Relative entropy for the finite volume approximation of hyperbolic systems H´ el` ene Mathis Supported by LRC Manon, CEA, UPMC Paris 6 and Universit´ e de Nantes, LMJL HYP 2012 H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Motivations: model adaptation Join work with: C. Canc` es, F. Coquel, E. Godlewski, N. Seguin Simulation of compressible water flows in pressurized water reactor Several models of two-phase flows Local properties of the flow � appropriate model Models linked by asymptotic limits � relaxation ◮ Fine model: complex system of equations, stiff source term ◮ Coarse model: simplified equations Coupling technics at interface Optimal position of the coupling interface ◮ Minimize the use of the fine model (reference solution) ◮ Minimize global error due to coupling Dynamical model adaptation H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Model adaptation, an example Phase transition model in 2D Fine model: thermodynamical equilibrium not reached (Euler equations + mass fraction evolution + closure law) Coarse model: thermodynamical equilibrium (Euler equations + closure law) Legend: blue = liquid, red=gas Fine Adapted Coarse Indicator (blue=fine, red=coarse) Simulation H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Determine an error indicator Error indicator to perform adaptation U f solution of the fine model, U c solution of the coarse model, U h c finite volume approximation of U c Indicator = � U f − U h c � ≤ � U f − U c � + � U c − U h c � c and U h , 0 Indicator ( t ) controlled by U 0 f , U 0 c State of art Scalar case [Kruzhkov 70 ; Kuznetsov 76 ; Lucier 86 ; Bouchut, Perthame 98 ; Kr¨ oner, Ohlberger 00] System case ◮ Chapman-Enskog expansion (adaptation [Mathis, Seguin 11]) Relative entropy [Di Perna 79, Dafermos 05, Tzavaras 05,... ] H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Fine and coarse models Fine model : hyperbolic system with relaxation [Chen, Levermore, Liu 94] d ∂ α F α ( w ) = 1 t > 0 , x ∈ R d ∂ t w + � ε R ( w ) , ( M f ) α =1 x ∈ R d w ( x , 0) = w 0 ( x ) , w = ( u , v ) T ∈ R n , u ∈ R m , n > m F , R smooth enough Stiff source term R : R ( w ) = 0 ⇔ w = w eq ( u ) Map w eq (assumed to be L w eq − Lipschitz continuous) H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Fine and coarse models Coarse model : in the limit ε → 0, hyperbolic system d t > 0 , x ∈ R d � ∂ t u + ∂ α g α ( u ) = 0 , ( M c ) α =1 x ∈ R d u ( x , 0) = u 0 ( x ) , where g ( u ) = P F ( w eq ( u )) P ∈ M n × m such that ∀ w ∈ R n P R ( w ) = 0 , ∀ w ∈ R n P w = u , (Hyp 1) ∀ u ∈ R m P w eq ( u ) = u , H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Error indicator Error indicator Indicator = � w − w eq ( u h ) � 2 ≤ 2 � w − w eq ( u ) � 2 + 2 L w eq � u − u h � 2 Outline 1 Modeling error estimate � w − w eq ( u ) � ◮ estimate in ε between smooth solutions of ( M f ) and ( M c ), relative entropy, [Tzavaras 06] 2 Numerical error estimate � u − u h � ◮ FV scheme, convergence towards entropy solution, error estimate [Chainais 99 ; Eymard, Gallou¨ et, Herbin 00], relative entropy 3 Conclusion and prospects H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Modeling error estimate: assumptions [Tzavaras 06] Non-degeneracy dim Ker( ∇ w R ( w eq ( u ))) = m (Hyp 2) dim Im( ∇ w R ( w eq ( u ))) = n − m The system ( M f ) is endowed with a β -convex entropy η : R n → R ∇ η ( w ) T ∇ w F ( w ) = ∇ w ξ ( w ) T (Hyp 3) where ξ α : R n → R , α = 1 , . . . , d is the entropy flux The pair η − ξ satisfy d ∂ α ξ α ( w ) = 1 � ∂ t η ( w ) + ε ∇ w η ( w ) ∗ R ( w ) α =1 where ∗ denotes the scalar product in R n H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Modeling error estimate: assumptions [Tzavaras 06] Entropy dissipation ∇ w η ( w ) ∗ R ( w ) ≤ 0 (Hyp 4) Entropy-entropy flux for ( M c ): restriction of η − ξ on { w ∈ R n , R ( w ) = 0 } induces ∇ u η ( u ) T ∇ u g ( u ) = ∇ u ξ ( u ) T (Hyp 5) where g ( u ) = P F ( w eq ( u )) For u ∈ R m regular solution of ( M c ), the pair η − ξ satisfies d � ∂ t η ( u ) + ∂ α ξ α ( u ) = 0 α =1 