Computation of entropic measure-valued solutions for Euler equations

Transcription

1 Computation of entropic measure-valued solutions for Euler equations Eitan Tadmor Center for Scientific Computation and Mathematical Modeling (CSCAMM) Department of Mathematics, Institute for Physical Science & Technology University of Maryland Laboratoire Jacques-Louis Lions Universite Pierre et Marie Curie, June 7 6 Collaborators: U. Fjordholm R. Käppeli S. Mishra

2 Prologue. Perfect derivatives and conservative differences u }{{} u ν+ u ν uu }{{} ( ) uν+ u ν u ν but... u 3 u }{{} u 3 ν ( uν+ u ν b a b a u d = boundary terms of u ( u ν+ u ν ) = boundary terms (u ) {}}{ uu d = boundary terms of u telescoping sum {}}{ ( u ν u ν+ u ν u ν b a ) ) = quadratic boundary terms 4 (u4 ) {}}{ u 3 u d = quartic boundary terms of u no perfect deriv. {( }}{ u 3 ν u ν+ u ν uν 3 ) no cancellation

3 Prologue. Perfect derivatives and conservative differences ( ) Set u 3 ( u 4 ν+ uν 4 4 ( 3 u 4 ν+ uν 4 4 uν+ 3 u4 ν uν 4 u3 ν uν 3 uν 3 ( 3 u 4 ν+ u 4 u3 ν 4 ν uν+ 3 u3 ν uν+ 3 u4 ν uν 4 u3 ν uν 3 uν 3 u4 ν u 4 ν u 3 ν u 3 ν = 3 ( u 3 4 ν+ uν) ( 3 uν+ 4 u4 ν uν+ 3 u3 ν ) ) boundary terms of u ) = ) boundary terms of u 4 How do we come up with ( )? Can we conserve general multipliers η (u)u = η(u) Applications entropic solution for nonlinear conservation laws

4 Strong and weak entropic solutions Euler equations with pressure law p := (γ ) ( E ρ (u + v ) ) : ρ ρu ρv ρu t ρv + ρu + p ρuv + ρuv ρv + p = E u(e + p) v(e + p) Nonlinear conservation laws: u t + f(u) = ; u := (ρ, ρu, ρv, E) balance of conservative variables u over spatial variables = (,,...) Strong solutions pointwise values u(, t) Weak solutions give up the certainty in pointvalues Instead observe the cell averages u(, t) := d u(, t)+ dt [ t+ t τ=t f ( u(+ ), τ) dτ Compute cell averages and numerical flues: t+ t τ=t f + / / u(y, t)dy ( u( ) ], τ) dτ = d dt u ν(t) + f ν+ f ν =

5 Weak solutions and entropy stability η (u), u t + f(u) = entropy {}}{ η(u) t + perfect derivatives? { }}{ η (u), f(u) = η(u) is an entropy iff η (u), f(u) = F (u) Entropy stability: { = η(u) t + F (u) Euler equations specific entropy S = ln(pρ γ ): η(u) {}}{ ( ρs) t + F (u) {}}{ ( ρus) { = Computation impose the entropy stability requirement d dt η(u ν(t)) + F ν+ F { ν =

6 D radial shock tube problem u L, r Radial initial data u () = + ɛ(δu )() u R, > r [ ] [ (δu )() sin(π ) Perturbation of steady-state ɛ = : = (δv )() sin(π ) Convergence: Fi ɛ =. keep mesh refinement...ρ ɛ at t =.4: ] (a) 8 (b) 56 (c) (d) u u / L Figure: Approimate density of the perturbed Sod problem (ɛ =.) computed with TeCNO scheme at t =.4 Stability as ɛ...

