Relaxation approximation for hyperbolic fluid systems

Relaxation approximation

for hyperbolic fluid systems

Nicolas Seguin

Frédéric Coquel Edwige Godlewski

Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie – Paris 6

Multimat 07, Praha

Relaxation approximation for hyperbolic fluid systems – p. 1/32

Outline of the talk

⊲ The relaxation phenomenon⊲ Principles⊲ Bibliography

⊲ Theoretical relaxation approximation⊲ The case of the Burgers equation⊲ Gas dynamics equations⊲ Hyperbolic fluid systems

⊲ Numerical schemes using the relaxation approximation⊲ Finite volume schemes and Godunov-type schemes⊲ Numerical relaxation⊲ Properties of relaxation schemes

⊲ Conclusion


The relaxation phenomenon

General relaxation

⊲ Friction, chemical reactions, external forces...

ε characteristic time

⊲ Convergence when t→ +∞ to the equilibrium

∇p = ρg z , u1 = u2 ...

⊲ Convergence when ε→ 0 to the equilibrium

c = Ceq(ρ) , u1 = u2 ...

⊲ Equilibrium = algebraic relations

=⇒ reduction of the number of PDE’s


The relaxation phenomenon

Relaxation in hyperbolic systems

Relaxation system

{

∂tU + ∂x F(U, V) = 0

∂tV + ∂xG(U, V) = R(U, V)/ε(R)

Equilibrum: R(U, V) = 0

R(U, V) = 0 ⇐⇒ V = Veq(U)

Equilibrium system ∂tU + ∂xF(U, Veq(U)) = 0 (E )

(only formal compatibility)


The relaxation approximation

Principles:

⊲ Theoritical and numerical approximation of hyperbolic systems

⊲ Proposition of a relaxation system (R) for a given hyperbolic system (E )

⊲ (R) must be simpler to solve than (E ) −→ approximate Riemann solver

⊲ Convergence: (R) −−→ε→0

(E ) ?...

Some existing works:

⊲ Whitham, Liu, Chen-Levermore-Liu, Natalini, Hanouzet-Natalini, Yong...General formalism: Relaxation ≈ Dissipation

⊲ Jin-Xin, Serre, BianchiniGlobal linearisation: (R) is a linear hyperbolic system

⊲ Suliciu, Coquel-Perthame, Coquel et al, Bouchut...Linearize only the nonlinear terms

Here: Relaxation approximation (R) for a large class of physical systems of (E )


The Jin-Xin approximation for a conservation law

Scalar conservation law:(E ) ∂tu + ∂x f (u) = 0

Nonlinear PDE

Approximation by the relaxation system:

(R)∂tu + ∂xv = 0

∂tv + a2 ∂xu =1

ε( f (u)− v)

Linear system of PDE’s + source term (a positive constant).




Nonlinear PDE


(R)∂tu + ∂xv = 0

∂tv + a2 ∂xu =1

ε( f (u)− v)

Linear wave equation, with a the soundspeed




Nonlinear PDE


(R)∂tu + ∂xv = 0

∂tv + a2 ∂xu =1

ε( f (u)− v)

Relaxation source term:

When ε→ 0, we formally obtain v = f (u) and the first PDE of (R) becomes

∂tu + ∂x f (u) = 0


Hilbert expansion for the Jin-Xin approximation

Solution near the equilibrium of the relaxation system (R) :

vε = f (uε) + ε vε1 + ε2 vε

2 + ...

∂tuε + ∂xvε = 0,

∂tvε + a2 ∂xuε =

1

ε( f (uε)− vε).

Then, the relaxation system becomes

∂tuε + ∂x f (uε) = −ε ∂xvε

1 +O(ε2)

∂t f (uε) + a2 ∂xuε = −vε1 +O(ε)





2 + ...

∂tuε + ∂xvε = 0,


1



f ′(uε)× ∂tuε + ∂x f (uε) = −ε ∂xvε

1 +O(ε2)

=⇒ ∂t f (uε) + ( f ′(uε))2 ∂xuε = O(ε)

∂t f (uε) + a2 ∂xuε = −vε1 +O(ε)





2 + ...

∂tuε + ∂xvε = 0,


1




1 +O(ε2)

(1) ∂t f (uε) + ( f ′(uε))2 ∂xuε = O(ε)

(2) ∂t f (uε) + a2 ∂xuε = −vε1 +O(ε)

(1)− (2) vε1 =

[

( f ′(uε))2 − a2]

∂xuε +O(ε)





2 + ...

