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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

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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

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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

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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 Coarse Adapted Indicator (blue=fine, red=coarse)

Simulation H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Determine an error indicator

Error indicator to perform adaptation Uf solution of the fine model, Uc solution of the coarse model, Uh

c finite volume approximation of Uc

Indicator = Uf − Uh

c ≤ Uf − Uc + Uc − Uh c

Indicator(t) controlled by U0

f , U0 c and Uh,0 c

State of art Scalar case [Kruzhkov 70 ; Kuznetsov 76 ; Lucier 86 ; Bouchut, Perthame 98 ; Kr¨

  • ner, 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

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Fine and coarse models

Fine model: hyperbolic system with relaxation [Chen, Levermore, Liu 94]      ∂tw +

d

  • α=1

∂αFα(w) = 1 εR(w), t > 0, x ∈ Rd w(x, 0) = w0(x), x ∈ Rd (Mf ) w = (u, v)T ∈ Rn, u ∈ Rm, n > m F, R smooth enough Stiff source term R: R(w) = 0 ⇔ w = weq(u) Map weq (assumed to be Lweq−Lipschitz continuous)

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Fine and coarse models

Coarse model: in the limit ε → 0, hyperbolic system      ∂tu +

d

  • α=1

∂αgα(u) = 0, t > 0, x ∈ Rd u(x, 0) = u0(x), x ∈ Rd (Mc) where g(u) = PF(weq(u)) P ∈ Mn×m such that PR(w) = 0, ∀w ∈ Rn Pw = u, ∀w ∈ Rn Pweq(u) = u, ∀u ∈ Rm (Hyp 1)

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Error indicator

Error indicator Indicator = w − weq(uh)2 ≤ 2w − weq(u)2 + 2Lwequ − uh2 Outline

1 Modeling error estimate w − weq(u) ◮ estimate in ε between smooth solutions of (Mf ) and (Mc), relative

entropy, [Tzavaras 06]

2 Numerical error estimate u − uh ◮ 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

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Modeling error estimate: assumptions [Tzavaras 06]

Non-degeneracy dim Ker(∇wR(weq(u))) =m dim Im(∇wR(weq(u))) =n − m (Hyp 2) The system (Mf ) is endowed with a β-convex entropy η : Rn → R ∇η(w)T∇wF(w) = ∇wξ(w)T (Hyp 3) where ξα : Rn → R, α = 1, . . . , d is the entropy flux The pair η − ξ satisfy ∂tη(w) +

d

  • α=1

∂αξα(w) = 1 ε∇wη(w) ∗ R(w) where ∗ denotes the scalar product in Rn

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Modeling error estimate: assumptions [Tzavaras 06]

Entropy dissipation ∇wη(w) ∗ R(w) ≤ 0 (Hyp 4) Entropy-entropy flux for (Mc): restriction of η − ξ on {w ∈ Rn, R(w) = 0} induces ∇uη(u)T∇ug(u) = ∇uξ(u)T (Hyp 5) where g(u) = PF(weq(u)) For u ∈ Rm regular solution of (Mc), the pair η − ξ satisfies ∂tη(u) +

d

  • α=1

∂αξα(u) = 0

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Modeling error estimate

Theorem ([Tzavaras 06])

Assume hypothesis (Hyp 1)-(Hyp 5). Let (wε) be a family of regular solutions of (Mf ) and u a regular solution of (Mc) on Rd × [0, T] with initial conditions (wε

0) and u0. Assume that wε, weq(uε) and weq(u) are in

Bweq ⊂ Rn. Assume it exists ν = ν(weq) such that

− (∇wη(w) − ∇wη(weq(u))) ∗ (R(w) − R(weq(u))) ≥ ν|w − weq(u)|2 (Hyp 6)

for w, weq(u) ∈ Bweq with u = Pw. Then for r > 0 it exists A, B, s such that:

  • |x|<r

|w − weq(u)|2dx ≤ A

  • |x|<r+st

|w(x, 0) − weq(u(x, 0)|2dx + Bε

  • H´

el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Modeling error estimate: relative entropy

