THE MATHEMATICAL THEORY OF SMALL-SCALE DEPENDENT SHOCK WAVES Philippe G. LeFloch Universit ´ e Pierre et Marie Curie, Paris Centre National de la Recherche Scientifique Blog: philippelefloch.org An example: the singular limit ε, κ → 0 ρ t +(ρu) x = 0 (ρu) t + ρu 2 + k ρ 2 x = ε u xx + κ (ρ 2 ρ xx ) x I ε = κ = 0: Euler system (shock formation, weak solutions) I ε 2 ’ κ → 0: singular limit problem (diffusive-dispersive regime) I κ = αε 2 → 0 : (vanishing) diffusive-dispersive shock waves depending upon α !
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THE MATHEMATICAL THEORY OF SMALL-SCALE ...THE MATHEMATICAL THEORY OF NONCLASSICAL SHOCK WAVES 1. Diffusive-dispersive models (non-convexity, entropy inequality) 2. Nonclassical Riemann
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THE MATHEMATICAL THEORY OFSMALL-SCALE DEPENDENT SHOCK WAVES
Philippe G. LeFlochUniversite Pierre et Marie Curie, Paris
Centre National de la Recherche ScientifiqueBlog: philippelefloch.org
An example: the singular limit ε, κ→ 0
ρt + (ρu)x = 0
(ρu)t +(ρu2 + k ρ2
)x
= εuxx + κ (ρ2ρxx)x
I ε = κ = 0: Euler system (shock formation, weak solutions)I ε2
→ 0 : (vanishing) diffusive-dispersive shock wavesdepending upon α !
Conservation laws with vanishing diffusion, dispersion, etc.
uεt + f(uε)x = R(εuεx , ε2uεxx , . . .)x
I u = limε→0 uε: shock wave solutions to ut + f(u)x = 0I Second-order: εuεxx (viscosity). Lax’s theory of shock waves (entropy
condition, compressive shocks)I Third- or higher-order: α ε2 uεxxx (capillarity)
Oscillations near shocks, driven by dispersive effects, delicatecompetition between “small scales”
I Classical compressive + nonclassical undercompressive shocks (orsubsonic phase boundaries).
The mathematical theory
I Dynamics of diffusive-dispersive shocks, nonlinear interactionsI Internal shock structure, analysis of diffusive-dispersive traveling wavesI Develop general mathematical methods for these singular limit problemsI Design numerical methods adapted to small-scale dependent shocks
THE MATHEMATICAL THEORY OF NONCLASSICAL SHOCK WAVES
I α , 0: nonclassical behavior (plot of r = (v2 + w2)1/2)PLF-Mishra
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−3
−2
−1
0
1
2
3
4
5
EC2:−−−−−−−−−−−−−−−−−−−−−−−−−−−
EC4:− − − − − − − − − −
EC6:− − − − − − −
EC8:o o o o o o o o
EC10:+ + + + + + + + +
Solutions depend on the (order of the) scheme
FOR ALL THESE MODELSI Complex wave patternsI Different ratio/regularizations/schemes yield different solutionsI Non-convex flux-function and a single entropy inequality
THE MATHEMATICAL THEORY OF SMALL-SCALE DEPENDENTSHOCKSI Include macro-scale effects without resolving the small-scalesI No “universal” admissibility criterion, but rather “several
hyperbolic theories”I Each being determined by specifying a physical regularization
−→ KINETIC RELATION for undercompressive shocks (Truskinovsky,Abeyaratne-Knowles, PLF, Shearer, etc.)
−→ DLM FAMILY of PATHS for nonconservative hyperbolic systemsut + A(u)ux = 0 (Dal Maso-LeFloch-Murat)
−→ ADMISSIBLE BOUNDARY SETS (PLF-Dubois, PLF-Joseph,Serre) for the boundary value problem for hyperbolic problems
2. THE NONCLASSICAL RIEMANN SOLVER
ut + f(u)x = 0I Concave-convex flux
u f ′′(u) > 0 (for u , 0)
f ′′′(0) , 0, limu→±∞
f ′(u) = +∞
I Tangent function ϕ\ : R→ R and its inverse ϕ−\
f ′(ϕ\(u)) =f(u) − f
(ϕ\(u)
)u − ϕ\(u)
, u , 0
uu
!
