THE MATHEMATICAL THEORY OF SMALL-SCALE ...THE MATHEMATICAL THEORY OF NONCLASSICAL SHOCK WAVES 1....

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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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Transcript of THE MATHEMATICAL THEORY OF SMALL-SCALE ...THE MATHEMATICAL THEORY OF NONCLASSICAL SHOCK WAVES 1....

  • THE MATHEMATICAL THEORY OFSMALL-SCALE DEPENDENT SHOCK WAVES

    Philippe G. LeFlochUniversité 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 ' κ→ 0: singular limit problem (diffusive-dispersive regime)I κ = α ε2 → 0 : (vanishing) diffusive-dispersive shock waves

    depending upon α !

  • Conservation laws with vanishing diffusion, dispersion, etc.

    uεt + f(uε)x = R(εuεx , ε

    2uεxx , . . .)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

    1. Diffusive-dispersive models(non-convexity, entropy inequality)

    2. Nonclassical Riemann solver with entropy-compatible kinetics(general theory, single jump initial data, kinetic function)

    3. Kinetic functions based on traveling waves(shock structure, specific models)

    4. The initial value problem for arbitrary initial data(finite total variation, ux ∈M, generalized TV functional, Glimm

    method)

    5. Schemes with well-controled dissipation (WCD)(entropy conservative discrete flux, equivalent equation)

    6. Computing kinetic functions(effect of the parameter α)

    7. The zero diffusion-dispersion limit(finite energy: u ∈ L2, weak convergence, conserved quantities)

  • First developments. Materials undergoing phase transitions

    wt − vx = 0vt − σ(w)x = ε vxx − α ε2 wxxx

    v : velocity w > −1 : deformation gradient σ(w) : stressε : viscosity α ε2 : capillarity

    I Slemrod (1984, etc): self-similar solutionsI Shearer (1986, etc.): Riemann problemI Truskinovsky (1987, etc): kinetic relationI Abeyaratne & Knowles (1990, etc): trilinear equation, nucleation

    I PLF, Propagating phase boundaries. Formulation of the problem andexistence via the Glimm scheme, Arch. Rational Mech. Anal. 123 (1993)

    I formulation for general hyperbolic systemsI weak solutions with finite total variationI Cauchy problem and the Glimm method

  • 1. DIFFUSIVE-DISPERSIVE MODELS

    Vanishing linear diffusion-dispersion.I Conservation law ut + f(u)x = εuxx + κuxxx

    (Shearer et al., Hayes-PLF, LeFloch, Bedjaoui-PLF)

    I Classical/nonclassical solutionsI κ > ε2 (dominant dispersion)

    high oscillations, weak convergence (Lax, Levermore)I κ = α ε2 (balanced regime)

    strong convergence, mild oscillations, nonclassical, depend on α

    I Entropy inequality with entropy flux F ′(u) = uf ′(u)12

    (u2

    )t+

    (F(u)

    )x= −D + Cx , D = ε |ux |2 ≥ 0

    C = εuux + κ(u uxx − (1/2)u2x

    )In the limit ε, κ→ 0, we obtain:

    (u2/2

    )t+ F(u)x ≤ 0

  • Nonclassical wave patterns

    For instance for ut + (u3)x = 0

    -4

    -3

    -2

    -1

    0

    1

    2

    3

    4

    5

    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 , -6-5

    -4

    -3

    -2

    -1

    0

    1

    2

    3

    4

    5

    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

    two shocks shock + rarefaction

    Solutions that are distinct from the ones selected by the standard(Oleinik) entropy conditions

  • Generalized Camassa-Holm model.I Conservation law (with β > 0)

    ut + f(u)x = εuxx + κ(utxx + 2ux uxx + u uxxx

    )Shallow water model for wave breaking

    I Limiting solutions when κ = α ε2: solutions are similar to, but donot coincide with, the ones obtained with the lineardiffusion-dispersion model.

    Weak solutions to ut + f(u)x depend on the underlying small-scalephysics

    I Entropy inequality:12

    (u2 + κ |ux |2

    )t + F(u)x = − ε |ux |2 + Cx

    In the limit ε, κ→ 0, we obtain again 12(u2

    )t+ F(u)x ≤ 0

    The inequality(u2/2

    )t

    + F(u)x ≤ 0 is insufficient in order to formulatea suitable theory of (nonclassical) weak solutions to the hyperbolicconservation law.