∂tuε + ∂xvε = 0,


1




1 +O(ε2)

vε1 =

[

( f ′(uε))2 − a2]

∂xuε +O(ε)





2 + ...

∂tuε + ∂xvε = 0,


1



∂tuε + ∂x f (uε) = ε ∂x

(

[

a2 − ( f ′(uε))2]

∂xuε)

+O(ε2).





2 + ...

∂tuε + ∂xvε = 0,


1



∂tuε + ∂x f (uε) = ε ∂x

(

[

a2 − ( f ′(uε))2]

∂xuε)

+O(ε2).

Equation (E ) + diffusion if

a > supu| f ′(u)| (Whitham stability condition)


Relaxation approximation for the gas dynamics equations

Euler equations in the barotropic case:

∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + p(v)) = 0,

où v = 1/ρ.

Decoupling of the linear and nonlinear parts:

⊲ Switch to Lagrangian coordinates: Dt := ∂t + u∂x and ∂y = v ∂x.

⊲ Relaxation approximation of the nonlinear part.

⊲ Switch to Eulerian coordinates.




∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + p(v)) = 0.

où v = 1/ρ.


⊲ Switch to Lagrangian coordinates: Dt := ∂t + u∂x et ∂y = v ∂x

Dtv− ∂yu = 0,

Dtu + ∂y p(v) = 0.

p(v) : nonlinear part




∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + p(v)) = 0.

where v = 1/ρ.


⊲ Relaxation approximation of the nonlinear part:

Dtv− ∂yu = 0,

Dtu + ∂yπ = 0,

Dtπ + a2∂yu =1

ε(p(v)− π).




∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + p(v)) = 0.

where v = 1/ρ.


⊲ Switch to Eulerian coordinates:

∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + π) = 0,

∂t(ρπ) + ∂x(ρuπ + a2u) =1

ερ (p(v)− π).



Relaxation approximation of the Euler equations:

(E )

∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + p(v)) = 0.

ε→0←−−

(R)

∂tρ + ∂x(ρu) = 0,

∂t(ρu) + ∂x(ρu2 + π) = 0,

∂t(ρπ) + ∂x(ρuπ + a2u) = 1ε ρ (p(v)− π).

⊲ Stability condition (Whitham) :

a2> sup

v|p′(v)|.

⊲ The relaxation system is linearly degenerate (but not linear),all the waves are contact discontinuities =⇒ easy to solve.

Suliciu, Coquel et al, Bouchut...


Rewriting the relaxation system

A new pressure law can be derived from the relaxation system (R) :

(R)

Dtv− ∂yu = 0

Dtu + ∂yπ = 0

Dtπ + a2∂yu = 0

⊲ Introduce a new variable T , such that

DtT = 0

⊲ Consider the new pressure law π = π(T , v). Then, the 3rd eq of (R) gives

∂vπ(T , v) = −a2

=⇒ π(T , v) = −a2v + f (T )

⊲ We require π(v, v) = p(v), then a simple choice is

π(T , v) = p(T ) + a2(T − v)


Rewriting the relaxation system

The previous relaxation system

(R)

Dtv− ∂yu = 0

Dtu + ∂yπ = 0

Dtπ + a2∂yu = 1ε (p(v)− π)

becomes

(R)

Dtv− ∂yu = 0

Dtu + ∂yπ(T , v) = 0

DtT ≃1ε (v−T )

with π(T , v) = p(T ) + a2(T − v). In Eulerian coordinates

(R)

∂tρ + ∂x(ρu) = 0

∂t(ρu) + ∂x(ρu2 + π) = 0

∂t(ρT ) + ∂x(ρuT ) = 1ε (1− ρT )


Fluid systems

Class of systems of conservation laws introduced by Després

If a hyperbolic system of conservation laws satisfies

⊲ The entropy s verifies Dts = 0 in Lagrangian coordinates

⊲ The unknowns are (v, u, e) ∈ Rn−1−d ×R

d ×R and e = |u|2/2 + ǫ(v, s)

⊲ Invariance by Galilean tranformations

⊲ Reversibility for smooth solutions

then, it can be written in Lagrangian coordinates under the form

Dtv− N∂yu = 0

Dtu− NT∂y[∂ve] = 0

Dts = 0

where N is a rectangular (n− 1− d)× d constant matrix.


Relaxation approximation of fluid systems (isentropic case)

In the isentropic case, ∂ve(v, u) = ǫ′(v). The fluid system becomes

Dtv− N∂yu = 0

Dtu− NT∂y[ǫ′(v)] = 0

Same structure as the gas dynamics equations !



