N ONLINEAR HYPERBOLIC PDE : C OUPLING TECHNIQUES USING DIFFUSION - - PowerPoint PPT Presentation

NONLINEAR HYPERBOLIC PDE : COUPLING TECHNIQUES USING DIFFUSION MECHANISMS

Benjamin BOUTIN

IRMAR, Université Rennes 1, France

Journées EDP Normandie Rouen, October 25th-26th, 2011

OUTLINE

1

Position of the problem

2

Geometrical framework for the state coupling Weak form of boundary conditions Existence result

3

Thin interface regime Reformulation of the problem DAFERMOS regularization Interfacial layers of RIEMANN-DAFERMOS solutions

4

Thick interface regime Theoretical results Numerical observations

• B. Boutin

Coupling techniques. . . 25/10/11 2/22

OUTLINE

1

Position of the problem

2

Geometrical framework for the state coupling Weak form of boundary conditions Existence result

3

Thin interface regime Reformulation of the problem DAFERMOS regularization Interfacial layers of RIEMANN-DAFERMOS solutions

4

Thick interface regime Theoretical results Numerical observations

• B. Boutin

Coupling techniques. . . 25/10/11 2/22

CONTEXT OF THE STUDY

Enceinte de confinement Liquide Vapeur Pompe G´ en´ erateur de vapeur (´ echangeur de chaleur) Turbine Alternateur Tour de refroidissement

• u rivi`

ere ou mer Circuit de refroidissement Condenseur Pompe Pompe Cœur Cuve Barres de contrˆ

• le

Pressuriseur Caloporteur chaud (330 ◦C) Caloporteur froid (280 ◦C) Vapeur d’eau

NEPTUNE project

PhD Thesis (27/11/09) CEA Saclay and University Paris 6 under the supervision of P. G. LEFLOCH Other collaborators :

• A. AMBROSO, F. COQUEL, C. CHALONS, E. GODLEWSKI,
• F. LAGOUTIÈRE, P. A. RAVIART, N. SEGUIN
• B. Boutin

Coupling techniques. . . 25/10/11 3/22

PRACTICAL MOTIVATIONS

Code coupling : space Fixed interface x = 0

Code A Code B

information

Aims :

• mathematically well-posed
• numerically solvable
• physically satisfactory

Possible information to transmit Conservation of some physical quantities Continuity of relevant variables Knowledge of steady states

Example : Two Euler systems with distinct EOS

             ∂tρ + ∂x(ρv) = 0, ∂t(ρv) + ∂x(ρv2 + p±) = 0, ∂t(ρe) + ∂x(ρve + p±v) = 0.

• B. Boutin

Coupling techniques. . . 25/10/11 4/22

PRACTICAL MOTIVATIONS

Code coupling : space Fixed interface x = 0

Code A Code B

information

Aims :

• mathematically well-posed
• numerically solvable
• physically satisfactory

Possible information to transmit Conservation of some physical quantities Continuity of relevant variables Knowledge of steady states

Example : Two Euler systems with distinct EOS

             ∂tρ + ∂x(ρv) = 0, ∂t(ρv) + ∂x(ρv2 + p±) = 0, ∂t(ρe) + ∂x(ρve + p±v) = 0.

• B. Boutin

Coupling techniques. . . 25/10/11 4/22

CONSERVATIVE VS. NON-CONSERVATIVE COUPLING

Two strictly hyperbolic PDE problems :

       ∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0, u(x, t) ∈ RN, x ∈ R. Flux coupling f−(u(0−, t)) = f+(u(0+, t)), t > 0. (Discontinuous flux conservative coupling)

AUDUSSE, PERTHAME, SEGUIN, VOVELLE, TOWERS, KARLSEN, RISEBRO

Rankine-Hugoniot jump relation Convenient entropy condition at the interface State coupling u(0−, t) = u(0+, t), t > 0.

• r more generally

θ−(u(0−, t)) = θ+(u(0+, t)),

t > 0.

θ± change of variable.

(Non-conservative coupling)

GODLEWSKI, RAVIART

No physical entropy criterion at the interface (⇒ study of the microscopical dynamic) Difficulty : Both frameworks are usually inconciliable !

Example (cont.) : (AMBROSO et al. 2005) conservation of ρE and preservation of steady states p, v = cst are not conciliable.

• B. Boutin

Coupling techniques. . . 25/10/11 5/22

CONSERVATIVE VS. NON-CONSERVATIVE COUPLING

Two strictly hyperbolic PDE problems :

       ∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0, u(x, t) ∈ RN, x ∈ R. Flux coupling f−(u(0−, t)) = f+(u(0+, t)), t > 0. (Discontinuous flux conservative coupling)

AUDUSSE, PERTHAME, SEGUIN, VOVELLE, TOWERS, KARLSEN, RISEBRO

Rankine-Hugoniot jump relation Convenient entropy condition at the interface State coupling u(0−, t) = u(0+, t), t > 0.

