1. No category

How to compute small-scale dependent shock waves.
The role of the kinetic relation
Philippe G. LeFloch
University of Paris 6 & CNRS
http://philippelefloch.wordpress.com
1. INTRODUCTION
Hyperbolic or hyperbolic-elliptic systems of conservation laws with
singular perturbation (diffusion, dispersion, etc)
ε
utε + f (u ε )x = R(εuxε , ε2 uxx
, . . .)x .
I
I
Characterize the limit u := limε→0 u ε . Admissible discontinuities ?
ε
Second-order: ε uxx
. Entropy conditions.
Lax, Oleinik, Volpert, Kruzkov, Dafermos, Wendroff, TP Liu.
I
ε
Add also a third order: α ε2 uxxx
, and even higher-order.
Oscillations near shocks and competition between small scales.
I
Classical compressive + nonclassical undercompressive shocks (or
subsonic phase boundaries).
......... Entropy inequality supplemented with a kinetic relation.
PHYSICAL MODELS
I
Continuum physics of complex flows.
Van der Waals fluid, phase transitions, Nonlinear elasticity, thin
liquid film, generalized Camassa-Holm, MHD with Hall effect.
MATHEMATICAL THEORY
I Nonclassical Riemann solver with entropy-compatible kinetics
I Kinetic functions determined by traveling waves
I Existence via Glimm-type scheme with generalized TV functionals
I Zero diffusion-dispersion limits
NUMERICAL APPROXIMATION
I Schemes with controled dissipation
(finite differences, entropy conservative, equivalent equations)
I Computing kinetic functions
GENERALIZATION
I
I
I
The case of two inflection points
Riemann solver with kinetics and nucleation
(splitting-merging phenomena)
DLM theory – Kinetic relations for nonconservative systems
(Dal Maso-LeFloch-Murat and LeFloch-TP Liu)
Typical model. Materials undergoing phase transitions
w t − vx = 0
vt − σ(w )x = ε vxx − α ε2 wxxx
v : velocity
ε : viscosity
w > −1 : deformation gradient
α ε2 : capillarity
σ(w ) : stress
I
Slemrod (1984, etc): self-similar solutions
I
Shearer (1986, etc.): Riemann problem with α = 0.
I
Truskinovsky (1987, etc)
I
Abeyaratne & Knowles (1990, etc): Riemann problem
I
LeFloch (ARMA 1993, etc): mathematical formulation in BV, and Cauchy
problem via Glimm scheme.
Active research on undercompressive shocks.
I
Collaborators. Bedjaoui, Piccoli, Shearer, Joseph, Mohamadian, Laforest,
Mishra.
(Former) postdocs and students. Hayes, Kondo, Baiti, Rohde, Mercier,
Correia, Thanh, Chalons, Boutin.
I
Related works (one- and multi-dimensional stability, thin films, traffic
flows).
T.-P. Liu, Kulikovskij, Marchesin, Plohr, H.T. Fan, Benzoni,
Metivier-Williams-Zumbrun, Colombo, Bertozzi-Shearer, Corli-Tougeron.
2. PHYSICAL MODELS
Approximation by diffusion.
I
Conservation law:
ut + f (u)x = ε b(u) ux
I
U : R → R,
F (u) :=
Ru
x
with b > 0.
f 0 (v ) U 0 (v ) dv
U(u)t + F (u)x = −ε b(u) U 00 (u) |ux |2 + Cx .
´
`
C := ε b(u) U(u)x x
I
Entropy inequalities:
If U is convex, the formal limit u = limε→0 u ε satisfies
U(u)t + F (u)x ≤ 0.
I
Classical entropy solutions: entropy criterion given by Kruzkov
“
”
|u − k|t + sgn(u − k)(f (u) − f (k)) ≤ 0,
x
Also equivalent to a pointwise version given by Oleinik.
k ∈ R.
Model with linear diffusion and dispersion.
I
Conservation law (with β > 0)
ut + f (u)x = β uxx + γ uxxx .
(Shearer et al., Hayes-PLF, Bedjaoui-PLF)
I
Single entropy inequality:
`
´
u 2 /2 t + F (u)x = −D + Cx ,
D := β |ux |2 ≥ 0,
`
´
C := β uux + γ u uxx − (1/2)ux2 .
` 2 ´
In the (formal) limit β, γ → 0 one gets
u /2 t + F (u)x ≤ 0,
but no sign for general convex entropies!
ut + f (u)x = β uxx + γ uxxx .
Classical/nonclassical solutions:
I
γ << β 2 (dominant diffusion):
I
γ >> β 2 (dominant dispersion):
high oscillations, weak convergence (Lax, Levermore)
I
γ := κ β 2 (balanced regime):
strong convergence, mild oscillations, nonclassical, depend on κ
classical entropy solutions
ut + (u 3 )x = 0
5
5
4
4
3
3
2
2
1
1
0
0
-1
-2
-1
-3
-2
-4
-3
-4
-5
0
0.5
1
1.5
2
2.5
Two shocks
3
3.5
4
4.5
5
,
-6
0
0.5
1
1.5
2
2.5
3
3.5
4
A shock + A rarefaction
The Riemann solutions are distinct from the ones selected by Oleinik’s
entropy inequalities.
4.5
5
Thin liquid film model.
I
Conservation law (with β, γ > 0)
ut + (u 2 − u 3 )x = −γ (u 3 uxxx )x
Effect of surface tension
(Bertozzi-Shearer, Zumbrun, Otto-Westdickenberg, PLF-Mohamadian)
I
Single entropy inequality:
u log u − u
t
+ F (u)x = −D + Cx ,
with D = γ |(u 2 ux )x |2 ≥ 0, thus in the limit γ → 0
u log u − u t + F (u)x ≤ 0.
0.8
0.75
0.7
0.65
0.6
0.55
0.5
u
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
250
500
x
750
Oscillations concentrated near shocks.
Generalized Camassa-Holm model.
I Conservation law (with β > 0)
ut + f (u)x = β uxx + γ utxx + 2ux uxx + u uxxx
Shallow water model for wave breaking
(Bressan-Constantin, Karlsen-Coclite, Raynaud, PLF-Mohamadian)
I
Single entropy inequality:
`
´
(u 2 + β |ux |2 )/2 t + F (u)x = − β |ux |2 + Cx .
Here, numerical solutions are similar to, but do not coincide with, the
ones obtained with the linear diffusion-dispersion model.
Limiting solutions depend on the regularization.
Van der Waals fluids.
I
Two conservation laws (also the energy equation may be included):
vt − ux = 0
ut + p(v )x = β(v ) ux
v : specific volume
β(v ): viscosity
p(v , T ) = vRT
−
−b
type otherwise.
I
a
.
v2
x
+ γ 0 (v )
vx2
− γ(v ) vx x x
2
u : velocity
γ(v ) : capillarity
Hyperbolic when T sufficiently large, but of mixed
Single entropy inequality:
ε(v ) +
vy2 u2
+ γ(v )
+ p(v ) u x = − β(v ) ux2 + Cx .
2
2 t
Nonclassical behavior for Van der waals fluids
3.5
0.2
lambda=1e-5
lambda=1e-1
lambda=0.75
lambda=1e-5
lambda=1e-1
lambda=0.75
0
3
-0.2
2.5
-0.4
-0.6
2
-0.8
1.5
-1
-1.2
1
-1.4
0.5
-0.6
-0.4
-0.2
0
0.2
0.4
Specific volume v
0.6
,
-1.6
-0.6
-0.4
-0.2
0
0.2
Velocity u
Solutions depend upon the ratio λ = (viscosity)2 /capillarity.
0.4
0.6
Ideal magnetohydrodynamics with Hall effect.
(simplified version)
vt + ((v 2 + w 2 ) v )x = ε vxx + α wxx
wt + ((v 2 + w 2 ) w )x = ε wxx − α vxx
(v , w ): transverse components of the magnetic field.
ε: magnetic resistivity, α: Hall parameter (solar wind).
(1/2) v 2 + w 2 t + (3/4) (v 2 + w 2 )2 x = − ε (vx2 + wx2 ) + Cx .