  • Van der Waals fluids.I Two coupled conservation laws

    vt − ux= 0

    ut + p(v)x =(ε(v) ux

    )x

    +(κ′(v)

    v2x2−

    (κ(v) vx

    )x

    )x

    pressure law p(v ,T) = RTv−b − av2

    I Entropy inequalitye(v) + u22 + κ(v) v2x2

    t

    +(p(v) u

    )x

    = − ε(v) u2x + Cx

    e = e(v) internal energy.

    In the limit ε(v), κ(v)→ 0, we obtain(e(v) + u

    2

    2

    )t+(p(v) u

    )x≤ 0

  • Nonclassical behavior for Van der Waals fluids

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    -0.6 -0.4 -0.2 0 0.2 0.4 0.6

    lambda=1e-5lambda=1e-1lambda=0.75

    , -1.6

    -1.4

    -1.2

    -1

    -0.8

    -0.6

    -0.4

    -0.2

    0

    0.2

    -0.6 -0.4 -0.2 0 0.2 0.4 0.6

    lambda=1e-5lambda=1e-1lambda=0.75

    Specific volume v Velocity u

    Weak solutions depend upon the ratio λ = (viscosity)2/capillarity

  • Thin liquid film model. (Bertozzi, Shearer, Münch, Levy)

    I Conservation law (with ε, κ > 0)ut + (u2 − u3)x = ε(u3ux)x − κ (u3 uxxx)x

    x

    u

    0 250 500 75000.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.8

    x

    u

    0 250 500 75000.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.8

    I Entropy inequality(u log u − u

    )t

    + F(u)x = −D + CxD = εu3u2x + κ |(u2 ux)x |2 ≥ 0

    In the limit ε, κ→ 0, we obtain(u log u − u

    )t

    + F(u)x ≤ 0

  • Ideal magnetohydrodynamics with Hall effect.I (v ,w): transverse components of the magnetic field

    vt + ((v2 + w2) v)x = ε vxx + α�wxxwt + ((v2 + w2) w)x = εwxx − α� vxx

    ε: magnetic resistivity, α: Hall parameterI Entropy inequality

    (1/2)(v2 + w2

    )t

    + (3/4)((v2 + w2)2

    )x

    = − ε (v2x + w2x ) + CxI α = 0: classical behavior

    Brio, Hunter, Freistühler, Pitman, Panov, Wu, Kennel

    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 tsatisfying 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 entropyinequalities

    U(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 > 0A 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

    L1 continuous dependence property (t ∈ [0,T ], compact K ⊂ R)‖u(t) − v(t)‖L1(K) ≤ C(T ,K) ‖u(0) − v(0)‖L1(K)

    Generalizations

    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 p = 1 : Hayes - PLF (1997)ϕ[′(0) = ϕ[0(0) = −1

  • Phase plane analysis. (LeFloch-Bedjaoui, 2001 & 2004)

    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)

  • Admissible shocks

    S(u−) ={u+ / there exists a TW connecting u±

    }Theorem. (Bedjaoui - PLF, 2001 & 2004).(i) Kinetic function ϕ[ : R→ R, Lipschitz continuous, strictlydecreasing,

    S(u) ={ϕ[(u)

    }∪

    (ϕ](u),u

    ], u > 0

    ϕ[0(u) < ϕ[(u) ≤ ϕ\(u), u > 0

    (ii) Threshold function A \ such thatI 0 ≤ p ≤ 1/3 :

    A \ : R→ [0,∞) Lipschitz continuous, A \(0) = 0ϕ[(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, lim

    β→+∞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, Birkhäuser (2002)

    Baiti - PLF - Piccoli 2001, LeFloch 2002, PLF - Laforest 2010, 2015)

  • 5. SCHEMES WITH WELL-CONTROLED DISSIPATION

    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∆xα given by some finite difference

    schemes together with its limit vα = lim∆x→0 u∆xα and itsassociated kinetic function ψ[α

    Essential observation vα , uα ψ[α , ϕ[αI schemes in conservative form, satisfying a discrete version of the

    entropy 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[α

    Hayes & 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) = �uxx + α �2 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 expandedin ∆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

    duidt

    +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 ≤ 2pp∑

    j=−pjαj = 1,

    p∑j=−p

    j lαj = 0, l , 1

    p∑j=−p

    j2βj = 2,p∑

    j=−pj lβj = 0, l , 2

    p∑j=−p

    j3γj = 6,p∑

    j=−pj lγj = 0, l , 3

    Stability Condition on c (ensures good approximation for shocks oflarge strength)

    σ < αc2 + c

    where σ is the local wave speed.