Let w ∈ Rm be a regular solution of (Mf ) Let u ∈ Rn be a regular solution of (Mc) The relative entropy H : Rn × Rn → R is given by H(w, weq(u)) = η(w) − η(weq(u)) − ∇wη(weq(u)) ∗ (w − weq(u)) Relative entropy flux Hα : Rn × Rn → R Q(w, weq(u)) = ξ(w)−ξ(weq(u))−∇wη(weq(u))∗(F(w)−F(weq(u))) Relative entropy positive definite (convexity of η) H(w, weq(u)) ≥ β 2 |w − weq(u)|2

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Modeling error estimate: main lines of the proof

Relative entropy identity ∂tH(w, weq(u)) +

d

  • α=1

∂αQα(w, weq(u)) + 1 εD = J1 + J2 where J1 = −

d

  • α=1

∇2

uη(u)∂αu ·

  • gα(u) − gα(u) − ∇ugα(u) · (u − u)
  • J2 = −

d

  • α=1

∇2

uη(u)∂αu · P

  • Fα(w) − Fα(weq(u))
  • D =(∇wη(w) − ∇wη(weq(u))) ∗ (R(w) − R(weq(u)))

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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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

  • |x|<r

H(w, weq(u))dx + 1 ε t

  • |x|<r+s(t−τ)

D dxdτ ≤

  • |x|<r+st

H(w0, weq(u0))dx + t

  • |x|<r+s(t−τ)

J1 + J2 dxdτ Use assumption (Hyp 6) and the β−convexity of η Control of the terms |J1| and |J2| |J1| ≤ C(η, F, weq, ∂αu)|w−weq(u)|2 |J2|+1 εD ≤ εC(η, F, weq, ∂αu) Conclude via a Gronwall Lemma Remark: ∂αu known a priori (initial data)

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Numerical error estimate

Numerical error estimate u − uh Scalar case: FV scheme, convergence towards entropy solution, error estimate [Chainais 99 ; Eymard, Gallou¨ et, Herbin 00 ; Kr¨

  • ner,

Ohlberger 00], System case: relative entropy      ∂tu +

d

  • α=1

∂αgα(u) = 0 u(x, 0) = u0(x) (Mc)

1 u0 ∈ L∞(Rd) 2 g Lipschitz continuous 3 Entropy-entropy flux η − ξ, η β−convex H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Finite volume scheme

Mesh T h = sup{diam(K), K ∈ T } < ∞ ahd ≤ m(K) E: set of interfaces N(K): neighbouring cells of K σKL K L nKL Time step: tn = n∆t, ∀n ∈ N Numerical scheme u0

K =

1 m(K)

  • K

u0 un+1

K

= un

K −

∆t m(K)

  • L∈N

GKL(un

K, un L)

uh(x, t) = un

K, if x ∈ K, tn ≤ t ≤ tn+1

(S)

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Numerical flux: assumptions

Numerical flux: GKL ∈ C(Rm × Rm, Rm) : (u, v) → GKL(u, v) Lipschitz continuous, conservative, stable under CFL condition, consistant 1 m(σKL)GKL(u, u) = g(u) · nKL Entropy flux ξKL(u, v), conservative Entropy inequality by interface [Harten, Lax, Van Leer 83 ; Bouchut 04]

ξKL(uK, uL) − m(σKL)ξ(uK) · nKL ≤ − 1 λ

  • η
  • uK − λ
  • GKL(uK, uL)

−m(σKL)G(uK) · nKL

  • − η(uK)
  • Numerical solution: (un

K) ∈ O ⊂ Rm, O open bounded subset

Weak BV estimate [Eymard, Gallou¨ et, Herbin, 00]

  • n
  • E

∆t |GKL(un

K, un L) − m(σKL)g(un K) · nKL| ≤ CWBV

√ h

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Numerical entropy flux

Lemma

Under assumptions on the numerical flux GKL and η properties, the numerical entropy flux ξKL satisfies the following properties:

1 Consistance: ξKL(u, u) = m(σKL)ξ(u) · nKL 2 Lipschitz continuity 3 Discrete entropy inequality

m(K) ∆t (η(un+1

K

) − η(uk

n)) +

  • L∈N

ξKL(un

K, un L) ≤ 0

4 Weak BV estimate

  • k
  • E

∆t

  • ξKL(un

K, un L) − m(σKL)ξ(un K) · nKL

  • ≤

˜ CWBV √ h

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Relative entropy

Let u ∈ Rm be a regular solution of (Mc) and uh its approximation. The relative entropy H : Rm × Rm → R is given by H(u, uh) = η(uh) − η(u) − ∇uη(u) · (uh − u) Relative entropy flux Qα : Rm × Rm → R Q(uh, u) = ξ(uh) − ξ(u) − ∇uη(u) · (g(uh) − g(u)) Link between relative entropy and Em: H(u, uh) ≥ β 2 |u − uh|2

