ENTROPY STABLE ENO SCHEMES

LiteratureEntropy Stable ENO Scheme

Stability Results for ENONumerical Experiments

Arbitrarily High-Order Essentially Non-OscillatoryEntropy Stable Schemes For Systems Of

Conservation Laws

Hamed Zakerzadeh

German Research School for Simulation Sciences

January 18, 2013

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Outline

1 Literature

2 Entropy Stable ENO SchemeIntroductionEntropy conservative fluxesNumerical diffusion operators

3 Stability Results for ENOENO ProcedureThe sign property for ENO reconstructionBoundedness of relative jumps of ENO reconstructionENO interpolation

4 Numerical ExperimentsSod shock tubeShock-entropy wave interaction

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Literature

[FMT, ’11] Fjordholm, Ulrik S., Siddhartha Mishra, and EitanTadmor.ENO reconstruction and ENO interpolation are stable,Foundations of Computational Mathematics, (2011): 1-21.

[FMT, ’12a] Fjordholm, Ulrik S., Siddhartha Mishra, and EitanTadmor.Entropy stable ENO scheme,“HYP2010: 13th International Conference on Hyperbolic Problems,Beijing, 2010, vol 1, Contemporary Appl. Math., Higher Ed. Press 17(2012) 12-27.

[FMT, ’12b] Fjordholm, Ulrik S., Siddhartha Mishra, and EitanTadmor.Arbitrarily High-order Accurate Entropy Stable EssentiallyNon-oscillatory Schemes for Systems of Conservation Laws,

SIAM Journal on Numerical Analysis 50.2 (2012): 544-573.

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Two important outcomes...

Jump of the reconstructed ENO atinterfaces has the same sign asunderlying cell averages.

This jump is upper-bounded interms of the jump of the underlyingcell averages.

Combine the ENO withentropy conservative fluxes

⇓New entropy stable ENOschemes of arbitrary order

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Two important outcomes...

Jump of the reconstructed ENO atinterfaces has the same sign asunderlying cell averages.

This jump is upper-bounded interms of the jump of the underlyingcell averages.

Combine the ENO withentropy conservative fluxes

⇓New entropy stable ENOschemes of arbitrary order

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IntroductionEntropy conservative fluxesNumerical diffusion operators

Continuous Settings

System of conservation laws

ut + f(u)x = 0 ∀(x, t) ∈ R,u(x, 0) = u0(x) ∀x ∈ R.

Here, u : R× R+ 7→ Rm is the vector of unknowns and f isthe (non-linear) flux vector.

Extra admissibility criteria: Entropy Condition

“continuous solutions of non-linear equations exist onlyfor a limited time interval. Such solutions can,nevertheless, be continued for all future times asdiscontinuous solutions which satisfy the conservationlaws in the integrated sense. . . [they] are not determineduniquely by their initial values.”[Lax, ’71]

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Continuous Settings

Assume that there exists a convex function E : Rm 7→ R andfunctions v, Q s.t.

v = ∂uE, ∂uQ = 〈v, ∂uf〉.

Then, just multiplying by vT shows that smooth solutions ofconservation law satisfy the entropy identity.

Et +Qx = 0.

But, the entropy has to be dissipated at shocks → entropyinequality.

Et +Qx ≤ 0.

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Conservative finite difference schemes

Consider a uniform Cartesian mesh.

The conservative finite difference (finite volume) methodupdates point values (cell averages in Ii) of the solution uresulting in the semi-discrete scheme:

dtui(t) = − 1

∆x(Fi+1/2(t)− Fi−1/2(t)),

with numerical flux Fi+1/2 = F(ui(t),ui+1(t)) computedfrom the (approximate) solution of the Riemann problem atthe interface.

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Objectives

No entropy stability results for high-order numerical schemes basedon the TVD, ENO, WENO, and DG procedures, are available[FMT, ’12b].

We present a class of schemes that are

(formally) arbitrarily order accurate

entropy stable for any system of conservation laws

essentially non oscillatory around discontinuities

computationally efficient

called

TeCNO⇐= Entropy conservative fluxes + Numerical diffusion

(using ENO reconstruction procedure)

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Entropy conservative fluxes

Theorem 2.1. [Tadmor, ’87b] Two-point entropy conservative flux(Tadmor’s flux)

Assume that a consistent numerical flux F̃i+1/2 satisfies

JvKTi+1/2F̃i+1/2 = JψKi+1/2.