When α = 0:
Brio, Hunter, Freist¨
uhler, Pitman, Panov, Wu, Kennel.
PLF-Mishra — Radius variable r = (v 2 + w 2 )1/2 .
When α 6= 0:
5
4
3
2
EC2:−−−−−−−−−−−−−−−−−−−−−−−−−−−
1
EC4:− − −
− −
EC6:−
−
−
EC8:o o o
o
−
−
o
−
−
o
−
−
o
− −
−
o
0
EC10:+ + + + + + + + +
−1
−2
−3
0
0.1
0.2
0.3
0.4
0.5
0.6
Solutions depend on the (order of the) scheme.
0.7
0.8
0.9
1
Further regularizations and models.
I
Buckley-Leverett equation for two-phase flows in porous media
(Hayes-Shearer, van Duijn, Peletier, Pop, Y. Wang)
I
Quantum hydrodynamics
I
Phase field models
Suliciu model
I
(Marcati, Jerome)
(Tzavaras, Bouchut, Frid).
I
Non-local, integral and fractional regularization terms
Kissling, Karlsen, PLF).
I
Discrete molecular models
(Rohde,
(Truskinovsky, Weinan E, etc).
FOR ALL THESE MODELS
I
Complex wave patterns
I
Different ratio/regularizations/schemes yield different solutions.
I
Non-convex flux-functions. A single entropy inequality.
FOR CONVEX FLUX-FUNCTIONS
I
One entropy is sufficient (Panov; Delellis, Otto, Westdickenberg)
I
Shocks are regularization-independent.
I
Classical entropy solutions (compressive shocks, Lax inequalities).
WHAT WE NEED
I
Include macro-scale effects without resolving the small-scales.
I
Encompass nonclassical entropy solutions (containing
undercompressive shocks having fewer impinging characteristics).
No “universal” admissibility criterion
but “several hyperbolic theories”,
each being determined by specifying a physical regularization.
.................. KINETIC RELATION
3. NONCLASSICAL RIEMANN SOLVER
For simplicity in the presentation, consider
ut + f (u)x = 0
I
Concave-convex flux
u f 00 (u) > 0
f 000 (0) 6= 0,
I
for u 6= 0
lim f 0 (u) = +∞
u→±∞
Tangent function ϕ\ : R → R and its inverse ϕ−\
´
`
f (u) − f ϕ\ (u)
f (ϕ (u)) =
,
u − ϕ\ (u)
0
\
ul
! (u )
l
!N(ul)
! (ul )
! (u )
l
u 6= 0
Weak solutions.
I
u ∈ L∞ in the sense of distributions
ZZ u ϕt + f (u) ϕx dxdt = 0
for every smooth, compactly supported function ϕ.
I
If u ∈ BVloc ∩ L∞ (bounded variation), then ut and ux are locally
bounded measures and
ut + f (u)x = 0
as an equality between measures.
Shock wave solutions. For (u− , u+ ) ∈ R2 , consider
(
u− ,
x < λ t,
u(t, x) =
u+ ,
x > λ t.
The Rankine-Hugoniot relation
−λ (u+ − u− ) + f (u+ ) − f (u− ) = 0
determines the shock speed
λ=
f (u− ) − f (u+ )
=: a(u− , u+ ).
u− − u+
Oleinik entropy inequalities for shocks.
Recalled here for the sake of comparison only:
f (u+ ) − f (u− )
f (v ) − f (u+ )
≤
v − u+
u+ − u−
for all v between u− and u+ .
I
Equivalent to the Kruzkov’s entropy inequalities.
I
Equivalent also to imposing all of the entropy inequalities
U(u)t + F (u)x ≤ 0,
U 00 > 0,
F 0 (u) := f 0 (u) U 0 (u).
A single entropy inequality.
U(u)t + F (u)x ≤ 0,
E (u− , u+ ) := −
U 00 > 0,
F 0 (u) := f 0 (u) U 0 (u)
f (u− ) − f (u+ )
U(u+ ) − U(u− ) + F (u+ ) − F (u− )
u− − u+
≤0
Observe that
Z
u+
E (u− , u+ ) = −
u−
U 00 (v ) (v −u− )
f (v ) − f (u ) f (u ) − f (u ) −
+
−
−
dv .
v − u−
u+ − u−
Zero entropy dissipation function ϕ[0 : R 7→ R.
E (u, ϕ[0 (u)) = 0,
(ϕ[0
◦
ϕ[0 )(u)
ϕ[0 (u) 6= u
( when u 6= 0)
= u.
This follows from (for u− 6= u+ ):
∂u+ E (u− , u+ ) = b(u− , u+ ) ∂u+ a(u− , u+ ),
b(u− , u+ ) := U(u− ) − U(u+ ) − U 0 (u+ ) (u− − u+ ) > 0,
∂u+ a(u− , u+ ) =
f 0 (u+ ) − a(u− , u+ )
.
u+ − u−
(
ul ,
u(x, 0) =
ur ,
Riemann problem.
A single entropy inequality allows for:
I
Classical compressive shocks
u− > 0,
ϕ\ (u− ) ≤ u+ ≤ u−
satisfying Lax shock inequalities
f 0 (u− ) ≥
f (u+ ) − f (u− )
≥ f 0 (u− ).
u+ − u−
x <0
x >0
I
Nonclassical undercompressive shocks
u− > 0,
ϕ[0 (u− ) ≤ u+ ≤ ϕ\ (u− ),
having all characteristics passing through
f (u ) − f (u )
+
−
.
min f 0 (u− ), f 0 (u− ) ≥
u+ − u−
The cord connecting u− to u+ intersects the graph of f .
I
Rarefaction waves.
Lipschitz continuous solutions u depending only upon ξ := x/t:
−ξ u(ξ)ξ + f (u(ξ))ξ = 0,
thus
u(t, x) := (f 0 )−1 (x/t),
provided f 0 is invertible on the interval under consideration.
Precisely, a rarefaction consists of two constant states separated by a
smooth part:


x < t f 0 (u− ),
u− ,
0
−1
u(t, x) = (f ) (x/t), t f 0 (u− ) < x < t f 0 (u− ),


u+ ,
x > t f 0 (u+ ),
provided
f 0 (u− ) < f 0 (u+ )
and
f 0 is strictly monotone on the interval limited by u− and u+ .
Evolutionary vs. non-evolutionary.
I
Compressive shocks arise from smooth initial data.
For instance, using the method of characteristics
`
´
u(t, x) = u 0, x − t f 0 (u(t, x)) ,
one sees that the implicit function theorem generally fails t is too large.
I
Compressive shocks arise also from singular limits.
For instance
ut + f (u)x = ε uxx .
I
Undercompressive shocks arise from singular limits only.
Provided oscillations take place, for instance with
ut + f (u)x = ε uxx + αε2 uxxx .
The Riemann problem admits (up to) a one-parameter family of solutions
satisfying a single entropy inequality.
One can combine an arbitrary nonclassical shock plus a classical one.
An additional admissibility criterion is needed.
Entropy-compatible kinetic function.
I A monotone decreasing, Lipschitz continuous function ϕ[ : R 7→ R
ϕ[0 (u) < ϕ[ (u) ≤ ϕ\ (u),
I
0
u>0
The kinetic relation u+ = ϕ[ (u− ) singles out one nonclassical shock.
Example. f (u) = u 3 , ϕ\ (u) = −u/2, ϕ[0 (u) = −u.
2nd order scheme
4th order scheme
classical solution
TW solution
extreme nonclasssical solution
-2
ul
! (u )
l
-4
!N(ul)
-6
! (ul)
-8
! (u )
l
-10
-12
-14
N+R
-16
2
4
6
8
10
12
14
,
N+C
C
R
ul
ur
Observe that:
I
Extremal choices:
ϕ[ = ϕ\
ϕ[ = ϕ[0
(classical solution, all convex entropies)
(dissipation-free, one entropy equality)
I
Equivalently, prescribe the entropy dissipation rate.
I
The property (ϕ[0 ◦ ϕ[0 )(u) = u implies the contraction property
|ϕ[ ϕ[ (u) | < |u|, u 6= 0.
Notation: Companion (threshold) function ϕ] : R → R
Nonclassical Riemann solver.