  • (2p + 1)-point, conservative, semi-discrete schemes

    ddt

    uj = −1h

    (g∗j+1/2 − g∗j−1/2

    )I uj = uj(t) is an approximation of u(xj , t), and h > 0 is the mesh

    lengthI The discrete flux

    g∗j+1/2 = g∗(vj−p+1, · · · , vj+p), vj = ∇U(uj)

    must be consistent with the exact flux g

    g∗(v , . . . , v) = g(v).

    Entropy conservative flux.I Second-order entropy conservative flux Tadmor 1984I Third-order entropy conservative flux LeFloch-Rohde 2000I Arbitrarily high order, discrete in time LeFloch-Mercier-Rohde

    2000

  • Theorem (Second-order, Tadmor, 1984). Two-point numerical flux

    g∗(v0, v1) =∫ 1

    0g(v0 + s (v1 − v0)) ds, v0, v1 ∈ RN

    where v is the entropy variable associated with a strictly convexentropy.I Entropy conservative scheme, satisfying

    ddt

    U(uj) +1h

    (G∗j+1/2 −G∗j−1/2

    )= 0

    with

    G∗(v0, v1) =12

    (G(v0) + G(v1)) +12

    (v0 + v1) g∗(v0, v1)

    − 12

    (v0 · g(v0) + v1 · g(v1)

    )I Second-order accurate, with (conservative) equivalent equation

    ∂tu + ∂x f(u) =h2

    6∂x

    (− g(v)xx +

    12

    vx · ∂xDg(v))

  • Theorem. (Third-order, PLF - Rohde, 2000) Given any symmetricN × N matrices B∗(v−p+2, · · · , vp), the (2p + 1)-point schemeassociated with

    g∗(v−p+1, · · · , vp) =∫ 1

    0g(v0 + s (v1 − v0)) ds

    − 112

    ((v2 − v1) · B∗(v−p+2, · · · , vp)

    − (v0 − v−1) · B∗(v−p+1, · · · , vp−1))

    is entropy conservative, with entropy flux

    G∗(v−p+1, · · · , vp) =12

    (v0 + v1) · g∗(v−p+1, · · · , vp)

    − 12

    (ψ∗(v−p+2, · · · , vp)+ψ∗(v−p+1, · · · , vp−1)

    ).

    When p = 2 and B∗(v , v , v) = B(v)(

    = Dg(v)), this five-point scheme

    is third-order, at least.

  • 6. COMPUTING KINETIC FUNCTIONS. 6.1 Cubic conservation law

    ut + (u3)x = εuxx + α ε2 uxxx

    Kinetic function Scaled entropy dissipationφ(s)/s2 (versus shock speed s)

    uL

    u M

    2 4 6 8 10 12 14 16 18 20

    -18

    -16

    -14

    -12

    -10

    -8

    -6

    -4

    -2 Forth orderSixth orderEighth orderTenth orderExact

    Shock speed

    Scaledentropydissipation

    100 200 300

    -0.5

    -0.4

    -0.3

    -0.2

    -0.1

    Forth orderSixth orderEighth orderTenth orderExact

  • 6.2 The kinetic relation for the Camassa-Holm model

    ut + (u3)x = εuxx + α ε2(utxx + 2ux uxx + u uxxx

    )Theory.

    I Well-posedness for the initial-value problemBressan, Constantin, Karlsen, Coclite, Raynaud

    I Kinetic relations via traveling wave analysis: open problem

    Numerical investigationI Existence of a kinetic function ? Globally monotone ?

    I Relation with the linear diffusive-dispersive model ?

  • Shocks with moderate strength.

    uL0.5 0.75 1 1.25 1.5

    -1.6-1.5-1.4-1.3-1.2-1.1-1

    -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2 Fourth order

    Sixth orderEighth orderTenth orderEntropy bounds

    The kinetic functions for the linear diffusion-dispersion andCamassa-Holm models almost coincide for shocks with moderatestrength.