Then the scheme with numerical flux F̃i+1/2 is entropy conservative, itscomputed solutions satisfy the discrete entropy equality

dtE(ui(t)) = − 1

(Q̃i+1/2 − Q̃i−1/2

with the numerical entropy flux

Q̃i+1/2 = vTi+1/2F̃i+1/2 − ψi+1/2.

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Entropy conservative fluxes

Theorem 2.1. (continued)

Then we can show that at least one solution for (1) exists as

F̃i+1/2 =

ξ=0f(vi+1/2(ξ)

vi+1/2(ξ) := vi + ξ∆vi+1/2, ∆vi+1/2 := vi+1 − vi,

that gives the second-order accuracy for the scheme.

It may be very hard to evaluate the path integral.=⇒ find simple and inexpensive solution for specific systems

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Example: Euler equations

ρρuE

, f(u) =

ρuρu2 + p(E + p)u

,where E = p

γ−1 + 12ρu

2. Also an entropy-entroy flux pair is

E =ρs

γ − 1, Q =

−ρusγ − 1

with s = log(p)− γlog(ρ) as thermodynamic entropy. So

γ−sγ−1 −

2pρup−ρp

, ψ(u) = ρu.

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Example: Euler equations

Ismail and Roe [Roe, ’09] have constructed an explicit solution forthe Euler equations. Define

, (a)lni+1/2 :=JaKi+1/2

Jlog(a)Ki+1/2.

So, the entropy conservative flux F̃i+1/2 = [F̃ 1i+1/2F̃

2i+1/2F̃

3i+1/2]

F̃ 1i+1/2 = z2i+1/2(z

3)lni+1/2,

F̃ 2i+1/2 =

z3i+1/2

z1i+1/2

+z2i+1/2

z1i+1/2

F̃ 1i+1/2,

F̃ 3i+1/2 =

z2i+1/2

z1i+1/2

(γ + 1

γ − 1

(z3)lni+1/2

(z1)lni+1/2

+ F̃ 2i+1/2

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High-order entropy conservative fluxes

Following the procedure of LeFloch, Mercier, and Rohde [Lefloch,’02] we construct a linear combination of Tadmor’s flux=⇒ 2pth-order accurate entropy conservative fluxes for any p ∈ N.

Theorem 2.2. [Lefloch, ’02]

For p ∈ N assume that αp1, . . . , αpp solve the the p linear equations

p∑r=1

rαpr = 1,

p∑i=1

i2s−1αpr = 0 (s = 2, . . . , p),

and define

F̃2pi+1/2 =

p∑r=1

r−1∑s=0

F̃(ui−s,ui−s+r).

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Then the finite difference scheme with flux F̃2p is

consistentF̃2p(ui, . . . ,ui) = f(ui),

2pth-order accurate, i.e for sufficiently smooth solutions u

(F̃2p(ui−p+1, . . . ,ui+p)− F̃2p(ui−p, . . . ,ui+p−1)

)= ∂xf(ui) +O(∆x2p),

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and entropy conservative, satisfies the discrete entropyidentity by

Q̃2pi+1/2 =

p∑r=1

r−1∑s=0

Q̃(ui−s,ui−s+r)

p∑r=1

r−1∑s=0

((vi−s + vi−s+r)

T F̃(ui−s,ui−s+r)

− (ψi−s + ψi−s+r)).

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Numerical diffusion operators

The entropy conservative schemes will produce high-frequencyoscillations near shocks=⇒ add some dissipative mechanism for entropy=⇒ entropy stable schemes whose solutions satisfy a discreteentropy inequality

dtE(ui(t)) +

(Q̂i+1/2 − Q̂i−1/2

)≤ 0.

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First-order numerical diffusion operators

We modify by 1

Fi+1/2 = F̃i+1/2 −1

2Di+1/2JvKi+1/2 (1)

Lemma 2.1. [Tadmor, ’87b]

The scheme flux (1) entropy stable, its solutions satisfy

dtE(ui) +

∆x(Q̂i+1/2 − Q̂i−1/2)

= − 1

4∆x(JvKTi+1/2Di+1/2JvKi+1/2 + JvKTi−1/2Di−1/2JvKi−1/2) ≤ 0,

Q̂i+1/2 = Q̃i+1/2 −1

2vTi+1/2Di+1/2JvKi+1/2.

