N ONLINEAR HYPERBOLIC PDE : C OUPLING TECHNIQUES USING DIFFUSION MECHANISMS Benjamin B OUTIN IRMAR, Université Rennes 1, France Journées EDP Normandie Rouen, October 25th-26th, 2011
O UTLINE Position of the problem 1 Geometrical framework for the state coupling 2 Weak form of boundary conditions Existence result Thin interface regime 3 Reformulation of the problem D AFERMOS regularization Interfacial layers of R IEMANN -D AFERMOS solutions 4 Thick interface regime Theoretical results Numerical observations B. Boutin Coupling techniques. . . 25/10/11 2/22
O UTLINE Position of the problem 1 Geometrical framework for the state coupling 2 Weak form of boundary conditions Existence result Thin interface regime 3 Reformulation of the problem D AFERMOS regularization Interfacial layers of R IEMANN -D AFERMOS solutions 4 Thick interface regime Theoretical results Numerical observations B. Boutin Coupling techniques. . . 25/10/11 2/22
C ONTEXT OF THE STUDY Enceinte de confinement N EPTUNE project Vapeur d’eau Pressuriseur Cuve Barres de contrˆ ole Turbine Vapeur G´ en´ erateur de vapeur Alternateur (´ echangeur de chaleur) Caloporteur Tour de chaud refroidissement (330 ◦ C) Condenseur Pompe ou rivi` ere ou mer Cœur Liquide Circuit de refroidissement Caloporteur Pompe froid (280 ◦ C) Pompe PhD Thesis (27/11/09) Other collaborators : CEA Saclay and University Paris 6 A. A MBROSO , F. C OQUEL , C. C HALONS , E. G ODLEWSKI , under the supervision of P. G. L E F LOCH F. L AGOUTIÈRE , P. A. R AVIART , N. S EGUIN B. Boutin Coupling techniques. . . 25/10/11 3/22
P RACTICAL MOTIVATIONS Code coupling : Aims : information • mathematically well-posed • numerically solvable Code A Code B • physically satisfactory space Fixed interface x = 0 Possible information to transmit Example : Two Euler systems with distinct EOS Conservation of some physical quantities ∂ t ρ + ∂ x ( ρ v ) = 0 , Continuity of relevant variables ∂ t ( ρ v ) + ∂ x ( ρ v 2 + p ± ) = 0 , Knowledge of steady states ∂ t ( ρ e ) + ∂ x ( ρ ve + p ± v ) = 0 . B. Boutin Coupling techniques. . . 25/10/11 4/22
P RACTICAL MOTIVATIONS Code coupling : Aims : information • mathematically well-posed • numerically solvable Code A Code B • physically satisfactory space Fixed interface x = 0 Possible information to transmit Example : Two Euler systems with distinct EOS Conservation of some physical quantities ∂ t ρ + ∂ x ( ρ v ) = 0 , Continuity of relevant variables ∂ t ( ρ v ) + ∂ x ( ρ v 2 + p ± ) = 0 , Knowledge of steady states ∂ t ( ρ e ) + ∂ x ( ρ ve + p ± v ) = 0 . B. Boutin Coupling techniques. . . 25/10/11 4/22
C ONSERVATIVE VS . NON - CONSERVATIVE COUPLING Two strictly hyperbolic PDE problems : ∂ t u + ∂ x f − ( u ) = 0 , x < 0 , t > 0 , u ( x , t ) ∈ R N , x ∈ R . ∂ t u + ∂ x f + ( u ) = 0 , x > 0 , t > 0 , Flux coupling A UDUSSE , P ERTHAME , S EGUIN , V OVELLE , T OWERS , K ARLSEN , R ISEBRO f − ( u ( 0 − , t )) = f + ( u ( 0 + , t )) , t > 0 . Rankine-Hugoniot jump relation Convenient entropy condition at the interface (Discontinuous flux conservative coupling) State coupling G ODLEWSKI , R AVIART u ( 0 − , t ) = u ( 0 + , t ) , t > 0 . No physical entropy criterion at the interface or more generally ( ⇒ study of the microscopical dynamic) θ − ( u ( 0 − , t )) = θ + ( u ( 0 + , t )) , t > 0 . θ ± change of variable. (Non-conservative coupling) Difficulty : Both frameworks are usually inconciliable ! Example (cont.) : ( A MBROSO et al. 2005 ) conservation of ρ E and preservation of steady states p , v = cst are not conciliable. B. Boutin Coupling techniques. . . 25/10/11 5/22
C ONSERVATIVE VS . NON - CONSERVATIVE COUPLING Two strictly hyperbolic PDE problems : ∂ t u + ∂ x f − ( u ) = 0 , x < 0 , t > 0 , u ( x , t ) ∈ R N , x ∈ R . ∂ t u + ∂ x f + ( u ) = 0 , x > 0 , t > 0 , Flux coupling A UDUSSE , P ERTHAME , S EGUIN , V OVELLE , T OWERS , K ARLSEN , R ISEBRO f − ( u ( 0 − , t )) = f + ( u ( 0 + , t )) , t > 0 . Rankine-Hugoniot jump relation Convenient entropy condition at the interface (Discontinuous flux conservative coupling) State coupling G ODLEWSKI , R AVIART u ( 0 − , t ) = u ( 0 + , t ) , t > 0 . No physical entropy criterion at the interface or more generally ( ⇒ study of the microscopical dynamic) θ − ( u ( 0 − , t )) = θ + ( u ( 0 + , t )) , t > 0 . θ ± change of variable. (Non-conservative coupling) Difficulty : Both frameworks are usually inconciliable ! Example (cont.) : ( A MBROSO et al. 2005 ) conservation of ρ E and preservation of steady states p , v = cst are not conciliable. B. Boutin Coupling techniques. . . 25/10/11 5/22