7 y y Kelvin-Helmholtz instability u L, H < H Initial state of three layers u () = u R, H or H fied u L,R ; perturbed interface H j (, ω) = j 4 + ɛy j(, ω), j =, ɛ = : u L ( ) [ 4, 3 4 ] + u R( ) [, ] [ 3 4,] is an entropic steady state Perturbed state u (, ω) = u L ( ) [H,H ] + u R ( ) [,H ] [H,]: Y j (, ω) = a n (ω) cos(b n (ω) + nπ ) an = so that ɛy j ɛ n density, sample, t = density, sample, t = Lack of convergence: Fi ω, ɛ =. keep mesh refinement...ρ ɛ at t = : KH (a) initial 8 data: amplitude(b) perturbation 56 (left) and phase (c) 5 perturbation (right)(d) with4 ɛ =.

8 The notion of entropic weak solution is not enough (d > ) (e) L rates vs. number of gridpoints with ɛ = (f) L error with respect to ɛ, at a fied 4 mesh Figure: L differences in density ρ at time t = for the Kelvin-Helmholtz problem These are NOT numerical artifacts but... contemplate lack of uniqueness [De Lellis, Szekelyhidi, E. Chiodaroli, O. Kreml, et. al.] Just because we cannot prove that compressible flows with prescribed initial values eist doesn t mean that we cannot compute them [La 7] The question is what information is encoded in our computations?

9 Back to weak solutions ν,t = δ u(,t) (certainty of strongly convergent averages) Ron DiPerna Answer: Young measures and measure valued solutions Recall weak convergence: u n u and f(u n ) f What is the relation f and f(u )? f (, t) depends linearly and positively on f: hence f (, t) = ν,t, f in terms of measure ν = ν,t Complete description in terms of Young measure ν,t Luc Tartar Certainty: strong convergence with Dirac Mass ν,t = δ u (,t) f = f(u ) Entropy measure valued (EMV) solutions: u(, t) ν,t : u t + f(u) = t ν,t, id + ν,t, f = η(u) t + F (u) t ν,t, η + ν,t, F ν,t is the (weighted) probability of having the value ν,t, id at (, t) Measure valued solutions give up the certainty in a given solution Instead observe averages in configuration space

10 Young measures as quantifiers of uncertainty (a) t = (b) t =.5 (c) t = (d) t =.5 (e) t = Figure: The approimate PDF for density ρ ν,t at = (.5,.7). More about this later... how do we compute these Entropic Measure-Valued solutions (EMVs)?

11 Computing Entropy Measure Valued (EMV) solutions (Formally) arbitrarily high-order of accuracy; Entropy stability as a selection principle Essentially non-oscillatory in the presence of discontinuities; Convergence for linear systems; How to realize (the configuration space of) EMVs? Computationally efficient; The class of the entropy Conservative Non-Oscillatory (TeCNO) schemes which compute faithful samples required for the ensemble of EMVs b,c Fjordholm, Mishra, ET, Arbitrarily high-order accurate entropy stable schemes..., SINUM b Fjordholm, Kappeli, Mishra & ET, Construction of approimate entropy measure valued solutions for hyperbolic systems of conservation laws, FoCM (5) c FMT, On the computation of measure-valued solutions, Acta Numerica (6)

12 Entropy inequality: PDEs numerical approimations η (u), u t + f(u) = Semi-discrete approimations: d dt u ν(t) + [ ] f ν+ f ν η (u), f(u) =F (u) = η(u) t + F (u) { = = f ν+ = f(u ν p+,..., u ν+p ) Entropy conservative discretization given η find f s.t. d f f dt u ν+ ν(t) + ν =? d dt η(u ν(t)) + F ν+ F ν { = Does so that = η (u ν ), f ν+ f ν ν η(u ν (t)) { =? Fν+ F ν } ν η(u ν ())

13 Entropy variables and entropy conservative schemes Fi an entropy η(u). Set Entropy variables: v v(u) := η (u). Conveity of η( ) implies that u v is -: v ν = η (u ν )... Entropy conservation: η (u ν ), f ν+ perfect difference { }}{ vν, f f ν+ ν if and only if f?= ν Fν+ F ν perfect difference { }}{ vν+ v ν, f ν+ : Entropy conservative flu: v ν+ v ν, f = ψ(v ν+ ν+ ) ψ(v ν ) Entropy flu potential: ψ(v) := v, f(v) F (u(v)) Abbreviation entropy conservative schemes: v ν+, f = ψ ν+ ν+