• r more generally

θ−(u(0−, t)) = θ+(u(0+, t)),

t > 0.

θ± change of variable.

(Non-conservative coupling)

GODLEWSKI, RAVIART

No physical entropy criterion at the interface (⇒ study of the microscopical dynamic) Difficulty : Both frameworks are usually inconciliable !

Example (cont.) : (AMBROSO et al. 2005) conservation of ρE and preservation of steady states p, v = cst are not conciliable.

• B. Boutin

Coupling techniques. . . 25/10/11 5/22

CONSERVATIVE VS. NON-CONSERVATIVE COUPLING

Two strictly hyperbolic PDE problems :

       ∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0, u(x, t) ∈ RN, x ∈ R. Flux coupling f−(u(0−, t)) = f+(u(0+, t)), t > 0. (Discontinuous flux conservative coupling)

AUDUSSE, PERTHAME, SEGUIN, VOVELLE, TOWERS, KARLSEN, RISEBRO

Rankine-Hugoniot jump relation Convenient entropy condition at the interface State coupling u(0−, t) = u(0+, t), t > 0.

• r more generally

θ−(u(0−, t)) = θ+(u(0+, t)),

t > 0.

θ± change of variable.

(Non-conservative coupling)

GODLEWSKI, RAVIART

No physical entropy criterion at the interface (⇒ study of the microscopical dynamic) Difficulty : Both frameworks are usually inconciliable !

Example (cont.) : (AMBROSO et al. 2005) conservation of ρE and preservation of steady states p, v = cst are not conciliable.

• B. Boutin

Coupling techniques. . . 25/10/11 5/22

CONSERVATIVE VS. NON-CONSERVATIVE COUPLING

Two strictly hyperbolic PDE problems :

       ∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0, u(x, t) ∈ RN, x ∈ R. Flux coupling f−(u(0−, t)) = f+(u(0+, t)), t > 0. (Discontinuous flux conservative coupling)

AUDUSSE, PERTHAME, SEGUIN, VOVELLE, TOWERS, KARLSEN, RISEBRO

Rankine-Hugoniot jump relation Convenient entropy condition at the interface State coupling u(0−, t) = u(0+, t), t > 0.

• r more generally

θ−(u(0−, t)) = θ+(u(0+, t)),

t > 0.

θ± change of variable.

(Non-conservative coupling)

GODLEWSKI, RAVIART

No physical entropy criterion at the interface (⇒ study of the microscopical dynamic) Difficulty : Both frameworks are usually inconciliable !

Example (cont.) : (AMBROSO et al. 2005) conservation of ρE and preservation of steady states p, v = cst are not conciliable.

• B. Boutin

Coupling techniques. . . 25/10/11 5/22

OUTLINE

1

Position of the problem

2

Geometrical framework for the state coupling Weak form of boundary conditions Existence result

3

Thin interface regime Reformulation of the problem DAFERMOS regularization Interfacial layers of RIEMANN-DAFERMOS solutions

4

Thick interface regime Theoretical results Numerical observations

• B. Boutin

Coupling techniques. . . 25/10/11 6/22

WEAK FORM OF BOUNDARY CONDITIONS

Half-CAUCHY problem (DUBOIS & LEFLOCH, 1988) x u0 t b(t)

∂tu + ∂xf(u) = 0 ”u(0+, t) = b(t) ”

u(0+, t) ∈ O(b(t)) Let W(x/t, ul, ur) be the self-similar entropy solution for the Riemann problem with data (ul, ur). Set O(b(t)) =

• W(0+, b(t),

u), u ∈ Ω

• .

→ well-posed problem.

Two half-CAUCHY problems sticked together (GODLEWSKI & RAVIART, 2004) x

∂tu + ∂xf−(u) = 0 ∂tu + ∂xf+(u) = 0

u(0−, t) = u(0+, t)

      

u(0−, t) ∈ O−(u(0+, t)), u(0+, t) ∈ O+(u(0−, t)).

• B. Boutin

Coupling techniques. . . 25/10/11 7/22

WEAK FORM OF BOUNDARY CONDITIONS

Half-CAUCHY problem (DUBOIS & LEFLOCH, 1988) x u0 t b(t)

∂tu + ∂xf(u) = 0 ”u(0+, t) = b(t) ”

u(0+, t) ∈ O(b(t)) Let W(x/t, ul, ur) be the self-similar entropy solution for the Riemann problem with data (ul, ur). Set O(b(t)) =

• W(0+, b(t),

u), u ∈ Ω

• .