For instance, suppose ul > 0.
I
ur ≥ ul : rarefaction wave.
I
ur ∈ [ϕ] (ul ), ul ): classical shock.
I
ur ∈ (ϕ[ (ul ), ϕ] (ul )): nonclassical shock ul , ϕ[ (ul ) + classical
shock ϕ[ (ul ), ur .
I
ur ≤ ϕ[ (ul ) : nonclassical shock ul , ϕ[ (ul ) + rarefaction wave.
1
1
u
u
0.8
0.5
0.6
0.4
0
0.2
-0.5
0
-0.2
-1
-0.4
-1.5
-0.6
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-2
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Conclusion.
Given a kinetic function ϕ[ compatible with an entropy, the Riemann
problem admits a unique solution, satisfying :
I hyperbolic conservation law with Riemann initial data
I
single entropy inequality plus a kinetic relation u+ = ϕ[ (u− )
L1 continuous dependence property.
ku(t) − v (t)kL1 (K ) ≤ C (T , K ) ku(0) − v (0)kL1 (K )
for all t ∈ [0, T ] and all compact set K ⊂ R.
But, the solutions contain “spikes” – some states are discontinuous.
Generalization (piecewise smooth solutions to the Riemann problem).
I
2 × 2 isentropic Euler equations and nonlinear elasticity or phase
transition system
– uniqueness if hyperbolic
– non-uniqueness if hyperbolic-elliptic
(Slemrod, Truskinovsky, Shearer et al., PLF-Thanh,
Hattori, Mercier-Piccoli, Corli-Tougeron)
I
N × N strictly hyperbolic systems of conservation laws.
Hayes-PLF (SIAM J. Math. Anal., 2000).
4. KINETIC FUNCTIONS ASSOCIATED WITH TRAVELING WAVES
TW analysis. For simplicity in the presentation, consider
ut + f (u)x = β |ux |p ux x + uxxx
f concave-convex, β > 0,
p ≥ 0.
Second-order ODE:
u(x, t) = u(y ),
y = x − λt
− λ (u − u− ) + f (u) − f (u− ) = β |u 0 |p u 0 + u 00
with boundary conditions
lim u(y ) = u±
y →±∞
and prescribed data u± , λ satisfying the Rankine-Hugoniot relation.
Remark. The small parameter ε has been removed by scaling. The
proper scaling is ε2 for the diffusion and ε for the dispersion.
Main issues. (LeFloch-Bedjaoui, 2001)
I
existence of classical / nonclassical traveling waves
I
Kinetic function ϕ[ associated to this model ?
I
Monotonicity ?
I
Behavior near u = 0 ?
Earlier result for the cubic flux. Explicit formulas
I
p=0:
Shearer et al. (1995)
I
p=1:
Hayes - PLF (1997).
0
Here, ϕ[ (0) = ϕ[0 (0) = −1.
Traveling wave analysis.
I
Phase plane analysis:
I
First-order system in the plane (u, u 0 ).
I
The equilibria are the solutions u1 < u2 < u3 = u− to
−λ (u − u− ) + f (u) − f (u− ) = 0.
For λ and u− fixed, there exist two non-trivial solutions.
I
Equilibria are saddle points (two real eigenvalues with opposite signs)
or nodes (two eigenvalues with same sign).
I
Existence of:
saddle-node connections from u− to u2 (classical shocks)
and
saddle-saddle connections from u− to u1 (nonclassical shocks).
I
Dependence upon u− , λ, β, p.
Admissible shocks
¯
˘
S(u− ) := u+ / there exists a TW connecting u±
Theorem. (Bedjaoui - PLF, 2001 & 2004).
(i) Kinetic function ϕ[ : R → R, Lipschitz continuous, strictly decreasing,
S(u) = ϕ[ (u) ∪ ϕ] (u), u ,
u>0
ϕ[0 (u) < ϕ[ (u) ≤ ϕ\ (u),
u>0
(ii) Threshold function A\ such that
I 0 ≤ p ≤ 1/3 :
A\ : R → [0, ∞) Lipschitz continuous,
ϕ[ (u) = ϕ\ (u)
I
iff β ≥ A\ (u)
p > 1/3 :
ϕ[ (u) 6= ϕ\ (u)
(u 6= 0)
A\ (0) = 0
(iii) Behavior of infinitesimally small shocks:
I
I
0
0
ϕ[ (0) = ϕ\ (0) = −1/2
A\ (0) = 0,
p = 0:
0
0 < p ≤ 1/3 :
A\ (0) = 0,
I
1/3 < p < 1/2 :
I
p = 1/2 :
0
ϕ[ (0) = −1/2
0
lim ϕ[ (0) = −1,
p > 1/2 :
ϕ[ (0) = −1/2
0
A\ (0±) = +∞
0
0
ϕ[ (0) ∈ ϕ−[
0 (0), −1/2 = (−1, −1/2)
β→0+
I
0
A\ (0±) 6= 0
0
lim ϕ[ (0) = −1/2
β→+∞
0
ϕ[ (0) = −1
Remark. Useful in the BV existence theory for the initial value problem.
Remark.
Explicit formula for the cubic flux f (u) = u 3 .
Recall that ϕ\ (u) = −u/2, ϕ[0 (u) = −u.
I
p = 1/2 : Linear kinetic function
ϕ[ (u) = −cβ u,
cβ ∈ (1/2, 1).
Conclusion.
To the augmented model one can associate a unique kinetic function
which is monotone and satisfies all the assumptions required in the
theory of the Riemann problem.
Generalizations.
I
2 × 2 Nonlinear elasticity/Euler equations (non-nec. monotone)
I
2 × 2 Van de Waals model (two inflection points) (multiple solutions)
(PLF-Bedjaoui, Truskinovsky, Benzoni, Shearer)
(Bedjaoui-Chalons-Coquel-PLF).
Partial results (on traveling waves).
I
Thin liquid film model
(Bertozzi, Shearer, M¨
unch).
I
Generalized Camassa-Holm model
(Constantin, Strauss, Lenells).
5. EXISTENCE THEORY for NONCLASSICAL ENTROPY SOLUTIONS
For simplicity, consider the conservation law with concave-convex flux
∂t u + ∂x f (u) = 0
u(x, 0) = u0 (x)
u0 ∈ BV (R),
f concave-convex.
That is, u0,x is a bounded measure and the total variation TV (u0 ) is the
mass of this measure.
Nonclassical Rieman solver based on a kinetic function ϕ[
Dafermos front tracking method.
Piecewise constant approximations
u h : R+ × R → R.
I
u0 replaced by a piecewise constant approximation.
I
At t = 0, solve a Riemann problem at each jump point.
I
Replace rarefaction waves by several small fronts, traveling with the
Rankine-Hugoniot speed.
I
Solve a new Riemann problem at each wave interaction.
Convergence ?
Main difficulties:
I
Show that the number of waves and interaction points remain finite.
(For scalar equations, at most two waves in each Riemann solution.)
I
Lack of monotonicity and possible increase of the (standard) total
variation.
I
For systems, lack of regularity of the wave curves.
Assumptions on the kinetic function.
I
ϕ[ : R → R: Lipschitz continuous, monotone decreasing.
I
The second iterate of ϕ[ is a strict contraction: for K ∈ (0, 1)
[
ϕ ◦ ϕ[ (u) ≤ K |u|,
u 6= 0.
Remark.
Since
[
ϕ ◦ ϕ[ (u) < |u|
for u 6= 0, this is only a condition at u = 0, about nonclassical shocks
with infinitesimally small strength.
Notion of generalized wave strength (Laforest-LeFloch, 2009).
(
u,
u > 0,
σ(u− , u+ ) = |ψ(u− ) − ψ(u+ )|,
ψ(u) =
[
ϕ0 (u), u < 0.
Properties.
I
Compare states with the same sign.
I
“Equivalence” with the standard strength:
C |u− − u+ | ≤ σ(u− , u+ ) ≤ C |u− − u+ |.
