluation of the dimensionless coefficients EOS from the pressure jump across
the discontinuity. In the case of an arbitrarily large pressure jump, the model yields
physically reasonable results. In real cases, the pressure jumps are not small on the
surfaces of nearest cells of the computational grid without viscosity.
The simplest multi-temperature shock wave (SW) structure in plasma was considered in [16]. This solution is a suitable test for the method. This work prevents
disadvantages of the strong SW test from the viewpoint of the physics in the last
publications [9–11]. To provide the correct simple physical description, one starts
from the heated ionized hydrogen plasma in the initial state. Thus, the temperature
after the strong SW becomes huge enough for the important role of the disregarded
radiation. Do the obtained mathematical results for strong SW contain physical applications? It is possible to give the answer on the base of qualitative estimates. Also by
means of introducing nonequilibrium radiation into the developed code, it is possible
to give the qualitative answer. EOS for the mixture of protons, electrons, and radiation is not ideal gas low in comparison with EOS for protons and electrons in fully
ionized hydrogen plasma. The introducing of radiation also illustrates an application
of the method to the general EOS. General EOS can contain domains with a negative
square of the sound speed c
2
≡ (dP/dρ) s at phase transitions. In these domains, the
gas enthalpy should be corrected to provide the nonnegative sound speed square.
The chapter is organized as follows. Section 9.2 provides a formulation of the
problem and the numerical method. Shock wave structure in hydrogen plasma is
discussed in Sect. 9.3. Discussion about radiation effects is given in Sect. 9.4.
Section 9.5 concludes the chapter.