→ well-posed problem.

Two half-CAUCHY problems sticked together (GODLEWSKI & RAVIART, 2004) x

∂tu + ∂xf−(u) = 0 ∂tu + ∂xf+(u) = 0

u(0−, t) = u(0+, t)

      

u(0−, t) ∈ O−(u(0+, t)), u(0+, t) ∈ O+(u(0−, t)).

• B. Boutin

Coupling techniques. . . 25/10/11 7/22

WEAK FORM OF BOUNDARY CONDITIONS

Half-CAUCHY problem (DUBOIS & LEFLOCH, 1988) x u0 t b(t)

∂tu + ∂xf(u) = 0 ”u(0+, t) = b(t) ”

u(0+, t) ∈ O(b(t)) Let W(x/t, ul, ur) be the self-similar entropy solution for the Riemann problem with data (ul, ur). Set O(b(t)) =

• W(0+, b(t),

u), u ∈ Ω

• .

→ well-posed problem.

Two half-CAUCHY problems sticked together (GODLEWSKI & RAVIART, 2004) x

∂tu + ∂xf−(u) = 0 ∂tu + ∂xf+(u) = 0

u(0−, t) = u(0+, t)

      

u(0−, t) ∈ O−(u(0+, t)), u(0+, t) ∈ O+(u(0−, t)).

• B. Boutin

Coupling techniques. . . 25/10/11 7/22

EXISTENCE RESULT FOR THE SCALAR COUPLED RIEMANN PROBLEM

Theorem (BB, CHALONS, RAVIART, 2010) There exists a (non-necessarily unique) solution for the coupled Riemann problem :

∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0, u(x, 0) =

      

uℓ, x < 0, ur, x > 0,

      

u(0−, t) ∈ O−(u(0+, t)), u(0+, t) ∈ O+(u(0−, t)).

• r more generally

      

u(0−, t) ∈ O−(θ−1

− (θ+(u(0+, t)))),

u(0+, t) ∈ O+(θ−1

+ (θ−(u(0−, t)))).

Important fact : For fluxes and Riemann data such that the non-uniqueness

• ccurs,

the captured nu- merical solution depends on the numerical scheme !

Related works :

CHALONS, RAVIART, SEGUIN (2 Euler systems with dif-

ferent EOS)

• B. Boutin

Coupling techniques. . . 25/10/11 8/22

OUTLINE

1

Position of the problem

2

Geometrical framework for the state coupling Weak form of boundary conditions Existence result

3

Thin interface regime Reformulation of the problem DAFERMOS regularization Interfacial layers of RIEMANN-DAFERMOS solutions

4

Thick interface regime Theoretical results Numerical observations

• B. Boutin

Coupling techniques. . . 25/10/11 9/22

REFORMULATION AS AN EXTENDED PDE SYSTEM

x

∂tu + ∂xf−(u) = 0 ∂tu + ∂xf+(u) = 0

u(0−, t) = u(0+, t) Color function : v(x, 0) =

       +1,

x > 0,

−1,

x < 0.

Extended PDE system

       ∂tu + A1(u, v)∂xu = 0, ∂tv = 0,

x ∈ R, t ≥ 0.

Example : A1(u, v) = 1 − v 2

∇f−(u) + 1 + v

2

∇f+(u),

supposed to be R-diagonalizable. Properties : The whole system is hyperbolic, in non-conservative form,

• system for u (with size N) is strictly hyperbolic
• system for (u, v) (with size N + 1) is non-strictly hyperbolic :

if A1(u, v) is singular, multiplicity of eigenvalue λ = 0 is greater than 2

→ resonance at the interface.

else u consists of RIEMANN invariants for 0-waves. The non-uniqueness reads now as a consequence of these properties

• B. Boutin

Coupling techniques. . . 25/10/11 10/22

REFORMULATION AS AN EXTENDED PDE SYSTEM

x

∂tu + ∂xf−(u) = 0 ∂tu + ∂xf+(u) = 0

u(0−, t) = u(0+, t) Color function : v(x, 0) =

       +1,

x > 0,

−1,

x < 0.

Extended PDE system

       ∂tu + A1(u, v)∂xu = 0, ∂tv = 0,

x ∈ R, t ≥ 0.