I
Continuity as u+ crosses ϕ] (u− ) during a transition from a single
crossing shock to a two wave pattern:
σ(u− , ϕ] (u− )) = u− − ϕ[0 ◦ ϕ] (u− )
= u− − ϕ[0 ◦ ϕ[ (u− ) + ϕ[0 ◦ ϕ[ (u− ) − ϕ[0 ◦ ϕ] (u− )
= σ(u− , ϕ[ (u− )) + σ(ϕ[ (u− ), ϕ] (u− )).
Generalized TV functional.
For a piecewise constant function u = u(t, ·) made of shock or
α
α
rarefaction fronts (u−
, u+
), the generalized total variation
X
α
α
V u(t) :=
σ(u−
, u+
)
α
is “equivalent” to the total variation
X
α
α
u−
− u+
.
TV u(t) :=
α
Classification of wave interaction patterns.
When two fronts (ul , um ) and (um , ur ) meet........................... 20 cases.
0
0
↓ ↓
↓
↑
Case CR-4. (C±
R− )–(N±
C−
)
Monotone to non-monotone:
ϕ[ (ul ) < ur < ϕ] (ul ) < um ≤ 0 < ul .
The standard total variation TV u h (t) increases, but V u h (t)
decreases.
0
↓ ↑
Case NC. (N±
C )–(C ↓ )
Non-monotone to monotone:
um = ϕ[ (ul ) and ϕ] (ul ) < ur < ϕ] (um ) < ul .
Both the standard total variation TV u h (t) and the generalized one
V u h (t) decrease.
Proposition (Laforest - PLF, 2009). The generalized total variation
functional V = V u h (t) is non-increasing along a sequence of front
tracking approximations.
More precisely, at each interaction

in
Cases RC-1, RC-3, CR-1, CR-2, CR-4,

−2 σ(R ),


−C σ(R in ),
Cases RC-2, RN,
V ≤
in
out
−2 σ(R ) − σ(R ) , Case CR-3,



0,
all other cases,
where R in and R out denote the incoming/outgoing rarefactions, and
Lip (ϕ[0 )−1 ◦ (id − ϕ[ )−1
C :=
.
Lip(ϕ[0 )
Theorem [Existence of nonclassical entropy solutions, Baiti-PLF-Piccoli
(2001), Laforest -PLF (2009)].
ku h (t)kL∞ (R) . ku0 kL∞ (R) ,
TV (u h (t)) . TV (u0 ),
ku h (t) − u h (s)kL1 (R) . |t − s|,
and u h converge in L1 to a weak solution satisfying the entropy inequality
and the kinetic relation.
Remarks.
I
Compactness follows from Helly’s compactness theorem.
I
Behavior near u = 0 important to prevent blow-up of the TV.
I
Assumption ϕ[ (0−) ϕ[ (0+) < 1, satisfied by kinetic functions generated
by nonlinear diffusion-dispersion
`
´
`
´
α b(u, ux ) |ux |p ux x + c1 (u) (c2 (u) ux )x x
0
0
provided p < 1/2.
I
Counter-example of blow-up of TV if ϕ[ (0) = −1.
Alternative strategy of proof (Baiti-PLF-Piccoli, 2001).
However:
I
Applies only to scalar equations.
I
Requires stronger conditions on the kinetic function.
Technique:
I
Decomposition into intervals where u h (t, ·) is alternatively
increasing/decreasing.
I
Use Fillipov-Dafermos’s theory of generalized characteristics to track
the maxima and minima.
I
Compute the total variation.
Wave interaction potential.
Q u(t) :=
X
β
β
α
α
σ(u−
, u+
) σ(u−
, u+
),
α,β approaching
provided not two of them being rarefactions.
2
It is bounded by C TV u(t) .
Remark. For classical entropy solutions, t 7→ Q(u h (t)) decreasing along a
sequence of front-tracking solutions.
Theorem (Nonclassical entropy solutions).
1. t 7→ Q(u h (t)) decreasing along a sequence of front-tracking solutions,
except at interactions ... → N + ...
2. The potential Q(u h (t)) is globally decreasing in the case of a
“splitting-merging” pattern (C → N + C , but later N + C → C ).
Generalization.
I
Kinetics and nucleation:
I
Hyperbolic systems:
PLF-Shearer (2005).
I
Perturbation of a nonclassical wave
PLF (1993), Corli-Tougeron (2002), Colombo-Corli (2002), Hattori
(2003), Laforest-PLF (2009).
I
The new functional opens the way to further investigations.
Uniqueness.
I
Classical setting:
I
Nonclassical setting:
Bressan-LeFloch (1997, tame variation),
extended by Bressan with Goatin and Lewicka.
Baiti-PLF-Piccoli (JDE, 2001)
L1 continuous dependence.
I
Classical setting :
I
Nonclassical setting:
Bressan et al., LeFloch et al., Liu-Yang.
open problem.
Further reading. Lect. in Math., ETH Z¨
urich, Birkh¨auser.
Download at http://www.ann.jussieu.fr/elefloch
6. ZERO DIFFUSION-DISPERSION LIMITS
I
Tartar’s compensated compactness method.
I
1D scalar equations:
I
2 × 2 elasticity system:
Camassa-Holm equation :
I
I
I
Schonbek
Hayes - PLF
PLF - Natalini
Hayes - PLF
Coclite - Karlsen
(1982)
(1997)
(1999)
(2000)
(2006)
DiPerna’s measure-valued solutions.
I
Multidimensional conservation laws:
Correia - PLF (1999)
Kondo - PLF (2001)
Holden - Karlsen - Mitrovic (2009)
I
With discontinuous flux:
Lions-Perthame-Tadmor’s kinetic formulation.
I
Multidimensional conservation laws:
Hwang - Tzavaras (2002)
Hwang (2004)
Kissling - PLF - Rohde (2009) (non-local regularization)
7. SCHEMES WITH CONTROLED DISSIPATION
For simplicity in the presentation, consider
∂t u + ∂x f (u) = ε uxx + α ε2 uxxx
I
uα : the limit when ε → 0.
I
ϕ[α : the associated kinetic function.
Can we design a scheme converging to uα ?
Glimm scheme and front tracking schemes.
I
Theoretical convergence results (Baiti - PLF - Piccoli)
I
Numerical experiments
Chalons and LeFloch, Interfaces and Free Boundaries (2003).
Level set techniques
I
Hou - PLF - Rosakis (JCP, 1999). Later extended by Merkle Rohde (2006).
I
Nonlinear elasticity model, with trilinear law in two spatial
dimensions.
I
Complex interfacial structure. Needles attached to the boundary.
Combination of differences and interface tracking
I
Hou - PLF- Zhang (JCP, 1996)
I
Boutin, Coquel, Lagoutiere, PLF (2008)
These methods ensure that the interface is sharp and (almost) exactly
propagated.
Finite difference schemes.
Hayes-LeFloch (SINUM, 1998).
I
uα∆x : numerical solution
vα := lim∆x→0 uα∆x : the limit of the scheme.
ψα[ : the numerical kinetic function.
I
Observation:
v α 6= u α ,
ψα[ 6= ϕ[α
Even if the scheme is “conservative”, “consistent”, ”high-order”, etc.
Small scale effects play a critical role in the selection of shocks.
The discrete dissipation 6= discrete dissipation.
Proposed criterion: ψα[ should be a good approximation of ϕ[α .
Schemes with controled dissipation.
I
I
I
High-order accurate, hyperbolic flux
High-order discretization of the augmented terms (diffusion,
dispersion)
Equivalent equation coincide with the augmented physical model, up
to a sufficiently high order of accuracy. For instance, for
∂t u + ∂x f (u) = uxx + α 2 uxxx
we require
∂t u + ∂x f (u) = ∆x uxx + α (∆x)2 uxxx + O(∆x)p ,
for p ≥ 3 at least.
References.
I
Hayes - PLF (SINUM, 1998) : scalar conservation laws
I
PLF - Rohde (SINUM, 2000) : third and fourth order
I
Chalons - PLF (JCP, 2001) : van der Waals fluids
I
PLF - Mohamadian (JCP, 2008) : very high-order schemes
Conjecture on the equivalent equation.