  • Shocks with large strength.

    uL50 100 150 200-200

    -175

    -150

    -125

    -100

    -75

    -50

    -25

    Fourth orderSixth orderEighth orderTenth orderEntropy bounds

    uL50 100 150 200-200

    -175

    -150

    -125

    -100

    -75

    -50

    -25

    Fourth orderSixth orderEighth orderTenth orderEntropy bounds

    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)

    )ρ2x

    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)

    I Asymptotic conditions: p(ρ) ∼ ργ, µ(ρ) ∼ ρa , κ(ρ) ∼ ρbwhen ρ→ 0 (vacuum) and ρ→ +∞ (unbounded density)

    I Coercivity condition on the capillarity κ(ρ) (see below)

    Germain-PLF, The finite energy method for compressible fluids. TheNavier-Stokes-Korteweg model, Comm. Pure Appl. Math. (2015)

  • Notion of weak solutions

    I Finite total energy

    supt≥0

    ∫ (12ρu2 + ρe(ρ) +

    12κ(ρ)ρ2x

    )(t , x) dx < +∞,

    where the internal energy e = e(ρ) is defined by e′(ρ) := p(ρ)ρ2 .

    I Finite total effective energy

    supt≥0

    ∫ (12ρũ2 + ρe(ρ) +

    12κ(ρ)ρ2x

    )(t , x) dx < +∞,

    where the effective velocity is ũ = u + µ(ρ)ρ2 ρx .

    This class of weak solutions allows for shock waves, vacuum regions,unbounded fluid density, etc.

    We have also some (singular) bounds on derivatives, deduced fromthe dissipation terms.

  • Effective Navier–Stokes–Korteweg system

    I Given a constant ω, we define the ω-effective velocity and theω-effective capillarity

    ũω = u + ωµ(ρ)

    ρ2ρx , κ̃

    ω = κ − ω(1 − ω)µ2

    ρ3

    I If ρ,u) is a solution to NSK, then (ρ, ũω) solves

    ρt + (ρũω)x =(ωµ(ρ)

    ρρx

    )x

    (ρ ũω)t +(ρ(ũω)2 + p(ρ)

    )x

    = Mω[ρ,u]x + Kω[ρ]x

    with modified viscosity and capillarity

    Mω[ρ,u] =µ(ρ)

    ρ

    ((1 − ω)ρ ũωx + ωρx ũω

    )Kω[ρ] =ρ κ̃ω(ρ)ρxx +

    12

    (ρκ̃ω′(ρ) − κ̃ω(ρ)

    )ρ2x

    Case ω = 1: Bresch-Desjardins, Mellet-Vasseur. Case ω = 1/2: Jüngel

  • The strong coercivity condition

    I Effective energy balance law

    Ẽ [ρ,u](t) +∫ t

    0D̃ [ρ,u](s) ds = Ẽ [ρ,u](0)

    D̃ [ρ,u](t) =∫R

    µ(ρ)ρ2 p′(ρ)ρ2x + µ(ρ)ρ κ(ρ)(ρ2xx + ζ(ρ)ρ4x)dx

    − 3ζ = 12κ′′

    κ+

    ρ

    )′′ ρµ

    I Strong coercivity (SC) condition

    There exists C0 > 0 such that for all function ρ = ρ(x) > 0

    (SC) D̃ [ρ,u](t) ≥ C0∫ (

    ρ2xx +ρ4xρ2

    )µ(ρ)κ(ρ)

    ρdx

  • The family of entropy pairsI The key compactness property

    NOT based on a “Sobolev embedding theorem” (a bound onhigh-order derivatives implies a compactness property forlow-order derivatives), but:

    based on the STRUCTURE of the Euler equations and an INFINITE list ofBALANCE LAWS. (DiPerna 1985, Germain-PLF 2015)

    I Balance laws generated from a Green kernel (χ, σ)(ηψ(ρ,u)

    )t

    +(qψ(ρ,u)

    )x≤ O(ε) for all convex ψ = ψ(v)

    (ηψ,qψ)(ρ,u) =∫R

    (χ, σ)(ρ,u, v)ψ(v) dv

    I Polytropic p(ρ) = kργ

    χ(ρ,u, v) =(ργ−1 − (v − u)2

    ) 3−γ2(γ−1)+

    σ(ρ,u, v) =(u +

    γ − 12

    (v − u)) (ργ−1 − (v − u)2

    ) 3−γ2(γ−1)+

    I Real fluids p(ρ) ' kργ: nonlinear superposition formula(expansion in a series of Bessel functions)