1D is an arbitrary S.P.D. matrix.

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First-order numerical diffusion operators

We will use diffusion matrices in the form of Di+1/2 = RΛRT .2

R is the matrix of eigenvectors of the flux Jacobian ∂uf and Λ is apositive diagonal matrix that depends on the eigenvalues of theflux Jacobian.

Roe-type diffusion operator

Λ = diag(|λ1|, . . . , |λm|),

where |λ1|, . . . , |λm| are eigenvalues of ∂uf(ui+1/2).

Rusanov -type diffusion operator

Λ = max(|λ1|, . . . , |λm|)IDm.

2We will see the discussion later on.

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High-order numerical diffusion operators

Remark

As the term JvKi+1/2 is of the order of ∆x, the scheme with flux(1) is in general only first-order accurate even if we use veryhigh-order entropy conservative flux.

=⇒ reconstruction of the entropy variables v

we define our higher-order numerical flux as

Fki+1/2 = F̃2pi+1/2 −

2Di+1/2〈v〉i+1/2, (2)

〈v〉i+1/2 = v−i+1−v+i , v+

i = vi(xi+1/2), v−i = vi(xi−1/2).

and vi(x) is the piecewise (k − 1)th-degree (kth-order)polynomial.

(choose 2p = k or (k + 1) for even (odd) k respectively)

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Remark

As the term JvKi+1/2 is of the order of ∆x, the scheme with flux(1) is in general only first-order accurate even if we use veryhigh-order entropy conservative flux.

=⇒ reconstruction of the entropy variables v

we define our higher-order numerical flux as

Fki+1/2 = F̃2pi+1/2 −

2Di+1/2〈v〉i+1/2, (2)

〈v〉i+1/2 = v−i+1−v+i , v+

i = vi(xi+1/2), v−i = vi(xi−1/2).

and vi(x) is the piecewise (k − 1)th-degree (kth-order)polynomial. (choose 2p = k or (k + 1) for even (odd) k respectively)

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might not be entropy stable? → modify the reconstruction

Lemma 2.2.

For each i ∈ Z, let {rli+1/2}ml=1 be a basis of Rm for each i and

define the non-singular Ri+1/2 = [r1i+1/2| . . . |rmi+1/2] and Λi+1/2 as

a non-negative diagonal matrix

Di+1/2 := Ri+1/2Λi+1/2RTi+1/2.

Let reconstruction polynomial vi(x) s.t. ∀i,∃ diagonal Bi+1/2 ≥ 0s.t.

〈v〉i+1/2 =(RTi+1/2

)−1Bi+1/2R

Ti+1/2JvKi+1/2. (3)

Then the scheme with numerical flux (2) is entropy stable.

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might not be entropy stable? → modify the reconstruction

Lemma 2.2.

For each i ∈ Z, let {rli+1/2}ml=1 be a basis of Rm for each i and

define the non-singular Ri+1/2 = [r1i+1/2| . . . |rmi+1/2] and Λi+1/2 as

a non-negative diagonal matrix

Di+1/2 := Ri+1/2Λi+1/2RTi+1/2.

Let reconstruction polynomial vi(x) s.t. ∀i,∃ diagonal Bi+1/2 ≥ 0s.t.

〈v〉i+1/2 =(RTi+1/2

)−1Bi+1/2R

Ti+1/2JvKi+1/2. (3)

Then the scheme with numerical flux (2) is entropy stable.

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Lemma 2.2. (continued)

i.e. its computed solutions satisfy the entropy dissipation estimate

dtE(ui) +

∆x(Q̂ki+1/2 − Q̂

ki−1/2) ≤ 0,

Q̂ki+1/2 = Q̃2pi+1/2 −

2vi+1/2Di+1/2〈v〉i+1/2.

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Remark (Role of Bi+1/2)

If the reconstructed variables satisfy (3) then

Fki+1/2 = F̃2pi+1/2 −

2Ri+1/2Λi+1/2Bi+1/2R

Ti+1/2JvKi+1/2

Remark (Discussion about choose of R and Λ)

R|Λ|R−1JuK ≈ R|Λ|R−1vuJvK = R|Λ|RT JvK.

refer to [Barth, ’98] and [Merriam, ’89].