I
[
PLF : As p → ∞ the kinetic function ψα,p
associated with a scheme
with equivalent equation
∂t u + ∂x f (u) = ∆x uxx + α (∆x)2 uxxx + O(∆x)p
converges to the exact kinetic function ϕ[α
[
lim ψα,p
= ϕ[α .
p→∞
I
PLF - Mohamadian (JCP, 2008) : Conjecture established
numerically for several models ! See below.
The role of entropy conservative schemes
I Higher order accurate, entropy conservative, discrete flux.
I Discrete version of the physically relevant entropy inequality. Hence,
preserve exactly (and globally in time) an approximate entropy
balance
Entropy variable.
System of conservation laws endowed with an entropy pair:
∂t u + ∂x f (u) = 0,
∂t U(u) + ∂x F (u) ≤ 0.
I
v (u) = ∇U(u): entropy variable. Suppose that U strictly convex or,
more generally, f (u) is a function of v .
I
Set f (u) = g (v ),
I
B(v ) is symmetric, since Dg (v ) = Df (u)D 2 U(u)−1 . So, there exists
ψ(v ) such that g = ∇ψ. In fact
F (u) = G (v ),
B(v ) = Dg (v ).
ψ(v ) = v · g (v ) − G (v ).
Consider (2p + 1)-point, conservative, semi-discrete schemes
1 ∗
d
∗
uj = −
gj+1/2 − gj−1/2
,
dt
h
I
uj = uj (t) is an approximation of u(xj , t), and h > 0 is the mesh
length
I
The discrete flux
∗
= g ∗ (vj−p+1 , · · · , vj+p ),
gj+1/2
must be consistent with the exact flux g
g ∗ (v , . . . , v ) = g (v ).
vj = ∇U(uj )
Theorem (Second-order, Tadmor, 1984). Two-point numerical flux
∗
Z
1
g (v0 + s (v1 − v0 )) ds,
g (v0 , v1 ) =
v0 , v1 ∈ R N ,
0
where v is the entropy variable associated with a strictly convex entropy.
I
Entropy conservative scheme, satisfying
d
1 ∗
∗
U(uj ) +
Gj+1/2 − Gj−1/2
= 0,
dt
h
with
1
1
G ∗ (v0 , v1 ) = (G (v0 ) + G (v1 )) + (v0 + v1 ) g ∗ (v0 , v1 )
2
2
1
−
v0 · g (v0 ) + v1 · g (v1 ) .
2
I
Second-order accurate, with (conservative) equivalent equation
∂t u + ∂x f (u) =
h2 1
∂x − g (v )xx + vx · ∂x Dg (v ) .
6
2
Theorem. (Third-order, PLF - Rohde, 2000) Given any symmetric N × N
matrices B ∗ (v−p+2 , · · · , vp ), the (2p + 1)-point scheme associated with
∗
Z
g (v−p+1 , · · · , vp ) =
1
g (v0 + s (v1 − v0 )) ds
0
−
1
(v2 − v1 ) · B ∗ (v−p+2 , · · · , vp )
12
− (v0 − v−1 ) · B ∗ (v−p+1 , · · · , vp−1 )
is entropy conservative, with entropy flux
1
G ∗ (v−p+1 , · · · , vp ) = (v0 + v1 ) · g ∗ (v−p+1 , · · · , vp )
2
1 ∗
−
ψ (v−p+2 , · · · , vp )+ψ ∗ (v−p+1 , · · · , vp−1 ) .
2
When p = 2 and B ∗ (v , v , v ) = B(v ) = Dg (v ) , this five-point scheme
is third-order, at least.
Generalization: arbitrarily high order LeFloch-Mercier-Rohde (SINUM,
2002).
8. COMPUTING KINETIC FUNCTIONS. Cubic conservation law
ut + (u 3 )x = ε uxx + α ε2 uxxx
Relatively small α:
Kinetic function
-2
Scaled entropy dissipation
φ(s)/s 2 (versus shock speed s)
Forth order
Sixth order
Eighth order
Tenth order
Exact
-6
-0.1
Scaled entropy dissipation
-4
uM
-8
-10
-12
-14
-16
-18
-0.2
-0.3
-0.4
-0.5
2
4
6
8
10
uL
12
14
16
18
20
Forth order
Sixth order
Eighth order
Tenth order
Exact
100
200
Shock speed
300
Cubic conservation law with (relatively) large diffusion
-2
Forth order
Sixth order
Eighth order
Tenth order
Exact
-6
-0.05
Scaled entropy dissipation
-4
uM
-8
-10
-12
-14
-16
-0.1
-0.15
-0.2
Forth order
Sixth order
Eighth order
Tenth order
Exact
-18
2
4
6
8
10
uL
12
14
16
Kinetic function
18
20
100
200
Shock speed
300
Scaled entropy dissipation
8. COMPUTING KINETIC FUNCTIONS. Camassa-Holm model
ut + (u 3 )x = ε uxx + α ε2 utxx + 2ux uxx + u uxxx
Theory.
I
Well-posedness for the initial-value problem
Bressan, Constantin, Karlsen, Coclite, Raynaud.
I
Kinetic relations via traveling wave analysis:
Numerical investigation
I
Existence of a kinetic function ? Globally monotone ?
I
Relation with the linear diffusive-dispersive model ?
open problem.
Shocks with moderate strength.
Fourth order
Sixth order
Eighth order
Tenth order
Entropy bounds
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2
-1.3
-1.4
-1.5
-1.6
0.5
0.75
1
uL
1.25
1.5
The kinetic functions
for the linear diffusion-dispersion and Camassa-Holm models
essentially coincide for shocks with moderate strength.
Shocks with large strength.
Fourth order
Sixth order
Eighth order
Tenth order
Entropy bounds
-25
-50
-50
-75
-75
-100
-100
-125
-125
-150
-150
-175
-175
-200
50
100
uL
150
200
Linear diffusion-dispersion model
Fourth order
Sixth order
Eighth order
Tenth order
Entropy bounds
-25
-200
50
100
uL
150
200
Camassa-Holm model
8. COMPUTING KINETIC FUNCTIONS. Van der Waals fluids.
Complex wave structure.
Initial data τL = 0.8, τR = 2, uR = 1 with variable left-hand data uL .
2.5
uL=0.2
2.25
uL=0.5
2
uL=.6
uL=0.7
1.75
uL=0.95
!v!
1.5
uL=1.1
1.25
uL=1.3
1
uL=1.4
uL=1.5
0.75
0.5
0
0.25
0.5
x
0.75
Better described... with the kinetic function.
1
Kinetic function.
For τ near to 1: existence and monotonicity.
3.6
3
lambda=0.4
lambda=0.5
lambda=0.7
Maxwell curve
3.4
2.8
Fourth order
Sixth order
Eighth order
Tenth order
3.2
2.6
3
2.4
2.8
u+
!2.2+
2.6
2
2.4
2.2
1.8
2
1.6
1.8
1.6
0.65
1.4
0.82
0.83
0.84
0.85
0.86
!u"
0.87
0.88
0.89
0.9
-
0.655
0.66
0.665
0.67
0.675
Varying the capillarity coefficient
0.68
0.685
Varying the order of the discretization.
8. COMPUTING KINETIC FUNCTIONS.
Magnetohydrodynamics with Hall effect.
LeFloch - Mishra (2009)
Kinetic function
Scaled entropy dissipation
−0.56
0
−0.58
−0.6
−0.62
−5
−0.64
−0.66
EC2−−−−−−−−−−−−−−−−−−−−−−−−−
−0.68
EC2:−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
EC4:−o−o−o−o−o−o−o−o−o−o−o
−10
EC6:−#−#−#−#−#−#−#−#−#−#
EC4:−o−o−o−o−o−o−o−o−o−o−o−o
−0.7
EC6:−#−#−#−#−#−#−#−#−#−#−#
EC8:−x−x−x−x−x−x−x−x−x
EC8:−x−x−x−x−x−x−x−x−x−x
−0.72
EC10:−+−+−+−+−+−+−+
EC10:−+−+−+−+−+−+−+−+
−0.74
−15
2
4
6
8
10
12
14
16
18
−0.76
0
20
,
50
100
150
200
250
300
350
Summary: schemes with controled dissipation.
I
I
No convergence to the analytical solution.