Example : A1(u, v) = 1 − v 2

∇f−(u) + 1 + v

2

∇f+(u),

supposed to be R-diagonalizable. Properties : The whole system is hyperbolic, in non-conservative form,

• system for u (with size N) is strictly hyperbolic
• system for (u, v) (with size N + 1) is non-strictly hyperbolic :

if A1(u, v) is singular, multiplicity of eigenvalue λ = 0 is greater than 2

→ resonance at the interface.

else u consists of RIEMANN invariants for 0-waves. The non-uniqueness reads now as a consequence of these properties

• B. Boutin

Coupling techniques. . . 25/10/11 10/22

REFORMULATION AS AN EXTENDED PDE SYSTEM

x

∂tu + ∂xf−(u) = 0 ∂tu + ∂xf+(u) = 0

u(0−, t) = u(0+, t) Color function : v(x, 0) =

       +1,

x > 0,

−1,

x < 0.

Extended PDE system

       ∂tu + A1(u, v)∂xu = 0, ∂tv = 0,

x ∈ R, t ≥ 0.

Example : A1(u, v) = 1 − v 2

∇f−(u) + 1 + v

2

∇f+(u),

supposed to be R-diagonalizable. Properties : The whole system is hyperbolic, in non-conservative form,

• system for u (with size N) is strictly hyperbolic
• system for (u, v) (with size N + 1) is non-strictly hyperbolic :

if A1(u, v) is singular, multiplicity of eigenvalue λ = 0 is greater than 2

→ resonance at the interface.

else u consists of RIEMANN invariants for 0-waves. The non-uniqueness reads now as a consequence of these properties

• B. Boutin

Coupling techniques. . . 25/10/11 10/22

DAFERMOS REGULARIZATION

Related works :

SLEMROD (comparison of the usual viscous regularization with the Dafermos regularization for

Burgers’ equation).

FAN, LIN, SCHECTER, SZMOLYAN

Nonlinear hyperbolic equation with unknown u ∈ Rn :

∂tu + A(u)∂xu = 0

An alternative viscous approximation :

∂tuǫ + A(uǫ)∂xuǫ = ǫt∂x(B(uǫ)∂xuǫ)

Motivation :

∂tuǫ + A(uǫ)∂xuǫ = ∂x(B(uǫ)∂xuǫ)

with the change of variable : ξ = x/t, T = ln t, u(t, x) = ˜ u(T, ξ)

∂T ˜

u +

• − ξId + A(˜

u)

• ∂ξ˜

u = ∂ξ

• B(˜

u)∂ξ˜ u

• Long time asymptotic behavior
• − ξId + A(˜

u)

• ∂ξ˜

u = ǫ ∂ξ

• B(˜

u)∂ξ˜ u

• B. Boutin

Coupling techniques. . . 25/10/11 11/22

DAFERMOS REGULARIZATION

Related works :

SLEMROD (comparison of the usual viscous regularization with the Dafermos regularization for

Burgers’ equation).

FAN, LIN, SCHECTER, SZMOLYAN

Nonlinear hyperbolic equation with unknown u ∈ Rn :

∂tu + A(u)∂xu = 0

An alternative viscous approximation :

∂tuǫ + A(uǫ)∂xuǫ = ǫt∂x(B(uǫ)∂xuǫ)

Motivation :

∂tuǫ + A(uǫ)∂xuǫ = ∂x(B(uǫ)∂xuǫ)

with the change of variable : ξ = x/t, T = ln t, u(t, x) = ˜ u(T, ξ)

∂T ˜

u +

• − ξId + A(˜

u)

• ∂ξ˜

u = ∂ξ

• B(˜

u)∂ξ˜ u

• Long time asymptotic behavior
• − ξId + A(˜

u)

• ∂ξ˜

u = ǫ ∂ξ

• B(˜

u)∂ξ˜ u

• B. Boutin

Coupling techniques. . . 25/10/11 11/22

DAFERMOS REGULARIZATION

Related works :

SLEMROD (comparison of the usual viscous regularization with the Dafermos regularization for

Burgers’ equation).

FAN, LIN, SCHECTER, SZMOLYAN

Nonlinear hyperbolic equation with unknown u ∈ Rn :

∂tu + A(u)∂xu = 0

An alternative viscous approximation :

∂tuǫ + A(uǫ)∂xuǫ = ǫt∂x(B(uǫ)∂xuǫ)

Motivation :

∂tuǫ + A(uǫ)∂xuǫ = ∂x(B(uǫ)∂xuǫ)

with the change of variable : ξ = x/t, T = ln t, u(t, x) = ˜ u(T, ξ)

∂T ˜

u +

• − ξId + A(˜

u)

• ∂ξ˜

u = e−T ∂ξ

• B(˜

u)∂ξ˜ u

• Long time asymptotic behavior
• − ξId + A(˜

u)

• ∂ξ˜

u = ǫ ∂ξ

• B(˜

u)∂ξ˜ u

• B. Boutin

Coupling techniques. . . 25/10/11 11/22

DAFERMOS REGULARIZATION

Related works :

SLEMROD (comparison of the usual viscous regularization with the Dafermos regularization for

Burgers’ equation).