14 #. Scalar eamples: f ν+ = ψ(v ν+) ψ(v ν ) v ν+ v ν Toda flow: u t + (e u ) = with Ep entropy pair: (e u ) t + (e u ) = η(u) = v(u) = e u, F (u) = eu and potential ψ(v) := vf F = v Entropy-conservative flu: f ν+ = ψ(v ν+) ψ(v ν ) v ν+ v ν = v ν+ v ν = v ν+ v ν (v ν + v ν+ ) = [ ] e uν + e u ν+ d Toda flow: dt u ν(t) = euν+(t) e uν (t) d e u ν(t) = Const. dt Inviscid Burgers: u t + ( u ) = quadratic entropy ( u ) t + ( 3 u3 ) = η(u) = u u3, v(u) = u, F (u) = 3 and potential ψ(v) := vf F = 6 u3 Entropy conservative 3 -rule : d dt u ν(t)= [ u ν+ u ] ν [ u ν u ν+ u ν ] u ν(t) =Const.

15 #. Linear equations Linear transport: How to discretize u such that u t + u = conserves the balance of quartic perfect derivatives ( u 4) t + ( u 4) =? Conservative differences of u and 4u 3 u = ( u 4) : η(u) = u 4, v = 4u 3, f (u) = u, F = u 4 and ψ(u) = 3u 4 = ψ ν+ = 3 v ν+ 4 u 3 ν+ u3 ν ( ) Indeed set u =ν 3 f ν+ 3 4 ( u 4 ν+ uν 4 uν+ 3 u3 ν ( 3 u 4 ν+ u 4 u3 ν 4 ν uν+ 3 u3 ν = 3 4 uν+ 4 u4 ν, u ( u 4 ν+ uν 4 4 u4 ν u 4 ν u 3 ν u 3 ν u4 ν u 4 ν u 3 ν u 3 ν f f ν+ ν u4 ν u 4 ν u 3 ν u 3 ν ) then uν+ 3 u3 ν ) boundary terms of u ) ( u 3 ν+ u 3 ν) ( u 4 ν+ u4 ν u 3 ν+ u3 ν = ) quartic boundary terms

16 #3. Second and higher-order accuracy The u 4 entropy conservative flu is nd-order accurate: f f ν+ ν f ν+ = 3 uν+ 4 u4 ν 4 uν+ 3 : u3 ν is nd-order approimation of u 8u 3 ν+ u f = ψ ν+ u = 3 uν+ 4 u4 ν 4 uν+ 3 3 u3 ν 4 6u ν+ u u ν+ The general case: epress the flu in the viscosity form viscosity form : f ν+ d dt u ν(t) = nd-order: = (f ν + f ν+ ) D ν+ (u ν+ u ν ) nd order centered differencing (Du ) {}}{ [ f (uν+ ) f (u ν ) ] + [{}} ]{ D u ν+ ν+ D u ν ν D := ( ( ν+ ξ ) f ( ) ) u 8 ν+ dξ u ν+ f ν+ ξ= The entropy-conservative flu is nd-order accurate Entropy conservative flues to any order of accuracy (LeFloch & Rohde )

17 #4. Systems: v ν+ v ν, f ν+ = ψ(vν+ ) ψ(v ν ) Choice of path: N linearly independent directions { r j} N Intermediate states { v j ν+ } N j= : Starting with v = v ν+ ν, and followed by ( v ν+ v ν+ v ν ) v j+ = v j + l j, v ν+ ν+ ν+ r j, j =,,..., N (v N+ = v ν+ ν+ ) [ET 3] The conservative scheme d dt u ν(t) = [ ] f f ν+ ν N ψ ( v j+ ) ( f ν+ ψ v j ) = ν+ ν+ j= l j l j, v ν+ is entropy conservative: v ν+ v ν, f ν+ = ψ(vν+ ) ψ(v ν ) j=