C ONSERVATIVE VS . NON - CONSERVATIVE COUPLING Two strictly hyperbolic PDE problems : ∂ t u + ∂ x f − ( u ) = 0 , x < 0 , t > 0 , u ( x , t ) ∈ R N , x ∈ R . ∂ t u + ∂ x f + ( u ) = 0 , x > 0 , t > 0 , Flux coupling A UDUSSE , P ERTHAME , S EGUIN , V OVELLE , T OWERS , K ARLSEN , R ISEBRO f − ( u ( 0 − , t )) = f + ( u ( 0 + , t )) , t > 0 . Rankine-Hugoniot jump relation Convenient entropy condition at the interface (Discontinuous flux conservative coupling) State coupling G ODLEWSKI , R AVIART u ( 0 − , t ) = u ( 0 + , t ) , t > 0 . No physical entropy criterion at the interface or more generally ( ⇒ study of the microscopical dynamic) θ − ( u ( 0 − , t )) = θ + ( u ( 0 + , t )) , t > 0 . θ ± change of variable. (Non-conservative coupling) Difficulty : Both frameworks are usually inconciliable ! Example (cont.) : ( A MBROSO et al. 2005 ) conservation of ρ E and preservation of steady states p , v = cst are not conciliable. B. Boutin Coupling techniques. . . 25/10/11 5/22
C ONSERVATIVE VS . NON - CONSERVATIVE COUPLING Two strictly hyperbolic PDE problems : ∂ t u + ∂ x f − ( u ) = 0 , x < 0 , t > 0 , u ( x , t ) ∈ R N , x ∈ R . ∂ t u + ∂ x f + ( u ) = 0 , x > 0 , t > 0 , Flux coupling A UDUSSE , P ERTHAME , S EGUIN , V OVELLE , T OWERS , K ARLSEN , R ISEBRO f − ( u ( 0 − , t )) = f + ( u ( 0 + , t )) , t > 0 . Rankine-Hugoniot jump relation Convenient entropy condition at the interface (Discontinuous flux conservative coupling) State coupling G ODLEWSKI , R AVIART u ( 0 − , t ) = u ( 0 + , t ) , t > 0 . No physical entropy criterion at the interface or more generally ( ⇒ study of the microscopical dynamic) θ − ( u ( 0 − , t )) = θ + ( u ( 0 + , t )) , t > 0 . θ ± change of variable. (Non-conservative coupling) Difficulty : Both frameworks are usually inconciliable ! Example (cont.) : ( A MBROSO et al. 2005 ) conservation of ρ E and preservation of steady states p , v = cst are not conciliable. B. Boutin Coupling techniques. . . 25/10/11 5/22
O UTLINE Position of the problem 1 Geometrical framework for the state coupling 2 Weak form of boundary conditions Existence result Thin interface regime 3 Reformulation of the problem D AFERMOS regularization Interfacial layers of R IEMANN -D AFERMOS solutions 4 Thick interface regime Theoretical results Numerical observations B. Boutin Coupling techniques. . . 25/10/11 6/22
W EAK FORM OF BOUNDARY CONDITIONS Half-C AUCHY problem ( D UBOIS & L E F LOCH , 1988 ) t Let W ( x / t , u l , u r ) be the self-similar entropy solution for ∂ t u + ∂ x f ( u ) = 0 the Riemann problem with data ( u l , u r ) . b ( t ) � � x u 0 Set O ( b ( t )) = W ( 0 + , b ( t ) , � . 0 u ) , � u ∈ Ω ” u ( 0 + , t ) = b ( t ) ” → well-posed problem. u ( 0 + , t ) ∈ O ( b ( t )) Two half-C AUCHY problems sticked together ( G ODLEWSKI & R AVIART , 2004 ) u ( 0 − , t ) = u ( 0 + , t ) u ( 0 − , t ) ∈ O − ( u ( 0 + , t )) , u ( 0 + , t ) ∈ O + ( u ( 0 − , t )) . ∂ t u + ∂ x f − ( u ) = 0 ∂ t u + ∂ x f + ( u ) = 0 x 0 B. Boutin Coupling techniques. . . 25/10/11 7/22
W EAK FORM OF BOUNDARY CONDITIONS Half-C AUCHY problem ( D UBOIS & L E F LOCH , 1988 ) t Let W ( x / t , u l , u r ) be the self-similar entropy solution for ∂ t u + ∂ x f ( u ) = 0 the Riemann problem with data ( u l , u r ) . b ( t ) � � x u 0 Set O ( b ( t )) = W ( 0 + , b ( t ) , � . 0 u ) , � u ∈ Ω ” u ( 0 + , t ) = b ( t ) ” → well-posed problem. u ( 0 + , t ) ∈ O ( b ( t )) Two half-C AUCHY problems sticked together ( G ODLEWSKI & R AVIART , 2004 ) u ( 0 − , t ) = u ( 0 + , t ) u ( 0 − , t ) ∈ O − ( u ( 0 + , t )) , u ( 0 + , t ) ∈ O + ( u ( 0 − , t )) . ∂ t u + ∂ x f − ( u ) = 0 ∂ t u + ∂ x f + ( u ) = 0 x 0 B. Boutin Coupling techniques. . . 25/10/11 7/22
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