FAN, LIN, SCHECTER, SZMOLYAN

Nonlinear hyperbolic equation with unknown u ∈ Rn :

∂tu + A(u)∂xu = 0

An alternative viscous approximation :

∂tuǫ + A(uǫ)∂xuǫ = ǫt∂x(B(uǫ)∂xuǫ)

Motivation :

∂tuǫ + A(uǫ)∂xuǫ = ∂x(B(uǫ)∂xuǫ)

with the change of variable : ξ = x/t, T = ln t, u(t, x) = ˜ u(T, ξ)

∂T ˜

u +

• − ξId + A(˜

u)

• ∂ξ˜

u = ǫ ∂ξ

• B(˜

u)∂ξ˜ u

• Long time asymptotic behavior
• − ξId + A(˜

u)

• ∂ξ˜

u = ǫ ∂ξ

• B(˜

u)∂ξ˜ u

• B. Boutin

Coupling techniques. . . 25/10/11 11/22

DAFERMOS REGULARIZATION

Parabolic regularization for selfsimilar solutions

∂tuǫ + A1(uǫ, vǫ)∂xuǫ = ǫt ∂x

• B0(uǫ, vǫ)∂xuǫ

, ∂tvǫ = ǫ2t ∂xxvǫ.

Local existence theorem (BB, COQUEL, LEFLOCH, 2011) Under smallness assumptions, the coupled RIEMANN problem admits a solution uǫ that converges to u ∈ BV and u is selfsimilar and entropy solution (over each half-space) of

       ∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0.

(Result obtained for the system case u ∈ RN with general coupling functions θ± close enough)

Sketch of proof : fixed point argument Following works of TZAVARAS, 1996 and JOSEPH, LEFLOCH, 2002 Estimate interactions between elementary waves of the solution Estimate interactions with the resonant wave

What happened at the interface ?

• B. Boutin

Coupling techniques. . . 25/10/11 12/22

DAFERMOS REGULARIZATION

Parabolic regularization for selfsimilar solutions

∂tuǫ + A1(uǫ, vǫ)∂xuǫ = ǫt ∂x

• B0(uǫ, vǫ)∂xuǫ

, ∂tvǫ = ǫ2t ∂xxvǫ.

Local existence theorem (BB, COQUEL, LEFLOCH, 2011) Under smallness assumptions, the coupled RIEMANN problem admits a solution uǫ that converges to u ∈ BV and u is selfsimilar and entropy solution (over each half-space) of

       ∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0.

(Result obtained for the system case u ∈ RN with general coupling functions θ± close enough)

Sketch of proof : fixed point argument Following works of TZAVARAS, 1996 and JOSEPH, LEFLOCH, 2002 Estimate interactions between elementary waves of the solution Estimate interactions with the resonant wave

What happened at the interface ?

• B. Boutin

Coupling techniques. . . 25/10/11 12/22

DAFERMOS REGULARIZATION

Parabolic regularization for selfsimilar solutions

∂tuǫ + A1(uǫ, vǫ)∂xuǫ = ǫt ∂x

• B0(uǫ, vǫ)∂xuǫ

, ∂tvǫ = ǫ2t ∂xxvǫ.

Local existence theorem (BB, COQUEL, LEFLOCH, 2011) Under smallness assumptions, the coupled RIEMANN problem admits a solution uǫ that converges to u ∈ BV and u is selfsimilar and entropy solution (over each half-space) of

       ∂tu + ∂xf−(u) = 0,

x < 0, t > 0,

∂tu + ∂xf+(u) = 0,

x > 0, t > 0.

(Result obtained for the system case u ∈ RN with general coupling functions θ± close enough)

Sketch of proof : fixed point argument Following works of TZAVARAS, 1996 and JOSEPH, LEFLOCH, 2002 Estimate interactions between elementary waves of the solution Estimate interactions with the resonant wave

What happened at the interface ?

• B. Boutin

Coupling techniques. . . 25/10/11 12/22

INTERFACIAL LAYERS OF RIEMANN-DAFERMOS SOLUTIONS

Blow-up for revealing interfacial layers of RIEMANN-DAFERMOS solutions : Uǫ(y) = uǫ(ǫy) and V ǫ(y) = vǫ(ǫy).