Practically useful schemes based on an analysis of the equivalent
equation.
The numerical kinetic function approaches the exact kinetic function.
I
Kinetic functions exist and are monotone for large classes of
physically relevant models:
I
I
thin liquid films, generalized Camassa-Holm, nonlinear phase
transitions, van der Waals fluids (small shocks),
magnetohydrodynamics.
Computing the kinetic function.
Useful to investigate:
I
I
I
Effects of the diffusion/dispersion ratio, regularization, order of
accuracy of the schemes.
Efficiency of the schemes.
Compare several physical models.
Kinetic functions associated with schemes.
Hayes - LeFloch (1997)
I
I
Beam-Warming scheme (for concave-convex flux) produces
non-classical shocks.
No such shocks observed with the Lax-Wendroff scheme.
All of this depends crucially on the sign of the numerical dispersion
coefficient !
Approximation of nonlinear hyperbolic systems in nonconservative form:
– Hou and PLF, Why nonconservative schemes converge to wrong
solutions. Error analysis, Math. of Comput. (1994).
– Berthon, Coquel, and PLF, unpublished notes (2003).
– Castro, PLF, Munoz-Ruiz, and Pares, J. Comput. Phys. (2008).
http://philippelefloch.wordpress.com
9. THE CASE OF TWO INFLECTION POINTS
Van der Waals fluids with viscosity and capillarity
τt − ux = 0
ut + p(τ )x = α(τ ) |τx |q ux
I
τ : specific volume
α : viscosity/capillarity,
I
Convex/concave/convex pressure law :
p 00 (τ ) ≥ 0,
p 00 (τ ) ≤ 0,
x
− τxxx
u : velocity
q≥0
τ ∈ (0, a) ∪ (c, +∞)
τ ∈ (a, c),
p 0 (a) > 0
2
1.8
1.6
p
I
1.4
1.2
1
0.8
1
2
!
3
Traveling wave analysis.
2-wave issuing from (τ0 , u0 ) at −∞ and with speed λ > 0:
λ (τ − τ0 ) + u − u0 = 0
λ (u − u0 ) − p(τ ) + p(τ0 ) = −α(τ )|τ 0 |q u 0 + τ 00 .
Phase plane analysis in the plane (τ, τ 0 ):
I
Second-order differential equation + an algebraic equation.
I
Fix a left-hand state τ0 and a speed λ within the interval where
there exist three other equilibria τ1 , τ2 , τ3 .
10
Pressure p
Line d with lambda = 0.80
Line d with lambda = 0.85
Line d with lambda = 0.90
8
6
4
2
0
-2
-4
0
2
4
6
8
10
12
Entropy inequality.
e 0 + Fe 0 = −α(τ )|τ 0 |q (u 0 )2 < 0
−λ U
with entropy
Z
e := −
U
τ
p(s) ds +
(τ 0 )2
u2
+
2
2
and entropy flux
Fe := u p(τ ) + λ (τ 0 )2 + u τ 00 − u α(τ ) |τ 0 |q u 0 .
Lemma. (Classification of the equilibrium points.)
I
For all q ≥ 0, the equilibria (τ0 , 0) and (τ2 , 0) are saddle points (two
real eigenvalues with opposite signs).
I
For q = 0 and i = 1, 3, the point (τi , 0) is :
I
I
I
a stable node (two negative eigenvalues) if p 0 (τi ) + λ2 ≤ (λα(τi ))2 /4
a stable spiral (two eigenvalues with the same negative real part and
with opposite sign and non-zero imaginary parts) if
p 0 (τi ) + λ2 > (λα(τi ))2 /4.
For q > 0 the equilibria (τ1 , 0) and (τ3 , 0) are centers (two purely
imaginary eigenvalues).
Theorem. (Bedjaoui, Chalons, Coquel, PLF, 2005). There exists a
decreasing sequence of diffusion/dispersion ratio αn = αn (τ0 , λ) → 0 for
n ≥ 0 such that:
I α(τ0 ) = αn
(nonclassical) TW with n oscillations connecting τ0 to
τ2 .
I
α(τ0 ) ∈ (α2m+2 , α2m+1 ) ∪ (α0 , +∞)
(classical) TW connecting τ0 to τ1 .
I
α(τ0 ) ∈ (α2m+1 , α2m )
(classical) TW connecting τ0 to τ3 .
Remark.
In the case of a single inflection point one would have a single critical
value α0 (τ0 , λ), only.
Nonclassical trajectories. Infinitely many, associated to a sequence
αn → 0:
1.8
Trajectory connecting tau0 and tau2 without oscillation
4
Trajectory connecting tau0 and tau2 with one oscillation (increasing part)
Trajectory connecting tau0 and tau2 with one oscillation (decreasing part)
1.6
3
1.4
2
1.2
1
1
0
0.8
-1
0.6
-2
0.4
-3
0.2
0
0.5
4
1
1.5
2
2.5
3
,
Trajectory connecting tau0 and tau2 with two oscillations (first increasing part)
Trajectory connecting tau0 and tau2 with two oscillations (decreasing part)
Trajectory connecting tau0 and tau2 with two oscillations (second increasing part)
-4
0
2
4
6
14
8
10
12
14
Trajectory connecting tau0 and tau2 without oscillation
Trajectory connecting tau0 and tau2 with one oscillation
Trajectory connecting tau0 and tau2 with two oscillations
3
12
2
10
1
8
0
6
-1
4
-2
2
-3
-4
0
2
4
6
8
10
12
14
,
0
-5
0
5
10
15
20
25
New features found with the van der Waals model.
I
Non-classical trajectories may be non-monotone.
I
Several kinetic functions: u+ = ϕ[α (u− ), that is, the right-hand side
is not unique.
I
Non-uniqueness for the Riemann problem.
10. RIEMANN SOLVER WITH KINETICS AND NUCLEATION
Joint work with M. Shearer (scalar equations) and M. Laforest (systems).
Available shocks after imposing a kinetic relation:
I
All nonclassical shocks (u− , u+ ) satisfy the kinetic relation
u+ = ϕ[ (u− ).
I
All classical shocks : ϕ\ (u− ) ≤ u+ ≤ u− .
Riemann problem. Given data ul , ur with ul > 0 there are actually still
two solutions for every ur < ϕ] (ul ):
I
Classical Riemann solution:
I
I
I
Single shock (if ur > ϕ\ (ul )),
or (right-characteristic) shock plus rarefaction (if ur < ϕ\ (ul )).
Nonclassical Riemann solution:
Undercompressive shock ul , ϕ[ (ul ) and faster wave ϕ[ (ul ), ur ,
which is
I
I
either a classical shock (if ur > ϕ[ (ul ))
or a rarefaction (if ur < ϕ[ (ul )).
Nucleation criterion.
I
Nucleation threshold:
Lipschitz continuous function ϕN satisfying
ϕ\ (ul ) ≤ ϕN (ul ) ≤ ϕ] (ul ) for ul > 0.
I
Impose the condition
If ur ≤ ϕN (ul ), the solution is nonclassical;
it is classical, otherwise.
Remarks.
I
A large initial jump always “nucleates”.
Similar criterion proposed in material science: nucleation in
austenite-martensite materials (Abeyaratne - Knowles)
I
Extremal choices :
ϕN = ϕ\ (fully classical)
ϕN = ϕ] (fully nonclassical)
Nonclassical Riemann solver with kinetics and nucleation.
I
Solution that satisfies the entropy inequality, the kinetic relation,
and the nucleation criterion.
I
Admissible waves:
Waves satisfying the entropy inequality, the kinetic relation and the
nucleation criterion.
Difficulties.
I
I
I
Nucleation solver not uniquely characterized by the set of admissible
waves.
Some Riemann problems have two solutions.
Nucleation solver not continuous in L1 w.r.t. the initial data (unless
ϕN ≡ ϕ] ). At the transition value ur = ϕN (ul ).
Relevance of the Riemann solver with nucleation.
I
Models with second-order diffusion/third-order dispersion.
Lax shocks lose TW profiles precisely when an undercompressive
shock admits a TW, so ϕN = ϕ] .
I
Models for which TW analysis does not lead to a unique Riemann
solver and instabilities and complex large-time behavior are observed.