18 #5. Entropy conservative Euler scheme (with W.-G. Zhong) f ν+ = N ψ ( v j+ ) ( ν+ ψ v j ) ν+ l j l j, v ν+ j= Entropy function: η(u) = ρs E/e S + γ + Euler entropy variables: v(u)=η u (u)= q/θ /θ Euler entropy flu potential ψ(v) = v, f F (u) = (γ )m path in phase-space: v = v ν, v j+ = v j + l j, v ν+ r j, v 4 = v ν+ {r j } 3 : three linearly independent directions in v-space (Riemann path) j= { l j } 3 : the corresponding orthogonal system j= { m j } 3 : intermediate values of the momentum along the path j= 3 f m j+ m j = (γ ) ν+ l j, v j= ν+ lj

19 #6. Affordable entropy conservative flues for Euler eqs [Ismail & Roe & (9)] D Euler eqs u t + f + g y = : Epress an entropy conservative flu f := (f, f, f 3, f 4 ) ν+ ρ in terms of z := u z ν+ := (z ν + z ν+ ) p v and the averages z ln z ν+ := ν+ p log(z) ν+ f ν+ f ν+ = (z ) ν+ (z 4 ) ln ν+ = (z 4) ν+ (z ) ν+ + (z ) ν+ (z ) ν+ f ν+ f 3 ν+ f 4 ν+ = (z 3) ν+ (z ) f ν+ ν+ = (z ) ν+ ( γ+ γ (z ) ln ν+ f + (z ν+ ) ν+ f + (z ν+ 3 ) ν+ f 3 ν+ satisfies the compatibility relation v ν+ v ν, f ν+ = ψ(v ν+ ) ψ(v ν ) )

20 ρ p u D Euler,density,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.,T ma =..6 D Euler,velocity,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.,T ma =..9 t=. t=.5 t=..4 t=. t=.5 t= D Euler,pressure,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.,T ma =..38 D Euler, entropy conservative scheme w/ 3rd R K mtd, entropy vs time,entropy = ρ ln(pρ γ ) t=..9 t=.5 t= U (total entropy) t Entropy conservative results for Euler s Sod problem: density, velocity, pressure & entropy. spatial grids, η(u) = ρ ln (pρ ) γ

21 ρ p u.9 D Euler,density,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.5,T ma =. t=. t=.5 t=..6.4 D Euler,velocity,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.5,T ma =. t=. t=.5 t= D Euler,pressure,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.5,T ma =. t=. t=.5 t= D Euler, entropy conservative scheme w/ 3rd R K mtd, entropy vs time,entropy = ρ ln(pρ γ ) U (total entropy) t Entropy conservative density, velocity, pressure & entropy w/4 spatial grids, η(u) = ρ ln (pρ ) γ Does not enforce physical solution with numerical viscosity

22 The question of entropy stability dissipation Entropy conservation: η(u) t + F (u) = Entropy decay due to shock discontinuities (La): η(u) t + F (u) Entropy decay is balanced by perfect derivatives: η(u(, t ))d η(u(, t ))d, t > t Q. How much entropy decay is enough? Entropic measure-valued solutions: d f f dt u ν+ ν(t)+ ν = ɛ(dv ) ; d dt η(u ν(t))+ F ν+ F ν =. ɛ dictates the small scale of dissipation...

23 #7. Entropy balance in Navier-Stokes (NS) eq s A semi-discrete scheme of NS equations u t + f(u) = ɛ (D( )v ) {}} vν { η (u ν ), d dt u ν(t) + ( ) ( f f = ɛ D ν+ ν ν+ Recall f ν+ is entropy-conservative: Euler flues conserve entropy: d dt d d NS dissipate: η(u η(u ν (t)) ν (t)) = = ɛ dt dt ν ν ν d ( ρ ν S ν ) = dt ν v ν+ v ν+, f ν+ D ν = ψ ν+ η(u ν (t)) = ψ ν+ v ν+, D ν+ v ν+ v ν ) = ν viscosity heat conduction {}}{{}}{ (λ + µ) ( q ν+ ) ( ) κ ( θ ν+ ) ( ) θ ν+ ν θ ν+ ν