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Relative entropy inequality

Theorem

Let u be a classical solution of (Mc) and uh an approximate solution defined by (S). Then, ∀ϕ ∈ Cc(Rd × R+)

  • Rd×R+ H(uh, u)∂tϕdxdt + Q(uh, u)divϕ +
  • Rd H(uh

0, u0)ϕ(x, 0)dx ≥

−

  • Rd×R+ |divϕ| + |∂tϕ|dµ −
  • Rd ϕ(x, 0)dµ0

−

  • Rd×R+ |div(∇uη(u)ϕ)| + |∂t(∇uη(u)ϕ)|dµ −
  • Rd ∇uη(u0)ϕ(x, 0)dµ0

−

  • Rd×R+ ϕ

d

  • α=1

∂αuTZα(uh, u)dxdt where Zα(uh, u) = (gα(uh) − gα)∇2

uη(u) − ∇ugα(u)(uh − u)

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Main lines of the proof

It exists µK, µK ∈ M(Rd × R+) and µ0, µ0 ∈ M(Rd) such that for ϕ ∈ C ∞

c (Rd × R+, R+)

  • Rd×R+ η(uh)∂tϕ + ξ(uh)divϕdxdt +
  • Rd η(uh

0)ϕ(x, 0)dx ≥

−

  • Rd×R+ (|divϕ| + |∂tϕ|) dµ −
  • Rd ϕ(x, 0)dµ0

and

  • Rd×R+ uh∂tϕ + g(uh)divϕdxdt +
  • Rd uh

0ϕ(x, 0)dx ≥

−

  • Rd×R+(|divϕ| + |∂tϕ|)dµ −
  • Rd ϕ(x, 0)dµ0

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Main lines of the proof

The measures µ and µ0 verify the following properties

1

If u0 ∈ BV (Rd) µ0(Rd) ≤ C(u0, a)h µ0(Rd) ≤ C(u0, a)h

2

∀r > 0, T > 0 µ(B(0, r) × [0, T]) ≤ C(GKL, CWBV , u0, r, T) √ h µ(B(0, r) × [0, T]) ≤ C(g, u0, r, T) √ h

Weak BV estimates and the numerical scheme definition (S)

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Numerical error estimate

Theorem

Let u be a classical solution of (Mc) on [0, T] taking values in a convex compact subset D ⊂ Rm, with initial data u0. Let uh be an approximate solution of (Mc) defined by (S). Then

  • |x|<r

|uh − u|2dx ≤ CeD

  • |x|<r+st

|uh

0 − u0|2dx

holds for any r > 0 and t ∈ [0, T] with positive constant s, C(η, D) and D(η, D, ∂αu) Main lines of the proof Consider relative entropy inequality (Kruzhkov procedure) Gronwall Lemma

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Global estimate

Corollary

Assume hypothesis (Hyp 1)-(Hyp 6). Let w be a regular solution of (Mf ) with initial conditions w0. Let u be a classical solution of (Mc) on [0, T] taking values in a convex compact subset D ⊂ Rm, with initial data u0. Let uh be an approximate solution of (Mc) defined by (S). Then

  • |x|<r

|w − weq(uh)|2dx ≤A

  • |x|<r+st

|w0 − weq(u0)|2dx + Bε

  • + CeD
  • |x|<r+st

|uh

0 − u0|2dx

holds for any r > 0 and t ∈ [0, T]

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP

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Conclusion and prospects

Conclusions Error estimate between hyberbolic system with relaxation and finite volume approximation of the hyperbolic limit Use of relative entropy Prospects Specify the regularity hypothesis

◮ existence of solution of (Mf ) [Yong 99 ; Hanouzet, Natalini 03] ◮ existence of solution of (Mc) [Dafermos 10]

Get precisely the constants [Kr¨

  • ner, Ohlberger 99]

Use this estimate to perform adaptation

H´ el` ene Mathis (LMJL) Relative entropy for hyperbolic systems HYP