u
ul
l
ul
l
l
l
! ( )
u! ( )
( )
! u( )
N
C RN+CN + R
r
Shock wave solutions.
u(t , x) =
u−, x < λ tu+, x > λ t
satisfying the Rankine-Hugoniot relation λ =f(u−)−f(u+)
u−−u+= a(u−,u+)
Standard Riemann solver based on the Oleinik entropy inequalitiesfor shocks.
f(v) − f(u+)
v − u+≤
f(u+) − f(u−)
u+ − u−for all v between u− and u+. Equivalent to imposing all of the entropy
inequalitiesU(u)t + F(u)x ≤ 0
U′′ > 0, F ′(u) = f ′(u) U′(u)
This condition characterizes shock generated by diffusion only
A single entropy inequality. This yields a much weaker condition
U(u)t + F(u)x ≤ 0, U′′ > 0, F ′(u) = f ′(u) U′(u)
E(u−,u+) = −f(u−) − f(u+)
u− − u+
(U(u+) − U(u−)
)+ F(u+) − F(u−)
≤ 0
Zero entropy dissipation function ϕ[0 : R 7→ R.
E(u, ϕ[0(u)) = 0, ϕ[0(u) , u ( when u , 0)
(ϕ[0 ◦ ϕ[0)(u) = u.
Solving the Riemann problem. u(x ,0) =
ul , x < 0ur , x > 0
A single entropy inequality allows for:I Classical compressive shocks
u− > 0, ϕ\(u−) ≤ u+ ≤ u−
satisfying Lax shock inequalities
f ′(u−) ≥f(u+) − f(u−)
u+ − u−≥ f ′(u+)
I Nonclassical undercompressive shocks
u− > 0, ϕ[0(u−) ≤ u+ ≤ ϕ\(u−)
having all characteristics passing through
min(f ′(u−), f ′(u+)
)≥
f(u+) − f(u−)
u+ − u−
The cord connecting u− to u+ intersects the graph of f .
I Rarefaction waves. Lipschitz continuous solutions u connectingtwo constant states and depending only upon ξ = x/t
Entropy-compatible kinetic functions.
I A monotone decreasing, Lipschitz continuous functionϕ[ : R 7→ R
ϕ[0(u) < ϕ[(u) ≤ ϕ\(u), u > 0
I Then, by definition, for each given left-hand state u− the kineticrelation u+ = ϕ[(u−) singles out a nonclassical shock.
I Notation: Companion (threshold) function ϕ] : R→ R
-16
-14
-12
-10
-8
-6
-4
-2
0
2 4 6 8 10 12 14
2nd order scheme4th order schemeclassical solution
TW solutionextreme nonclasssical solution
, uu
!
u
ul
l
ul
l
l
l
! ( )
u! ( )
( )
! u( )
N
C RN+CN + 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.
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
u
-2
-1.5
-1
-0.5
0
0.5
1
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
u
THE NONCLASSICAL RIEMANN SOLVER BASED ON ANENTROPY-SATISFYING KINETIC FUNCTION
Given a kinetic function ϕ[ compatible with an entropy U of a conser-vation law with concave/convex flux, the Riemann problem admits aunique solution, satisfying:I the single entropy inequalityI the kinetic relation u+ = ϕ[(u−) at each undercompressive shock
I 2 × 2 isentropic Euler equations and nonlinear elasticity or phasetransition system
uniqueness if hyperbolic, non-uniqueness if hyperbolic-elliptic)(Shearer et al., LeFloch, PLF-Thanh)
I N × N strictly hyperbolic systems of conservation laws.B.T. Hayes and P.G. LeFloch, Nonclassical shock waves and kineticrelations. Strictly hyperbolic systems, SIAM J. Math. Anal. (2000).
3. KINETIC FUNCTIONS BASED ON TRAVELING WAVES
For instance, consider conservation laws with nonlinear diffusion andlinear dispersion
ut + f(u)x = β(|ux |
p ux
)x
+ uxxx
f concave-convex, β > 0, p ≥ 0
Internal structure of shock waves:
I second-order ODE for traveling wave solutions u(x , t) = u(y)with y = x − λ t
− λ (u − u−) + f(u) − f(u−) = β |u′|p u′ + u′′
I with boundary conditions
limy→±∞
u(y) = u±
I prescribed data u±, λ satisfying the Rankine-Hugoniot relation
Three regimes.