ξ=0−

y=−∞

ξ=0+

y=+∞

u(0−) u(0+) U−∞ U+∞ y = ξ/ǫ

ǫ −ξdξu + dξf−(u) = 0 −ξdξη(u) + dξq−(u) ≤ 0 −ξdξu + dξf+(u) = 0 −ξdξη(u) + dξq+(u) ≤ 0 Viscous profile equation A1(U, V)Uy = (B0(U, V)Uy)y Entropy 0-shock waves at the interface f−(u(0−)) = f−(U−∞), q−(u(0−)) ≥ q−(U−∞), f+(U+∞) = f+(u(0+)), q+(U+∞) ≥ q+(u(0+)).

• B. Boutin

Coupling techniques. . . 25/10/11 13/22

INTERFACIAL LAYERS OF RIEMANN-DAFERMOS SOLUTIONS

Blow-up for revealing interfacial layers of RIEMANN-DAFERMOS solutions : Uǫ(y) = uǫ(ǫy) and V ǫ(y) = vǫ(ǫy).

ξ=0−

y=−∞

ξ=0+

y=+∞

u(0−) u(0+) U−∞ U+∞ y = ξ/ǫ

ǫ −ξdξu + dξf−(u) = 0 −ξdξη(u) + dξq−(u) ≤ 0 −ξdξu + dξf+(u) = 0 −ξdξη(u) + dξq+(u) ≤ 0 Viscous profile equation A1(U, V)Uy = (B0(U, V)Uy)y Entropy 0-shock waves at the interface f−(u(0−)) = f−(U−∞), q−(u(0−)) ≥ q−(U−∞), f+(U+∞) = f+(u(0+)), q+(U+∞) ≥ q+(u(0+)).

• B. Boutin

Coupling techniques. . . 25/10/11 13/22

EXHAUSTIVE STUDY OF SIMPLE CASES

f−(u) = u2/2 f+(u) = (u − 1)2/2

u ur c c

ur = 0 ur = c uℓ = 0 uℓ = c uℓ = ur uℓ + ur = 0 ur + uℓ = 2c

• B. Boutin

Coupling techniques. . . 25/10/11 14/22

EXHAUSTIVE STUDY OF SIMPLE CASES

f−(u) = u2/2 f+(u) = (u + 1)2/2

c c uℓ ur

ur = c/2 uℓ = c/2 uℓ = ur u

r

+ u

ℓ

= 2 c u

r

+ u

ℓ

= ur = c uℓ = 0

• B. Boutin

Coupling techniques. . . 25/10/11 15/22

OUTLINE

1

Position of the problem

2

Geometrical framework for the state coupling Weak form of boundary conditions Existence result

3

Thin interface regime Reformulation of the problem DAFERMOS regularization Interfacial layers of RIEMANN-DAFERMOS solutions

4

Thick interface regime Theoretical results Numerical observations

• B. Boutin

Coupling techniques. . . 25/10/11 16/22

THICK INTERFACE APPROACH

x = 0 x

∂tw + ∂xf−(w) = 0

v = −1

∂tw + ∂xf+(w) = 0

v = 1

−1 ≤ v ≤ 1

The coupling θ−(w(0−, t)) = θ+(w(0+, t)) is ensured by requiring the scheme to preserve u-constant states, where u = C0(w, v), C0(w, ±1) = θ±(w) Conservative form with smooth source term

∂tw + ∂xf(w, v) = ℓ(w, v)∂xv.

(1) Kružkov theorem (Lipschitz source term) Let w0 ∈ L1(R) ∩ L∞(R) and v ∈ W 2,∞(R) be CAUCHY data, then

∃!w ∈ L∞(R+, L1(R) ∩ L∞(R)) entropy solution of (1).

• B. Boutin

Coupling techniques. . . 25/10/11 17/22

THICK INTERFACE APPROACH

x = 0 x

∂tw + ∂xf−(w) = 0

v = −1

∂tw + ∂xf+(w) = 0

v = 1

−1 ≤ v ≤ 1

The coupling θ−(w(0−, t)) = θ+(w(0+, t)) is ensured by requiring the scheme to preserve u-constant states, where u = C0(w, v), C0(w, ±1) = θ±(w) Conservative form with smooth source term

∂tw + ∂xf(w, v) = ℓ(w, v)∂xv.

(1) Kružkov theorem (Lipschitz source term) Let w0 ∈ L1(R) ∩ L∞(R) and v ∈ W 2,∞(R) be CAUCHY data, then

∃!w ∈ L∞(R+, L1(R) ∩ L∞(R)) entropy solution of (1).

• B. Boutin

Coupling techniques. . . 25/10/11 17/22

THICK INTERFACE APPROACH

x = 0 x

∂tw + ∂xf−(w) = 0

v = −1

∂tw + ∂xf+(w) = 0

v = 1

−1 ≤ v ≤ 1

The coupling θ−(w(0−, t)) = θ+(w(0+, t)) is ensured by requiring the scheme to preserve u-constant states, where u = C0(w, v), C0(w, ±1) = θ±(w) Conservative form with smooth source term

∂tw + ∂xf(w, v) = ℓ(w, v)∂xv.

