Chapter 9
A Godunov-Type Method
for a Multi-temperature Plasma
with Strong Shock Waves and a General
Equation of State
Alexey G. Aksenov
Abstract A multi-temperature code for a multi-component gas dynamic is considered. The velocities of components with nonzero mass are assumed to be identical
to each other. The gas dynamic part is the Godunov-type method based on the efficient approximate solution of the Riemann problem operating with all components
of the homogeneous gas mixture. The method assumes the table equation of state
(EOS), but the system of the hydrodynamic equations should be hyperbolic. This
work contains the test of the method on a strong shock wave in hydrogen plasma,
so-called Shafranov’s solution. By taking into account the radiation component,
the chapter discusses the applicability of the two temperature models for the strong
shock wave in the hydrogen with the large temperatures behind a shock wave without
consideration of the radiation at a considered short timescale. General EOS for the
mixture of protons, electrons, and radiation differs from an ideal gas low EOS for
two components (protons and electrons) fully ionized hydrogen plasma.
9.1 Introduction
A multi-component gas of different substances α is described by a set of densities
ρ α (r, t) ≡ c α (r, t)ρ(r, t), where c α are concentrations, and internal energy densities
ρε α (r, t), where ε α is specific energy. All massive particles have identical velocities v(r, t) and temperatures, while the “massless” fast particles from viewpoint of
the total density ρ (electrons, photons) have their own temperatures. The equations
of state are P =
α P(ρ, ε α ), ε α = ε α (ρ, T α ). The components can exchange
energy, can transfer energy by heat conduction not associated with the transfer of the
massive particles, and can participate in reactions. Such problems arise in inertial
thermonuclear fusion [1], laser ablation experiments [2, 3], and astrophysics [4].
This is an intermediate case between the description based on the Boltzmann kinetic
A. G. Aksenov (B)
Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow 123056,
Russian Federation
e-mail: aksenov@icad.org.ru
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_9
115
A Godunov-Type Method
for a Multi-temperature Plasma
with Strong Shock Waves and a General
Equation of State
Alexey G. Aksenov
Abstract A multi-temperature code for a multi-component gas dynamic is considered. The velocities of components with nonzero mass are assumed to be identical
to each other. The gas dynamic part is the Godunov-type method based on the efficient approximate solution of the Riemann problem operating with all components
of the homogeneous gas mixture. The method assumes the table equation of state
(EOS), but the system of the hydrodynamic equations should be hyperbolic. This
work contains the test of the method on a strong shock wave in hydrogen plasma,
so-called Shafranov’s solution. By taking into account the radiation component,
the chapter discusses the applicability of the two temperature models for the strong
shock wave in the hydrogen with the large temperatures behind a shock wave without
consideration of the radiation at a considered short timescale. General EOS for the
mixture of protons, electrons, and radiation differs from an ideal gas low EOS for
two components (protons and electrons) fully ionized hydrogen plasma.
9.1 Introduction
A multi-component gas of different substances α is described by a set of densities
ρ α (r, t) ≡ c α (r, t)ρ(r, t), where c α are concentrations, and internal energy densities
ρε α (r, t), where ε α is specific energy. All massive particles have identical velocities v(r, t) and temperatures, while the “massless” fast particles from viewpoint of
the total density ρ (electrons, photons) have their own temperatures. The equations
of state are P =
α P(ρ, ε α ), ε α = ε α (ρ, T α ). The components can exchange
energy, can transfer energy by heat conduction not associated with the transfer of the
massive particles, and can participate in reactions. Such problems arise in inertial
thermonuclear fusion [1], laser ablation experiments [2, 3], and astrophysics [4].
This is an intermediate case between the description based on the Boltzmann kinetic
A. G. Aksenov (B)
Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow 123056,
Russian Federation
e-mail: aksenov@icad.org.ru
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_9
115