9 A Godunov-Type Method for a Multi-temperature Plasma …
117
9.2 Formulation of the Problem and the Numerical Method
The system in the fixed Euler coordinates is: the mass transfer equations for the
components
∂ρ α
∂t
+ divρ α v = ρ ˙
c α ,
(9.1)
the momentum conservation law
∂ρv
∂t
+ Div = 0,
(9.2)
and the energy density equations
∂ρ E α
∂t
+ div(ρ E α + P α )v + v(c α grad P − grad P α ) = div(κgrad T α ) + ρ Q α ,
(9.3)
where the energy densities E α = ε α + c α v
2
/2, the tensor Π i j = ρv i v j + Pδ i j , and
the equation of state P =
α P α (ρ, c, ε α ) with specific energies ε α (ρ, c, T α ). The
kinetic coefficients ˙
c α , κ α , and Q α depend on ρ, c, and T. The problem is computed
by applying the splitting on the physical processes and the dimensional splitting. The
heat conduction equations are solved using central difference approximations. As a
result, the system of partial differential equations (PDEs) is reduced to the ordinary
differential equations (ODEs) system for ˙
ε α,i . ODEs system is solved by applying
the implicit Gear’s method [17]. To describe the kinetics of reactions, ODEs system
for ˙
ρ α and ˙
ε α is solved in each grid cell also by Gear’s method. The transport of
“massless” particles in both transparent and opaque cases can be described in the
frame of diffusion with flux limiters.
The system in Euler variables involves the term v(c α grad P − grad P α ), which
is different from the divergence of a flux. This term requires a special treatment at
discontinuities. In the classical problem of SW in a hydrogen plasma, one has only
three conservation laws and, due to heat conduction, a piecewise-smooth temperature,
for which a differential equation can be used instead of a conservation law.
The hydrodynamic part of the code is based on a high-order explicit Godunov
scheme for single-temperature single-component gas dynamics [9, 12]. A local model
for EOS simplifies the solution of the Riemann problem and makes it possible to
obtain fluxes and partial pressures of the components in any flow region with discontinuities. Following [18], such model for a multi-component gas is constructed so as
it holds strictly in the case of weak discontinuities. The increment of the specific
entropy s across SW is a quantity of the third order of smallness with respect
to the pressure jump: O([P]
3
). Neglecting the entropy variation behind SW, one
computes the dimensionless coefficients γ α ≡ P α τ/ε α + 1 as functions of the state
ahead of SW and the total pressure behind SW. The local model for EOS is used
117
9.2 Formulation of the Problem and the Numerical Method
The system in the fixed Euler coordinates is: the mass transfer equations for the
components
∂ρ α
∂t
+ divρ α v = ρ ˙
c α ,
(9.1)
the momentum conservation law
∂ρv
∂t
+ Div = 0,
(9.2)
and the energy density equations
∂ρ E α
∂t
+ div(ρ E α + P α )v + v(c α grad P − grad P α ) = div(κgrad T α ) + ρ Q α ,
(9.3)
where the energy densities E α = ε α + c α v
2
/2, the tensor Π i j = ρv i v j + Pδ i j , and
the equation of state P =
α P α (ρ, c, ε α ) with specific energies ε α (ρ, c, T α ). The
kinetic coefficients ˙
c α , κ α , and Q α depend on ρ, c, and T. The problem is computed
by applying the splitting on the physical processes and the dimensional splitting. The
heat conduction equations are solved using central difference approximations. As a
result, the system of partial differential equations (PDEs) is reduced to the ordinary
differential equations (ODEs) system for ˙
ε α,i . ODEs system is solved by applying
the implicit Gear’s method [17]. To describe the kinetics of reactions, ODEs system
for ˙
ρ α and ˙
ε α is solved in each grid cell also by Gear’s method. The transport of
“massless” particles in both transparent and opaque cases can be described in the
frame of diffusion with flux limiters.
The system in Euler variables involves the term v(c α grad P − grad P α ), which
is different from the divergence of a flux. This term requires a special treatment at
discontinuities. In the classical problem of SW in a hydrogen plasma, one has only
three conservation laws and, due to heat conduction, a piecewise-smooth temperature,
for which a differential equation can be used instead of a conservation law.
The hydrodynamic part of the code is based on a high-order explicit Godunov
scheme for single-temperature single-component gas dynamics [9, 12]. A local model
for EOS simplifies the solution of the Riemann problem and makes it possible to
obtain fluxes and partial pressures of the components in any flow region with discontinuities. Following [18], such model for a multi-component gas is constructed so as
it holds strictly in the case of weak discontinuities. The increment of the specific
entropy s across SW is a quantity of the third order of smallness with respect
to the pressure jump: O([P]
3
). Neglecting the entropy variation behind SW, one
computes the dimensionless coefficients γ α ≡ P α τ/ε α + 1 as functions of the state
ahead of SW and the total pressure behind SW. The local model for EOS is used