(1) Kružkov theorem (Lipschitz source term) Let w0 ∈ L1(R) ∩ L∞(R) and v ∈ W 2,∞(R) be CAUCHY data, then

∃!w ∈ L∞(R+, L1(R) ∩ L∞(R)) entropy solution of (1).

• B. Boutin

Coupling techniques. . . 25/10/11 17/22

WELL-BALANCED NUMERICAL SCHEME

∂tw + ∂xh(w, x) = S(w, x)

x

wn

j

xj−1/2 xj+1/2 xj−1 xj+1 xj vj−1/2 vj+1/2 un

j−1

un

j

un

j+1

wn

j−1/2+

wn

j+1/2−

wn

j−1/2−

wn

j+1/2+

wn+1−

j−1/2+

wn+1−

j+1/2−

wn+1

j

reconstruction advection (CFL 1/2) projection Reconstruction : un

j solution de wn j = 1 2

• wn

j−1/2+:=

• C0(un

j , vj−1/2) + wn

j+1/2−:=

• C0(un

j , vj+1/2)

• Time evolution and projection :

wn+1

j

= wn

j − ∆t

∆x

• Hn

j+1/2 − Hn j−1/2

• − ∆t

∆x

• h(wn

j−1/2+, vj−1/2) − h(wn j+1/2−, vj+1/2)

• B. Boutin

Coupling techniques. . . 25/10/11 18/22

WELL-BALANCED NUMERICAL SCHEME

∂tw + ∂xh(w, x) = S(w, x)

x

wn

j

xj−1/2 xj+1/2 xj−1 xj+1 xj vj−1/2 vj+1/2 un

j−1

un

j

un

j+1

wn

j−1/2+

wn

j+1/2−

wn

j−1/2−

wn

j+1/2+

wn+1−

j−1/2+

wn+1−

j+1/2−

wn+1

j

reconstruction advection (CFL 1/2) projection Reconstruction : un

j solution de wn j = 1 2

• wn

j−1/2+:=

• C0(un

j , vj−1/2) + wn

j+1/2−:=

• C0(un

j , vj+1/2)

• Time evolution and projection :

wn+1

j

= wn

j − ∆t

∆x

• Hn

j+1/2 − Hn j−1/2

• − ∆t

∆x

• h(wn

j−1/2+, vj−1/2) − h(wn j+1/2−, vj+1/2)

• B. Boutin

Coupling techniques. . . 25/10/11 18/22

WELL-BALANCED NUMERICAL SCHEME

∂tw + ∂xh(w, x) = S(w, x)

x

wn

j

xj−1/2 xj+1/2 xj−1 xj+1 xj vj−1/2 vj+1/2 un

j−1

un

j

un

j+1

wn

j−1/2+

wn

j+1/2−

wn

j−1/2−

wn

j+1/2+

wn+1−

j−1/2+

wn+1−

j+1/2−

wn+1

j

reconstruction advection (CFL 1/2) projection Reconstruction : un

j solution de wn j = 1 2

• wn

j−1/2+:=

• C0(un

j , vj−1/2) + wn

j+1/2−:=

• C0(un

j , vj+1/2)

• Time evolution and projection :

wn+1

j

= wn

j − ∆t

∆x

• Hn

j+1/2 − Hn j−1/2

• − ∆t

∆x

• h(wn

j−1/2+, vj−1/2) − h(wn j+1/2−, vj+1/2)

• B. Boutin

Coupling techniques. . . 25/10/11 18/22

WELL-BALANCED NUMERICAL SCHEME

∂tw + ∂xh(w, x) = S(w, x)

x

wn

j

xj−1/2 xj+1/2 xj−1 xj+1 xj vj−1/2 vj+1/2 un

j−1

un

j

un

j+1

wn

j−1/2+

wn

j+1/2−

wn

j−1/2−

wn

j+1/2+

wn+1−

j−1/2+

wn+1−

j+1/2−

wn+1

j

reconstruction advection (CFL 1/2) projection Reconstruction : un

j solution de wn j = 1 2

• wn

j−1/2+:=

• C0(un

j , vj−1/2) + wn

j+1/2−:=

• C0(un

j , vj+1/2)

• Time evolution and projection :

wn+1

j

= wn

j − ∆t

∆x

• Hn

j+1/2 − Hn j−1/2

• − ∆t

∆x

• h(wn

j−1/2+, vj−1/2) − h(wn j+1/2−, vj+1/2)

• B. Boutin

Coupling techniques. . . 25/10/11 18/22

CONVERGENCE OF THE SCHEME

Well-balanced property / Coupling A numerical data w0

j satisfying u0 j := C0(w0 j , vj) = cst is a numerical steady state.