24 ρ p u.9 D N S,density,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.5 t=. t=.5 t=.. D N S,velocity,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.5 t=. t=.5 t= D N S,pressure,λ+µ=.8e 5,γ=.4,C v =76,κ=,entropy= ρ ln(pρ γ ), t=.5e 5, =.5 t=. t=.5 t=..38 D N S, entropy conservative scheme w/ 3rd R K mtd,entropy vs time,entropy = ρ ln(pρ γ ) U (total entropy) t Navier-Stokes equations for Sod s problem: viscosity but no heat conduction; 4 spatial grids. η(u) = ρ ln (pρ ) γ

25 ρ p u D Euler,density,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =.5,T ma =..4 D Euler,velocity,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =.5,T ma =..9 t=. t=.5 t=.. t=. t=.5 t= D Euler,pressure,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =.5,T ma =..38 D Euler, entropy conservative scheme w/ 3rd R K mtd, entropy vs time,entropy = ρ ln(pρ γ ) t=..9 t=.5 t= U (total entropy) t Entropy conservative results for Sod s problem: heat conduction but no viscosity; 4 spatial grids. η(u) = ρ ln (pρ ) γ

26 ρ p u D N S,density,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =..4 D N S,velocity,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =..9 t=. t=.5 t=.. t=. t=.5 t= D N S,pressure,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =..38 D N S, entropy conservative scheme w/ 3rd R K mtd,entropy vs time,entropy = ρ ln(pρ γ ) t=..9 t=.5 t= U (total entropy) t Navier-Stokes equations Sod problem: both viscosity and heat conduction; spatial grids, η(u) = ρ ln (pρ ) γ

27 ρ p u D N S,density,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =.5. D N S,velocity,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =.5.9 t=. t=.5 t=. t=. t=.5 t= D N S,pressure,λ+µ=.8e 5,γ=.4,C v =76,κ=.3,entropy= ρ ln(pρ γ ), t=.5e 5, =.5 t=. t=.5.9 t= D N S, entropy conservative scheme w/ 3rd R K mtd,entropy vs time,entropy = ρ ln(pρ γ ) U (total entropy) t Navier-Stokes equations Sod problem: viscosity and heat conduction; 4 spatial grids, η(u) = ρ ln (pρ ) γ

28 #8. TeCNO: Arbitrarily high-order entropy stable schemes Set numerical viscosity f ν+ f ν+ Any D ν+ is high order and f ν+ = f D ν+ ν+ (v ν+ v ν ) such that entropy stable: v ν+, D ν+ v ν+ < > will do for entropy dissipation; but... v ν+ p ; non-oscillatory? To overcome first order accuracy: D ν+ A class of arbitrarily high-order entropy-stable schemes: ( ) f ν+ := f (v) ν+ D ν+ v ν+, v ν+ := v ν+ v+ ν Entropy stability requires v ν+, D ν+ v ν+ <...? Interface values v ± : reconstructed by ENO (Harten, Engquist, Osher, Shu,..) (i) ENO reconstruction is highly accurate: D ν+ v ν+ v ν+ (ii) It is Essentially Non-Oscillatory (hence the acronym...) ; and... (iii) [FMT ] The sign property: sign v ν+ = sign v ν+ p ( )!

29 Back to computation of entropy measure valued solutions EMV solutions ν,t : t ν,t, id + ν,t, f = How to compute a realization of the Young measure ν,t? Every Young measure can be realized as a law of random variable: (Ω, Σ, P) : ν,t (E) = P(u(, t, ω) E) Initial data: Realize the law of σ = ν (,t=) by a random field u (, ω) Classical case atomic initial delta: σ = δ u () Beyond classical case uncertainty in initial measurement Compute the entropy measure valued solutions: To be well-defined, we need to make sure {u(, t, ω)} form a measurable ensemble in the right space (Ω, Σ, P) (X, B(X )) Scalar case X = L (Risebro, Schwab, Weber,...) Systems? Eample Back to D Euler realize steady KH as MC average: δ u () = Eu (, ω) M M u k (, ω), u (, ω) = a n cos(b n + nπ ) k=