I
Hyperbolic-elliptic systems modeling phase transitions.
In a range of data, the TW analysis allows for two distinct solutions
containing zero or two propagating phase boundaries.
I
Higher-order regularizations: thin liquid films, Camassa-Holm, etc.
Applications to the thin liquid film model.
I
Make direct numerical simulation of the augmented model with
diffusion and dispersion.
I
The nucleation function could be determined numerically. Find a
range where both classical and nonclassical connections are available.
Work in progress. To what extend the hyperbolic theory with kinetics and
nucleation is valid ?
I
Instabilities (one-wave / two-wave patterns) observed numerically.
I
Difficult to sort out the competition between viscosity, capillarity,
time and space step sizes, perturbation, etc.
I
Hyperbolic model valid in a range of parameters only.
Existence results.
I
I
I
Front tracking scheme applies with the solver with kinetics and
nucleation.
Same TV estimate. Same convergence theory.
Qualitative properties of solutions ? Quite different when the
nucleation criterion is effective.
Observation.
I
I
Instability phenomena.
Limiting front-tracking solution depends on the approximation of the
initial data u0 .
Perturbations of Riemann data.
lim u0 (x) = ul ,
x→−∞
lim u0 (x) = ur = ϕN (ul ).
x→+∞
Two possible asymptotic patterns as t → +∞ :
a single shock or a two-wave solution ?
Example 1. Two possible time-asymptotic behaviors.
(1)
(2)
ur ∈ (ϕN (ul ), ϕ] (ul )), um ∈ (ϕ\ (ul ), ϕN (ul )), um ∈ (ϕN (ul ), ur )
t
t
ul
ur
ul
u
u(1)
m
(2)
um
0
x
x
0
u
(1)
um
ur
f
! (ul)
f
(2)
um
!N(u )
=
!N(u )
l
l
ul
! (u )
l
u
ul
! (u )
l
ur
ur
C-C −→ C
N-C-C −→ N-C
u
Example 2. Order of wave interactions matters.
u1 = ϕ[ (ul ),
u2 = ϕ[ (u1 ),
ϕN (ul ) < ur < ϕ] (ul ).
f
ur
u
2
u1
u 1 = ! (ul)
u
u2
a
u0
l
u2
t = 0
b
u
u 2 = ! (u 1)
u0
ul
ul
a
x
b
ur
x
ur
u1
u1
u
u
ul
ul
t > 0
u2
x
x
ur
ur
u1
(i) a large, b − a small.
N-N-C −→ N-C
(ii) a small, b − a large.
N-N-C −→ C
Example 3. Splitting-merging pattern.
ϕ[ (ul ) < u2 < ϕN (ul ) < u1 < ϕ] (ul ) < ur < ϕN (ϕ[ (ul )) < ul
t
C
N
! (u )
l
ul
u2
u1
ul
u = ! (ul)
ur
C
N
u
ur
u
C
C
u1
u2
0
C-R-C −→ N-C-C −→ N-C −→ C
x
Splitting-merging initial data.
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Fix u∗ > 0 and u0 (x) = u0N (x) + v0 (x)
(
u∗ ,
x <0
N
u0 (x) :=
N
ϕ (u∗ ), x > 0
Two parameters in the problem : η << ε
TV (v0 ) < ε << 1
η := ϕ] (u∗ ) − ϕN (u∗ ) << 1
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The initial data u0N gives rise to two distinct solutions made of
admissible waves.
Example.
(
0, x < 0
v0 (x) =
δ, x > 0
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When δ > 0, a single classical shock C ↓ :
(
u∗ ,
x <st
↓
u (x, t) :=
N
ϕ (u∗ ) + δ, x > s t
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When δ < 0, a two-wave solution u ↓↑ consisting of an
undercompressive shock N ↓ plus a classical shock C ↑ .
Splitting/merging structure.
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One or two big waves at each time t > 0.
Solution close to a one-wave solution u ↓ or a two-wave solution u ↓↑
Notation: x = y (t) : locus of big shocks N ↓ and C ↓
x = z(t) : locus of the big shock C ↑
Main issue. Generalized TV functional for front tracking solutions
u h = u h (t, x)
t
C
N
x
=
y(
t
)
C
N
C
x =z ( t )
C
C
0
x
Generalized wave strength.
V (t) :=
X
σ(u− , u+ )
jumps(u− ,u+ )
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Generalized strengths of the big increasing classical shock located at
z = z(t)
σ C (u− , u+ ) := ϕ[ (u− ) − ϕ[ (u+ ) > 0.
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Nonclassical shock at y = u(t)
σ NC (u) := (u − ψ(u)) − (ϕ[ ◦ ψ(u) − ϕ[ ◦ ϕ[ (u))
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Standard definition for the big decreasing classical shock.
Properties.
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Continuity/decreasing properties above, since for u > 0
ψ(u) < ϕ[ ◦ ψ(u) < ϕ[ ◦ ϕ[ (u) < u.
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The generalized strength σ(u) is strictly positive.
Notation.
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Total strength of small waves in each region :
y h (t)
h
Vleft
(t) := TV−∞ (u h (t)),
z h (t)
h
Vmiddle
(t) := TVy h (t) (u h (t))
h
h
Vright
(t) := TVy+∞
h (t) (u (t))
h
h
h
V h (t) = Vleft
(t) + κ0 Vmiddle
(t) + κ0 Vright
(t)
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Total strength of big waves : W h (t)
Theorem [LeFloch - Shearer]
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Front tracking solutions u h = u h (x, t) have the splitting/merging
structure, with
V h (t) + κ2 W h (t) ≤ V h (0) + κ2 W h (0)
h
h
Vleft
(t) ≤ Vleft
(0),
h
h
Vright
(t) ≤ Vright
(0)
h
h
h
h
Vleft
(t) + κ1 Vmiddle
(t) ≤ Vleft
(t) + κ1 Vmiddle
(t)
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The limit h → 0 yields an exact, splitting/merging solution
u = u(x, t) made of admissible waves, only.
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At each splitting, V h (t) + κ2 W h (t) decreases by at least
ψ(u∗ ) − ϕN (u∗ ).
At each merging, it decreases by at least ϕ] (u∗ ) − ψ(u∗ ).
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If ϕN (u∗ ) 6= ϕ] (u∗ ), only finitely many mergings/splittings
and the solution eventually settles to a solution having a specified
(one-wave or two-wave) structure.
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When ϕN (u∗ ) = ϕ] (u∗ ), the splittings/mergings
may continue for all times.
Generalization to strictly hyperbolic systems. Laforest - PLF.
11. DLM THEORY – Kinetic relations for nonconservative systems
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Physical models for fluid mixtures (see below). Need averaging
procedure and simplifying assumptions.
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Underline the importance of small-scale phenomena for formulating
a well-posed hyperbolic theory.
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Notion of family of paths proposed by Dal Maso - PLF - Murat
(1990, 1995): Φ : [0, 1] × RN × RN × RN
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Φ(·; u− , u+ ) is a path connecting u− to u+
Φ(0; u− , u+ ) = u− ,
Φ(1; u− , u+ ) = u+ ,
`
´
TV[0,1] Φ(·; u− , u+ ) . |u+ − u− |.
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Φ is Lipschitz continuous in the graph distance
“
”
0
0
0
0
dist Φ(·; u− , u+ ), Φ(·; u−
, u+
) . |u− − u−
| + |u+ − u+
|.
Definition. Given u ∈ BV (R, RN ) and g a Borel function,
a
h there exists
i
unique measure called the nonconservative product µ = g (u) ∂x u
Φ
such that:
R
I If B is a Borel subset of C(u), then µ(B) :=
g (u) ∂x u.
B
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At a point of jump x of u, setting u± := u± (x)
Z
g (Φ(·; u− , u+ )) ∂s Φ(·; u− , u+ ).
µ( x ) :=
[0,1]
Remark.
I
I
A definite concept of weak solutions of nonconservative systems,
once a family of paths is prescribed.
In the conservative
case,
h
i this is consistent with the distributional
definition ∇h(u) ∂x u = ∂x h(u).
Φ
Theorem (Riemann problem, Dal Maso - PLF - Murat, 1990).