I β ∈ (0,+∞) : diffusion and dispersion kept in balanceI β = 0: dispersion onlyI β→ +∞: diffusion only
First results.For the cubic flux f = u3, one has ϕ\(u) = −u/2, ϕ[0(u) = −u, andclosed formulas are available:
I p = 0 : Shearer et al. (1995)ϕ[ is piecewise linear (with slope −1 and −1/2)
I p = 1/2 : Bedjaoui - PLFϕ[ is linear with slope cβ ∈ (−1/2,−1)
I existence of classical / nonclassical traveling wavesI Kinetic function ϕ[ associated to this model ?I Monotonicity ?I Behavior near u = 0 ?
KINETIC FUNCTIONS BASED ON TRAVELING WAVESTo a large class of augmented models, we are able to associate aunique kinetic function which is monotone and satisfies the assump-tions required in the theory of the Riemann problem.
Generalizations.
I 2 × 2 Nonlinear elasticity/Euler equations (non-nec. monotone)(Shearer et al., PLF-Bedjaoui)
I 2 × 2 Van de Waals model (two inflection points, multiplesolutions) (Bedjaoui-Chalons-Coquel-PLF)
(ii) Threshold function A \ such thatI 0 ≤ p ≤ 1/3 :
A \ : R→ [0,∞) Lipschitz continuous, A \(0) = 0
ϕ[(u) = ϕ\(u) iff β ≥ A \(u)
I p > 1/3 :ϕ[(u) , ϕ\(u) (u , 0)
(iii) Asymptotic behavior of infinitesimally small shocks:
I p = 0: ϕ[′
(0) = ϕ\′(0) = −1/2
A \(0) = 0, A \′(0±) , 0
I 0 < p ≤ 1/3 : ϕ[′
(0) = −1/2A \(0) = 0, A \′(0±) = +∞
I 1/3 < p < 1/2 : ϕ[′
(0) = −1/2
I p = 1/2 : ϕ[′
(0) ∈(ϕ−[0
′
(0),−1/2)
= (−1,−1/2)
limβ→0+
ϕ[′
(0) = −1, limβ→+∞
ϕ[′
(0) = −1/2
I p > 1/2 : ϕ[′
(0) = −1
4. THE INITIAL VALUE PROBLEM
The behavior of the kinetic function for arbitrarily small shocks isrequired in our general existence theory.
Glimm method.I Nonclassical Riemann solver as building blockI Random choice (equidistributed) or front tracking techniqueI Numerical experiments (Chalons - PLF 2003)
EXISTENCE THEORY FOR THE INITIAL VALUE PROBLEMI Theoretical convergence results in the strong L1 normI Uniform convergence at points of continuityI Convergence of left- and right-hand limit at discontinuitiesI TV(u(t , ·)) is uniformly bounded (but need not be decreasing,
generalized total variation functionals adapted to nonclassicalwave interactions)
PLF, Hyperbolic Systems of Conservation Laws. The theory of classical andnonclassical shock waves, Birkhauser (2002)
I Consider the limiting solutions uα = limε→0 uαε to a diffusive-dispersive conservation law
∂tu + ∂x f(u) = εuxx + α ε2 uxxx , u = uα,ε
together with the associated kinetic function ϕ[αI Consider numerical solutions u∆x
α given by some finite differenceschemes together with its limit vα = lim∆x→0 u∆x
α and itsassociated kinetic function ψ[α
Essential observation vα , uα ψ[α , ϕ[α
I schemes in conservative form, satisfying a discrete version of theentropy inequality (in the sense of Lax)
I small-scale effects drive the selection of shocksI discrete dissipation , continuous dissipation
Hayes-LeFloch criterion
ψ[α should be an accurate approximation of ϕ[αHayes & PLF, Nonclassical shocks and kinetic relations. Finite differenceschemes, SIAM Journal of Numerical Analysis (1998)
Schemes with well-controled dissipation.I Finite difference schemes in conservative form in the sense of
LaxI Entropy conservative flux associated with the hyperbolic system
I High-order accurateI Discrete version of the physically relevant entropy inequality
I High-order finite differences for the augmented terms (viscosity,capillarity, etc.), preserving the discrete entropy inequality
I Essential requirement: the equivalent equation (also called themodified equation) should coincide with the augmented physicalmodel, with very high accuracy.