Convergence of the 1D scheme (BOUTIN, COQUEL, LEFLOCH) The numerical solution converges to the unique KRUŽKOV solution of

∂tw + ∂xh(w, x) = S(w, x).

The proof : L∞ stability : (under CFL 1/2) mink u0

k ≤ un j ≤ maxk u0 k,

∀n ∈ N, j ∈ Z.

Consistency with some associated entropy inequalities : ∂tw + ∂xh(w, x) = S(w, x) No TV estimate (because of the reconstruction step) Weak estimate for the discrete derivatives Limit in the sense of entropy measure valued solutions of DIPERNA The limit in the sense of YOUNG measures is nothing but the unique KRUŽKOV solution, due to the uniqueness of entropy measure valued solution.

• B. Boutin

Coupling techniques. . . 25/10/11 19/22

CONVERGENCE OF THE SCHEME

Well-balanced property / Coupling A numerical data w0

j satisfying u0 j := C0(w0 j , vj) = cst is a numerical steady state.

Convergence of the 1D scheme (BOUTIN, COQUEL, LEFLOCH) The numerical solution converges to the unique KRUŽKOV solution of

∂tw + ∂xh(w, x) = S(w, x).

The proof : L∞ stability : (under CFL 1/2) mink u0

k ≤ un j ≤ maxk u0 k,

∀n ∈ N, j ∈ Z.

Consistency with some associated entropy inequalities : ∂tw + ∂xh(w, x) = S(w, x) No TV estimate (because of the reconstruction step) Weak estimate for the discrete derivatives Limit in the sense of entropy measure valued solutions of DIPERNA The limit in the sense of YOUNG measures is nothing but the unique KRUŽKOV solution, due to the uniqueness of entropy measure valued solution.

• B. Boutin

Coupling techniques. . . 25/10/11 19/22

NUMERICAL EXAMPLES

x uℓ ur u⋆ 1-parameter set of solutions u⋆ ∈ [uℓ, ur] v(x) = erf(x/η + ζ) + 1 2

.

• 0.2

0.2 0.4 0.6 0.8 1 1.2

• 0.1
• 0.05

0.05 0.1 interface - zeta = 0.0 interface - zeta = 0.5 interface - zeta =-0.5

• 1.5
• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 solution - zeta = 0.0 solution - zeta = 0.5 solution - zeta =-0.5 phi at t=0.0

Interfaces Corresponding solutions (N = 1000)

• B. Boutin

Coupling techniques. . . 25/10/11 20/22

NUMERICAL EXAMPLES

x uℓ ur x uℓ ur x uℓ ur v(x) = erf(x/η + ζ) + 1 2

.

• 0.2

0.2 0.4 0.6 0.8 1 1.2

• 0.1
• 0.05

0.05 0.1 interface - zeta = 0.0 interface - zeta = 0.5 interface - zeta =-0.5

• 2.5
• 2
• 1.5
• 1
• 0.5

0.5 1 1.5

• 1
• 0.5

0.5 1 solution - zeta = 0.0 solution - zeta = 0.5 solution - zeta =-0.5

Interfaces Corresponding solutions (N = 1000)

• B. Boutin

Coupling techniques. . . 25/10/11 21/22

CONCLUSIONS AND PERSPECTIVES

Boundary conditions formulation of the coupling condition :

✔ Existence result of solutions to the scalar coupled RIEMANN problem. ✔ Possible non-uniqueness of solutions, numerically established.

Extended PDE formalism and thin interface regime :

✔ DAFERMOS regularization as a selection criterion for nonconservative problems

and stability analysis of solutions to the coupled RIEMANN problem.

✔ Reduced non-uniqueness after viscous profile criterion at the interface.

c c uℓ ur ur = c/2 uℓ = c/2 uℓ = ur ur + uℓ = 2c ur + uℓ = 0 ur = c uℓ = 0

Thick interface regime :

✔ Well-posed problem. ✔ Well-balanced numerical scheme, preserving steady states of the coupling problem. ✔ Numerical sensitiveness of the solution according to the interface profile.

• 1.5
• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 solution - zeta = 0.0 solution - zeta = 0.5 solution - zeta =-0.5 phi at t=0.0

Perspectives :

• study of the sensitiveness of solution according to the interface profile
• kinetic relations to treat the nonconservative coupling and intermediate coupling methods
• the Dafermos regularization as a numerical prospective tool
• stability of interfacial Riemann-Dafermos profiles
• B. Boutin

Coupling techniques. . . 25/10/11 22/22