Given a nonconservative, strictly hyperbolic, genuinely nonlinear
system and a family of paths Φ, the Riemann problem admits an
entropy solution (in the DLM sense) satisfying Lax shock inequalities.
Properties.
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Generalized Hugoniot jump relations
Z 1
A(Φ(·; u− , u+ )) ∂s Φ(·; u− , u+ ) = 0.
−λu (u+ − u− ) +
0
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Wave curves Lipschitz continuous at the origin
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In contrast with Lax’s standard C 2 regularity result. See Bianchini Bressan, Iguchi - PLF, and Liu - Yang.
Theorem (Existence result, PLF - Liu, Forum Math. 1992).
Glimm scheme for nonconservative, strictly hyperbolic, genuinely
nonlinear system:
I u h = u h (t, x) have uniformly bounded TV
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converge to an entropy solution u in the DLM sense
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and for all but countably many times
h
i
h
i
A(u h ) ∂x u h (t) * A(u) ∂x u (t).
Φ
Φ
Remark.
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The theory of nonconservative systems / the theory of conservative
systems.
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Bianchini-Bressan’s theory via the vanishing viscosity method.
Paths subordinate to traveling waves
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The DLM family of paths encodes the information required for the
hyperbolic theory.
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No canonical choice. Need an augmented model
∂t u ε + A(u ε ) ∂x u ε = ∂x R ε ∂x u ε , ε2 ∂xx u ε , . . .
Definition [PLF, IMA Preprint 1989]. A DLM family of paths Φ is
subordinate to the family of traveling waves if, whenever u− , u+ are
connected by a traveling wave u, then Φ(·; u− , u+ ) ∼ u.
Theorem. [PLF, IMA Preprint 1989]. If Φ is a DLM family subordinate
to TW, and if u ε = u((x − λ t)/ε) is a TW, converging to
u(t, x) := u− for x < λ t,
u+ for x > λ t,
h
i
then A(u ε ) ∂x u ε (t) * A(u) ∂x u (t) weak-star, and u is a weak
Φ
solution in the DLM sense.
Remark. Existence of traveling waves : Sainsaulieu (1995), Schecter
(2000), Bianchini-Bressan (2002)
Example. One-dimensional nozzle flows.
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Evolution equations
∂t (aρ) + ∂x (aρv ) = 0
2
∂t (aρv ) + ∂x (aρv + a p(ρ)) − p(ρ) ∂x a = 0
a : R → R: piecewise Lipschitz continuous. Set u := (aρ, aρv ).
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Solutions obey the entropy inequality
∂t U(u, a) + ∂x F (u, a) ≤ 0
U(u, a) = a2 ρ
I
u2
+ aρ e(ρ),
2
F (u, a) = U(u, a) + p(ρ) u
See contributions by Bouchut; Gallou¨et, H´erard, and Seguin; PLF
and Thanh. For similar systems: Amadori, Gosse, Guerra, Jin, etc.
Example. Shallow water equations with topography.
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Evolution equations
∂t ρ + ∂x (ρv ) = 0,
ρ2 − g ρ ∂x Z = 0,
∂t (ρv ) + ∂x ρv 2 + g
2
ρ: mass density;
v : velocity of the fluid. Set u := (ρ, ρv ).
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Prescribed topography function Z : R → R, depending on x, solely
piecewise Lipschitz continuous. g : gravity constant.
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Entropy inequality
∂t U(u, a) + ∂x F (u, a) ≤ 0,
U(u, a) := ρE (v ) + ρZ ,
F (u, a) := ρ
e 0 (ρ) =
p(ρ)
,
ρ2
v3
+ ρv e(ρ) + p(ρ)v + ρv Z .
2
Example. Two-fluid mixtures.
α2 = 1 − α1 : fraction of the fluid 2
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α1 : fraction of the fluid 1.
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Evolution equations with stiff source-terms
∂t α1 + VI ∂x α1 = λ(p2 − p1 )
∂t (α1 ρ1 ) + ∂x (α1 ρ1 u1 ) = 0
∂t (α1 ρ1 u1 ) + ∂x (α1 ρ1 u12 + α1 p1 ) − PI ∂x α1 = λ (u2 − u1 ) + ε∂x (µ1 ∂x u1 )
∂t (α2 ρ2 ) + ∂x (α2 ρ2 u2 ) = 0,
∂t (α2 ρ2 u2 ) + ∂x (α2 ρ2 u22 + α2 p2 ) − PI ∂x α2 = −λ(u2 − u1 ) + ε∂x (µ2 ∂x u2 )
I
Pressure law pi = pi (ρi ) satisfying pi0 (ρi ) > 0.
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Relaxation parameter λ > 0 (large).
Inverse of Reynolds number ε (small).
I
Ransom and Hicks, Baer and Nunziato.
Investigated by Berthon, Coquel, Gallouet, H´erard, Nkonga, Seguin.
I
Constitutive functions:
VI : interfacial velocity.
PI : interfacial pressure.
For instance, Ransom and Hicks imposes
VI :=
I
1
(u1 + u2 ),
2
PI :=
1
(p1 + p2 ).
2
Independently of this choice and provided the non-resonance
condition holds
|VI − ui | =
6 ci (ρi ),
i = 1, 2,
then the system is admits five real eigenvalues
VI ,
ui ± ci (ρi ),
with ci2 (ρi ) := p 0 (ρi ) > 0, as well as a basis of right eigenvectors.
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Key issue: Closure laws for VI and PI
Entropy balance law ?
Compatibility condition
VI (p2 − p1 ) + PI (u2 − u1 ) = p2 u1 − p1 u2
ensuring that
U := α1 ρ1 E1 + α2 ρ2 E2 ,
Ei :=
ui2
+ ei (ρi )
2
is a mathematical entropy
∂t U + ∂x F = −λ(u2 − u1 )2 − λ(p2 − p1 )2 − D,
U := (α1 ρ1 E1 + α2 ρ2 E2 )
F := (α1 ρ1 E1 + α1 p1 )u1 + (α2 ρ2 E2 ) + α2 p2 )u2 ,
D := εµ1 (∂x u1 )2 + εµ2 (∂x u2 )2 − ε ∂x (µ1 α1 ∂x u1 + µ2 α2 ∂x u2 ).
A class of nonconservative systems endowed with an entropy.
Collaboration with C. Berthon and F. Coquel
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A class of nonconservative hyperbolic models with singular
source-term
∂t v + ∂x f (v , a) = g (v , a) ∂x a.
a: given, piecewise Lipschitz continuous function of x
I
Endowed with an entropy pair U, F satisfying the entropy inequality
∂t U(v , a) + ∂x F (v , a) ≤ 0.
I
Reformulation as a nonconservative system: following PLF (Preprint
IMA, 1989) for the nozzle flow equations. we introduce the extended
variable u := (v , a)
∂t v + ∂x f (v , a) − g (v , a) ∂x a = 0,
∂t a = 0,
I
Away from resonance, PLF-Liu existence theory applies.
Kinetic relations for nonconservative systems.
I
Impose a kinetic relation for shock (u− , u+ ) with speed Λ
−Λ (U(u+ ) − U(u− )) + F (u+ ) − F (u− ) = Φ(u− , Λ).
I
Typical example: Φ ≡ 0 for standing waves and, for other waves, Φ
coincides with the entropy dissipation of the given conservation laws
(with a constant).
In progress.
I
Riemann problem for classes of nonconservative systems with
imposed kinetic relations
I
Application and analysis of traveling waves to the two-fluid model.
I
Numerical discretization that “tune” the entropy dissipation rate.
I
Numerical experiments with the two-fluid model and plot the kinetic
function.
Some directions of research
I
Physics of phase transitions and multi-fluid phenomena
I
I
I
I
Mathematical issues
I
I
Liquid-vapor mixtures
Solid-solid phase transitions (smart shape-memory materials,
martensite-austenite, Cu-Al-Ni alloys)
Better description of the internal structure. Hidden variables.
Discrete models. Integral terms.
Stability of multi-dimensional interfaces
Numerical approximation
I
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Physically realistic situations
Multi-scale flows and multi-fluid mixtures