For instance for p ≥ 3 (at least)
∂tu + ∂x f(u) = εuxx + α ε2 uxxx
∂tu + ∂x f(u) = ∆x uxx + α (∆x)2 uxxx + O(∆x)p
where we wrote unj = u(tn, xj) = u(n∆t , j∆x) and formally expanded
in ∆t ,∆x → 0
A conjecture about the equivalent equation.
I PLF : As p →∞ the kinetic function ψ[α,p associated with ascheme with equivalent equation
∂tu + ∂x f(u) = ∆x uxx + α (∆x)2 uxxx + O(∆x)p
converges to the exact kinetic function ϕ[α
limp→∞
ψ[α,p = ϕ[α
References.
I Hayes - PLF (SINUM, 1998) scalar conservation lawsI PLF - Rohde (SINUM, 2000) third and fourth order schemesI Chalons - PLF (JCP, 2001) van der Waals fluidsI PLF - Mohamadian (JCP, 2008) very high-order schemesI Review paper: PLF and Mishra, Numerical methods with controled
dissipation for small-scale dependent shocks, Acta Numerica 23 (2014)
Class of 2p-th order WCD schemes
dui
dt+
1∆x
j=p∑j=−p
αj fi+j =c
∆x
j=p∑j=−p
βjui+j + αc2
∆x
j=p∑j=−p
γjui+j
2p-order accuracy for al 0 ≤ l ≤ 2p
p∑j=−p
jαj = 1,p∑
j=−p
j lαj = 0, l , 1
p∑j=−p
j2βj = 2,p∑
j=−p
j lβj = 0, l , 2
p∑j=−p
j3γj = 6,p∑
j=−p
j lγj = 0, l , 3
Stability Condition on c (ensures good approximation for shocks oflarge strength)
Linear diffusion-dispersion model Camassa-Holm model
6.3 The kinetic relation for Van der Waals fluids
Complex wave structure.
Initial data τL = 0.8, τR = 2, uR = 1 with variable left-hand data uL
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=1.5
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=0.5
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5uL=0.2
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=0.95
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=0.7
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=1.4
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=.6
x
v
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=1.1
x
!
0 0.25 0.5 0.75 10.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
uL=1.3
!
Better described... with the kinetic function
Kinetic function.
For τ near to 1: existence and monotonicity
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
0.65 0.655 0.66 0.665 0.67 0.675 0.68 0.685
lambda=0.4lambda=0.5lambda=0.7
Maxwell curve
u-
u +
0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.91.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Fourth orderSixth orderEighth orderTenth order
!+
!"
Varying the capillarity coefficient Varying the order of the discretization
Schemes with Well Controled Dissipation (WCD)I Robust and reliable schemes, validated by an analysis of the
equivalent equationI Numerical kinetic function approaching (with arbitrary accuracy)
the exact kinetic function limp→∞ ψ[α,p = ϕ[αI Schemes based on entropy conservative flux, ensuring the
correct sign on the entropy dissipation U(u)t + F(u)x ≤ 0
Approximation of the nonclassical entropy solutions with arbitraryaccuracy
I The kinetic function characterizes the shock dynamics and wasinvestigated for a large class of models.
I Computing the kinetic function provides a tool.I Effect of the diffusion/dispersion ratioI Effect of the regularizationI Order of accuracy of the schemeI Compare several physical models
7. THE ZERO-DIFFUSION-DISPERSION LIMIT
I Navier-Stokes-Korteweg system
ρt + (ρu)x = 0
(ρu)t +(ρu2 + p(ρ)
)x
=ε(µ(ρ) ux
)x
+ ε2(K [ρ]
)x
K [ρ] =ρκ(ρ)ρxx +12
(ρκ′(ρ) − κ(ρ)
)ρ2
x
I Convergence to the Euler system when ε→ 0
ρt + (ρu)x = 0
(ρu)t +(ρu2 + p(ρ)
)x
= 0
Rigorous convergence theorem for general functions p(ρ), µ(ρ), κ(ρ)