A isentropic fluid system writes

Dtv− N∂yu = 0


[ǫ′(v)] : nonlinear term



A isentropic fluid system writes

Dtv− N∂yu = 0


the relaxation approximation of this system writes

Dtv− N∂yu = 0

Dtu− NT∂y[π(T , v)] = 0

DtT = 1ε (v−T )

with the pressure law

π(T , v) = ǫ′(T ) + θ′(v−T )

where θ : Rn−1−d 7→ R is a quadratic function (θ′′ is a constant matrix).



The relaxation approximation of isentropic fluid systems writes

Dtv− N∂yu = 0

Dtu− NT∂yπ(T , v) = 0

DtT = 1ε (v−T )

with the pressure law

π(T , v) = ǫ′(T ) + θ′(v−T )

The stability condition (Whitham) is now

the matrix θ′′ − ǫ′′(v) is positive definite

(for Euler equations: a2> supv |p

′(v)|)


Relaxation approximation of fluid systems

Under the Whitham stability condition, we have:

Theorem.

The RHS of the relaxation approximation of fluid systems is such that:

⊲ The system is hyperbolic

⊲ All the fields are linearly degenerate=⇒ Global well-posedness for BV smooth and nonsmooth solutions

Theorem. [Yong, Dressel-Yong]

The relaxation approximation of fluid systems satisfies:

⊲ Global existence of smooth solutions for initial data near the equilibrium

⊲ Convergence towards the equilibrium for smooth solutions

⊲ Existence of traveling wave solutions

More results for nonsmooth solutions ? [Chen-Levermore-Liu, Serre...]


Numerical methods

Finite volume schemes for ∂tU + ∂x F(U) = 0:

Un+1i = Un

i −∆t

∆x

(

F (Uni , Un

i+1)−F (Uni−1, Un

i ))

where F (Ul, Ur) approximates the flux between Ul and Ur.

Godunov-type schemes by Harten-Lax-Van Leer:

Un+1i =

1

∆x

(

∫

∆x/2

0U (x/∆t, Un

i−1, Uni ) dx +

∫ 0

−∆x/2U (x/∆t, Un

i , Uni+1) dx

)

where U (x/t; Ul , Ur) is an approximate Riemann solver.

−→ Consistency, conservation and entropy properties.


Relaxation Riemann solver

Hyperbolic fluid systems

(E ) ∂tU + ∂x F(U) = 0 (R) ∂tW + ∂xG(W) =1

εR(W)

Define L and M(.) such that

U = L W, L M(U) = U, L G(M(U)) = F(U).

Proposition.

LetW be the exact Riemann solver of (R) . Then under the Whitham condition,

LW(x/t; M(Ul), M(Ur))

is an approximate Riemann solver for (E ) in the sense of Harten-Lax-Van Leer.

−→ Consistent, conservative and entropy satisfying numerical schemes.


Properties of relaxation schemes

Relaxation schemes

⊲ Application to hyperbolic fluid systems (Després)⊲ Gas dynamics equations (≡ HLLC, cf Bouchut)⊲ One-velocity one-pressure multifluid models⊲ Multi-temperature models⊲ Ideal MHD⊲ ...

⊲ Based on approximate Riemann solvers (Harten-Lax-Van Leer)

⊲ Only Riemann problems with contact discontinuities to solve

⊲ Consistent, conservative and entropy satisfying numerical schemes

⊲ Positivity preserving schemes (domain invariance by relaxation)

⊲ Finite volume interpretation for multiD applications


Numerical tests

⊲ Cavitation

⊲ Isothermal model of phase transition⊲ A box immersed in a liquid with high velocity⊲ Apparition of bubbles of vapor

⊲ Shallow water with topography

⊲ Topography dealt by the hydrostatic reconstruction [Audusse et al. 04]⊲ Difficult phenomenon of transition between dry and wet area⊲ Positivity of water height⇐⇒ conservation of mass


Conclusion

⊲ A new relaxation approximation

⊲ Application to hyperbolic fluid systems⊲ Linearly degenerate system⊲ ≡ diffusion for smooth solutions

⊲ New numerical schemes

⊲ Application to hyperbolic fluid systems⊲ Formalism of Harten-Lax-Van Leer⊲ Riemann solvers with only linearly degenerate fields⊲ Positive and entropy satisfying numerical schemes


Documents

Relaxation approximation for hyperbolic fluid systems