30 #9. Computation of EMV solutions cont d Evolution: compute the ensemble of entropic solutions u,k (, ω) u,k (, t, ω) Realize the EMVs by propagating the law of MC simulation ν,t = M δ M u,k (,t,ω), ν,t, g(λ) = g(u,k (, t, ω)) M k= This is an approimate entropic MV solution: for all φ(, t) C φ t ν,t, id + φ ν,t, f t + φ t ν,t, id + φ ν,t, f ddt + φ( ν, φ(, ) σ ) σ, id, id d = = ν,t n R + R d ν R d and for all conve entropy pairs (η, F ) with φ C φ t ν,t, η + φ ν,t, F t + φ t ν,t, η + φ ν,t, F ddt + φ( ν, φ(, ) σ ) σ, id, id d ν,t n R + R d ν R d Entropic ν,t with weak BV-bound weak lim ν,t k = ν,t is an EMV

31 On entropic measure-valued solutions... Entropic ν,t with weak BV-bound weak lim ν,t The scalar case Wasserstein W -stability: for all ξ s in R: ν,t, u(, t) ξ d σ, u () ξ d. R d R d In particular, if σ = δ u at t = then ν,t = δ u(,t) No uniqueness for non-atomic initial data = ν,t is an EMV Systems weak-strong W -stability: if u W, classical solution ν,t, u(, t) ξ d σ, u () ξ d. R d R d There is no uniqueness but compute the observables the mean: ν,t, λ Eu (, t, ω) the variance: ν,t, λ E ( u (, t, ω) Eu ), etc.,

32 #. The computed observables (a) 8 (b) 56 (c) 5 (d) 4 Figure: Approimate sample means of the density for the Kelvin-Helmholtz problem at t = and different mesh resolutions. All results are with 4 Monte Carlo samples (a) 8 (b) 56 (c) 5 (d) 4 Figure: Approimate sample variances of the density for the Kelvin-Helmholtz problem at t = and different mesh resolutions. All results are with 4 Monte Carlo samples.

33 Comments on the computation of EMVs MC slow convergence Eu (, t, ω) Eu,M (, t, ω) < M / Key point: Each ensemble was computed with M = 4 samples...! (a) Pointwise (b) Mean (c) Variance (d) W -rate Figure: Kelvin-Helmholtz. (a) The Cauchy rates at t = for the density (y-ais) for a single sample of the Kelvin-Helmholtz problem, vs. (b)+(c) different mesh resolutions (-ais) (d) Cauchy rates in the W distance at t = for the density (y-ais) with respect to different mesh resolutions (-ais) Richtmeyer-Meshkov instability: similar computations; other methods...

34 Computing EMVs: Young measures quantify uncertainty (a) t = (b) t =.5 (c) t = (d) t =.5 (e) t = (f) t = (g) t =.5 (h) t = (i) t =.5 (j) t = Figure: The approimate PDF for density ρ at = (.5,.7) (first row) and = (.5,.8) (second row) on a grid of 4 mesh points (a) n = 8 (b) n = 56 (c) n = 5 (d) n = 4 Figure: The approimate PDF for density ρ at the points = (.5,.7) (first row) and = (.5,.8) (second row) a series of meshes

35 Entropy stability theory for difference approimations of nonlinear conservation laws..., Acta Numerica 3 FMT, ENO reconstruction and ENO interpolation are stable, FoCM FMT Arbitrarily high order accurate entropy stable ENO schemes for systems of conservation laws, SINUM FM(K b )T, Construction of approimate entropy measure valued solutions for hyperbolic systems of conservation laws, J. FoCM (5) FMT, Computation of measure-valued solutions Acta Numerica (6) Fjordholm, Lanthaler, Mishra, Statistical solutions of hyperbolic conservation laws I: Foundations, ArXiv. Fjordholm, Mishra Tadmor; b R. Kappli

36 THANK YOU