Numerical methods for FCI B. Despr es+ Part IV X. Blanc - PowerPoint PPT Presentation

Numerical methods for FCI B. Despr´ es+ Part IV X. Blanc LJLL-Paris VI+CEA Multi-temperature fluid models Thanks to same Hele-Shaw models collegues as before plus C. Buet, H. Egly and R. Sentis B. Despr´ es+ X. Blanc LJLL-Paris VI+CEA Thanks to same collegues as before plus C. Buet, H. Egly and R. Sentis Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 1 / 27

FCI scenario Introduction Numerical discretization of T i − T e model Coupling with radiation Hele-Shaw models During the implosion, pure hydrodynamics is a very strong hypothesis. It is much more relevant to consider multi-temperature models. Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 2 / 27

Plan Introduction Numerical discretization of T i − T e model Coupling with One temperature for ions T i and one temperature for electrons T e : T i − T e model radiation Basic considerations One temperature for the matter, and one temperature for radiatiopn T r Hele-Shaw models Hele-Shaw model for the stability of the ablation front Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 3 / 27

The simplified T i − T e model Starting point is ( page 9 and page 23 of the notes ) 8 D t ρ + ρ ∇ · u = 0 , > > > ρ D t u + ∇ p = F r , Introduction > < ρ D t ε e + p e ∇ · . u − ∇ · ( χ e ∇ T e ) + W ei = Q r + S , > Numerical > > > : ρ D t ε i + p i ∇ · u + ∇ · ( χ i ∇ T i ) − W ei = 0 , discretization of T i − T e model The unknowns of this system are the density ρ ( x , t ) ∈ R + of the plasma, its velocity u ( x , t ) ∈ R 3 and its pressure p ( x , t ) ∈ R . We also have electronic and ionic values : pressures p e ( x , t ) , p i ( x , t ) ∈ R (with Coupling with p = p e + p i ), energies ε e ( x , t ) , ε i ( x , t ) ∈ R (with E = E e + E i ), and temperatures T e ( x , t ) , T i ( x , t ). radiation Here, The terms F r and Q r are the radiative sources, and S is an additional source term modelling the laser energy drop. Hele-Shaw This set of equations is closed by an adapted equation of state models ( ε e , p e , ε i , p i ) = F ( ρ, T e , T i ) . We will assume that the fluid is described by a perfect gas EOS p i = ( γ i ) ρ C vi T i = ( γ i ) ρε i , ε i = C vi T i and the electronic part is described by a perfect gas EOS p e = ( γ e ) ρ C ve T e = ( γ e ) ρε e , ε e = C ve T e . Since electrons are monoatomic 5 γ e = . 3 Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 4 / 27

Hydrodynamics of the T i − T e model The hydrodynamic part is 8 D t ρ + ρ ∇ · u = 0 , > > > > ρ D t u + ∇ p = 0 , < Introduction ρ D t ε e + p e ∇ · . u = 0 , > > > > ρ D t ε i + p i ∇ · u = 0 , : Numerical discretization This system is non conservative. For discontinuous functions a and b of T i − T e model the product a ∂ x b is not defined . Coupling with radiation What is a shock in such a system ? We need to transform it into a conservative system of conservation laws. Hele-Shaw models Universal principles : mass is preserved, momentum is preserved ad total energy is preserved. We get after convenient manipulations the correct conservative formulation (in 1D) 8 ∂ t ρ + ∂ x ( ρ u ) = 0 , > < “ ρ u 2 + p ” ∂ t ( ρ u ) + ∂ x = 0 , > ∂ t ( ρ e ) + ∂ x ( ρ ue + pu ) = 0 , : with 1 u 2 . p = p i + p e and e = ε i + ε e + 2 Question : Is there a fourth conservation laws ? Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 5 / 27

Conservation of the electronic entropy Convenient manipulations show that smooth solutions satisfy Introduction Numerical ∂ t ( ρ ( α S i + β S e )) + ∂ x ( ρ u ( α S i + β S e )) = 0 , ∀ α, β. discretization of T i − T e For discontinuous solutions, these relations are not equivalent. model The correct choice is α = 0 and β = 1. This is called the Born-Oppenheimer hypothesis. Coupling with It is related to the fact that me mi is small. radiation Zeldovith-Raizer, Cordier (PhD thesis 96), Degond-Luquin, Massot, . . . Hele-Shaw models Finally ∂ t ρ + ∂ x ( ρ u ) = 0 , 8 > “ ρ u 2 + p ” > ∂ t ( ρ u ) + ∂ x = 0 , < ∂ t ( ρ S e ) + ∂ x ( ρ uS e ) = 0 , > > : ∂ t ( ρ e ) + ∂ x ( ρ ue + pu ) = 0 . The mathematical entropy law writes ∂ t ( ρ S i ) + ∂ x ( ρ uS i ) ≥ 0 . Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 6 / 27