H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Modeling error estimate Theorem ([Tzavaras 06]) Assume hypothesis (Hyp 1) - (Hyp 5) . Let ( w ε ) be a family of regular solutions of ( M f ) and u a regular solution of ( M c ) on R d × [0 , T ] with initial conditions ( w ε 0 ) and u 0 . Assume that w ε , w eq ( u ε ) and w eq ( u ) are in B w eq ⊂ R n . Assume it exists ν = ν ( w eq ) such that − ( ∇ w η ( w ) − ∇ w η ( w eq ( u ))) ∗ ( R ( w ) − R ( w eq ( u ))) ≥ ν | w − w eq ( u ) | 2 (Hyp 6) for w , w eq ( u ) ∈ B w eq with u = P w. Then for r > 0 it exists A , B , s such that: �� � � | w − w eq ( u ) | 2 dx ≤ A | w ( x , 0) − w eq ( u ( x , 0) | 2 dx + B ε | x | < r | x | < r + st H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Modeling error estimate: relative entropy Let w ∈ R m be a regular solution of ( M f ) Let u ∈ R n be a regular solution of ( M c ) The relative entropy H : R n × R n → R is given by H ( w , w eq ( u )) = η ( w ) − η ( w eq ( u )) − ∇ w η ( w eq ( u )) ∗ ( w − w eq ( u )) Relative entropy flux H α : R n × R n → R Q ( w , w eq ( u )) = ξ ( w ) − ξ ( w eq ( u )) −∇ w η ( w eq ( u )) ∗ ( F ( w ) − F ( w eq ( u ))) Relative entropy positive definite (convexity of η ) H ( w , w eq ( u )) ≥ β 2 | w − w eq ( u ) | 2 H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Modeling error estimate: main lines of the proof Relative entropy identity d ∂ α Q α ( w , w eq ( u )) + 1 � ∂ t H ( w , w eq ( u )) + ε D = J 1 + J 2 α =1 where d � ∇ 2 � � J 1 = − u η ( u ) ∂ α u · g α ( u ) − g α ( u ) − ∇ u g α ( u ) · ( u − u ) α =1 d � ∇ 2 � � J 2 = − u η ( u ) ∂ α u · P F α ( w ) − F α ( w eq ( u )) α =1 D =( ∇ w η ( w ) − ∇ w η ( w eq ( u ))) ∗ ( R ( w ) − R ( w eq ( u ))) H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Modeling error estimate: main lines of the proof Consider the weak form of the relative entropy identity (Kruzhkov procedure) Let r > 0, t > 0, s such that sH + Q x | x | > 0 � t � H ( w , w eq ( u )) dx + 1 � D dxd τ ε | x | < r 0 | x | < r + s ( t − τ ) � t � � ≤ H ( w 0 , w eq ( u 0 )) dx + J 1 + J 2 dxd τ | x | < r + st 0 | x | < r + s ( t − τ ) Use assumption (Hyp 6) and the β − convexity of η Control of the terms | J 1 | and | J 2 | | J 2 | +1 | J 1 | ≤ C ( η, F , w eq , ∂ α u ) | w − w eq ( u ) | 2 ε D ≤ ε C ( η, F , w eq , ∂ α u ) Conclude via a Gronwall Lemma Remark: ∂ α u known a priori (initial data) H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Numerical error estimate Numerical error estimate � u − u h � Scalar case: FV scheme, convergence towards entropy solution, error estimate [Chainais 99 ; Eymard, Gallou¨ et, Herbin 00 ; Kr¨ oner, Ohlberger 00], System case: relative entropy d ∂ t u + � ∂ α g α ( u ) = 0 ( M c ) α =1 u ( x , 0) = u 0 ( x ) 1 u 0 ∈ L ∞ ( R d ) 2 g Lipschitz continuous 3 Entropy-entropy flux η − ξ , η β − convex H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Finite volume scheme Mesh T h = sup { diam( K ) , K ∈ T } < ∞ ah d ≤ m ( K ) L E : set of interfaces K n KL N ( K ): neighbouring cells of K σ KL Time step : t n = n ∆ t , ∀ n ∈ N Numerical scheme 1 � u 0 K = u 0 m ( K ) K ∆ t � ( S ) u n +1 = u n G KL ( u n K , u n K − L ) K m ( K ) L ∈N K , if x ∈ K , t n ≤ t ≤ t n +1 u h ( x , t ) = u n H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
Numerical flux: assumptions Numerical flux : G KL ∈ C ( R m × R m , R m ) : ( u , v ) → G KL ( u , v ) Lipschitz continuous, conservative, stable under CFL condition, consistant 1 m ( σ KL ) G KL ( u , u ) = g ( u ) · n KL Entropy flux ξ KL ( u , v ), conservative Entropy inequality by interface [Harten, Lax, Van Leer 83 ; Bouchut 04] ξ KL ( u K , u L ) − m ( σ KL ) ξ ( u K ) · n KL ≤ − 1 � � � u K − λ G KL ( u K , u L ) η λ �� � − m ( σ KL ) G ( u K ) · n KL − η ( u K ) Numerical solution : ( u n K ) ∈ O ⊂ R m , O open bounded subset Weak BV estimate [Eymard, Gallou¨ et, Herbin, 00] K ) · n KL | ≤ C WBV � � ∆ t | G KL ( u n K , u n L ) − m ( σ KL ) g ( u n √ h n E H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP
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