Shock relations As a consequence the ionic entropy increases at shocks while the electronic entropy is constant as shocks S + > S − S + = S − Introduction , e . i e i Numerical Exercise : prove it. discretization of T i − T e Solution model 8 − σ ( ρ R − ρ L ) + ( ρ R u R − ρ L u L ) = 0 , Coupling with < ` ´ ` ´ − σ ρ R S e , R − ρ L S e , L + ρ R u R S e , R − ρ L u L S e , L = 0 , radiation − σ ` ρ R S i , R − ρ L S i , L ´ + ` ρ R u R S i , R − ρ L u L S i , L ´ > 0 . : Hele-Shaw models So ρ R ( u R − σ ) = ρ L ( u L − σ ). This is the constant mass flux D = ρ R ( u R − σ ) = ρ L ( u L − σ ). Therefore DS e , R = DS e , L and DS i , R > DS i , L . Assume a shock and the mass flux is positive D > 0. Then S e , R = S e , L and S i , R > S i , L . CQFD This behavior is absolutely fundamental : it explains that ions and electrons behave differently at shocks. In summary physical considerations show that is the correct eulerian system of conservation laws to analyze for the two temperature T i − T e model. Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 7 / 27

Lagrangian T i − T e In Lagrange variable in dimension one, one gets 8 ∂ t τ − ∂ m u = 0 , > > ∂ t u + ∂ m p = 0 , p = p i + p e , < Introduction ∂ t S e = 0 , > > ∂ t e + ∂ m ( pu ) = 0 . : Numerical discretization Set of T i − T e ∂ p i ∂ p i ∂ p i ∂ p e and ρ 2 c 2 = − ρ 2 c 2 ρ 2 c 2 i = − , e = − − model ∂τ | S i ∂τ | S e ∂τ | S i ∂τ | S e Coupling with The sound speed of the lagrangian system is ρ c where c 2 = c 2 i + c 2 e . radiation The natural Lagrangian scheme is now Hele-Shaw models Mj 8 ∆ t ( τ L j − τ n j ) − u ∗ + u ∗ = 0 , > j + 1 j − 1 > > 2 2 > > Mj > > ∆ t ( u L j − u n j ) + p ∗ − p ∗ > = 0 , < j + 1 j − 1 2 2 ( S e ) L j − ( S e ) n j = 0 , > > > > Mj > ∆ t ( e L j − e n j ) + p ∗ u ∗ − p ∗ u ∗ > = 0 , > > j + 1 j + 1 j − 1 j − 1 : 2 2 2 2 with the solver u ∗ = 1 2 ( u n j + u n 2 ρ c ( p n 1 j − p n 8 j +1 ) + j +1 ) j + 1 > > 2 > > p ∗ = 1 2 ( p n j + p n j +1 ) + ρ c 2 ( u n j − u n < j +1 ) , j + 1 2 > = 1 h ( ρ c ) n j + ( ρ c ) n i > > ( ρ c ) j + 1 . > 2 j +1 : 2 A pure Lagrangian scheme is such that f n +1 = f L for all f . Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 8 / 27

With source terms A simplified eulerian T i − T e system with source terms writes Introduction 8 ∂ t ρ + ∂ x ρ u = 0 Numerical ∂ t ρ u + ∂ x ( ρ u 2 + p i + p e ) = 0 > > > < discretization 1 ∂ t ρε i + ∂ x ρ u ε i + p i ∂ x u = τ ei ( T e − T i ) of T i − T e > > 1 ∂ t ρε e + ∂ x ρ u ε e + p e ∂ x u = τ ei ( T i − T e ) + ∂ x ( K e ∂ x T e ) . model > : Coupling with radiation The relaxation time is τ ei . The electronic diffusion coefficient is K e . We assume that Hele-Shaw ε i = C vi T i et ε e = C ve T e . models The rigorous way to write this is ∂ t ρ + ∂ x ρ u = 0 8 ∂ t ρ u + ∂ x ( ρ u 2 + p i + p e ) = 0 > > < . 1 1 ∂ t ρ S e + ∂ x ρ uS e = τ ei Te ( T i − T e ) + Te ∂ x ( K e ∂ x T e ) > > : ∂ t ρ e + ∂ x ( ρ ue + p i u + p e u ) = ∂ x ( K e ∂ x T e ) , where the unknowns are the density ρ , the momentum ρ u , the electronic entropy ρ S e and the total energy ρ e . Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 9 / 27

Numerical solution base on a splitting strategy First stage : solve the hydro. Second stage solve the remaining part Introduction 8 ∂ t ρ = 0 Numerical > > ∂ t ρ u = 0 > < discretization 1 ∂ t ρε i = τ ei ( T e − T i ) of T i − T e > > 1 ∂ t ρε e = τ ei ( T i − T e ) + ∂ x ( K e ∂ x T e ) . > model : Coupling with radiation We use the linear law ε i = C vi T i and ε e = C ve T e . Hele-Shaw models The numerical solution of the system can be computed with an implicit linear solver in case the gas is described by perfect gas equations of state. ρ n +1 = ρ L , 8 > > u n +1 = u L , > > > > ( Ti ) n +1 − ( Ti ) L > > j > ρ L C vi j τ ei (( T e ) n +1 1 − ( T i ) n +1 > = ) , > < ∆ t j j ( Te ) n +1 − ( Te ) L ρ L C ve j j τ ei (( T i ) n +1 1 − ( T e ) n +1 > = ) > > ∆ t j j > > > ( Te ) n +1 j +1 − ( Te ) n +1 ( Te ) n +1 − ( Te ) n +1 “ ” “ ” > Ke , i + 1 − Ke , i − 1 > > j − 1 j j > > 2 2 + . : ∆ x 2 Numerical methods for FCI Part IV Multi-temperature fluid models Hele-Shaw models p. 10 / 27