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Reconstruction procedure

Q: How to find such a reconstruction to satisfy positivity of matrixB?

Corollary 2.1.

Let vi, vi+1, v+i , v−i+1 be given and define the scaled entropyvariables as

w±i = RTi±1/2vi, w̃±i = RTi±1/2v±i .

Now, (3) reads

〈w̃〉i+1/2 = Bi+1/2〈w〉i+1/2 (sign property),

i.e. at each cell interface, the jump in reconstructed values has thesame sign as the jump in the underlying cell averages.

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Reconstruction procedure, How? (I)

Second-order reconstructionThe only symmetric TVD limiter that satisfies thesign property is the minmod limiter.Or use non-TVD limiters e.g. ENO limiter,second-order version of the ENO reconstruction procedure

φ(θ) =

{θ if − 1 ≤ θ ≤ 1,

1 else.

Higher-order versions of the ENO procedure?

Yes! With the results we are going to prove.Just accept for now!

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Reconstruction procedure, How? (II)

Define the point value µii = w−i and inductively

µij+1 = µij + δj+1/2 (j = i, i+ 1, . . . ),

µij−1 = µij − δj−1/2 (j = i, i− 1, . . . ).

similarly we define νii = w+i and

νij+1 = νij + δj+1/2 (j = i, i+ 1, . . . ),

νij−1 = νij − δj−1/2 (j = i, i− 1, . . . ).

These are those values we have to interpolate.

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Reconstruction procedure, How? (III)

The jump is is retained

JµiKj+1/2 = JνiKj+1/2 = δj+1/2 = 〈w〉j+1/2.

µij = νij + (µij − νij), ∀j.

So, the reconstruction is

pi(x) = qi(x) + (µij − νij), ∀x.

Hence, we find the reconstructed values as

w̃−i = pi(xi−1/2), w̃+i = qi(xi+1/2).

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Final step

Theorem 2.2. [FMT, ’12b]

For any k ≥ 1, let 2p = k (if k is even) or 2p = k + 1 (if k is odd).Define the entropy conservative flux F̃2p. Let 〈v〉 be defined bythe kth-order accurate ENO reconstruction procedure.Then the finite difference scheme with numerical flux (2) is

(k − 1)th-order accurate for smooth solutions;

entropy stable, computed solutions satisfy the discrete entropyinequality.

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ENO ProcedureThe sign property for ENO reconstructionBoundedness of relative jumps of ENO reconstructionENO interpolation

A bit of historical notes...

Essentially Non-Oscillatory method (ENO) method was initiallydesigned by Harten et al. [Harten, ’87]

s.t. for a given wj = w(xj), cell averages of a piecewise smoothfunction w(x), we construct R(x;w), a piecewise polynomialfunction of x of degree r − 1 s.t.

At all points x for which there is a neighbourhood where w issmooth,

R(x;w) = w(x) + e(x)hr +O(hr+1) (Accuracy),

R(xj ;w) = wj (Conservation),

TV (R(·;w)) ≤ TV (w) +O(hr) (ENO property).

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A bit of historical notes...

Recall Our Goal!

To show that the jump of the reconstructed ENO point values at eachcell interface

1 has the same sign as the jump of the underlying cell averages acrossthat interface.

2 can be upper-bounded in terms of the jump of the underlying cellaverages.

Compare with already existing stability results:

1 ENO property above is for sufficiently small h. [Harten, ’87]

2 ENO-SR for continuous f ∈ Cm(R\{a}) is globally rth-orderaccurate (sufficiently small h) and globally 2nd-order(∀h > 0), a gain of one order compared to linear method.[Arandiga, ’05]

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What is ENO?

Given cell averages {vi} ∈ Z over consecutive intervalsIi = [xi1/2, xi+1/2) s.t. vi := 1

|Ii|∫Iiv(x)dx, then

ENO: A v(x) =∑k

vk1Ik(x) 7→ RA v(x) =∑k

fk(x)1Ik(x),

where fk(x) are polynomials of degree p− 1 such that thepiecewise polynomial ENO reconstruction RA v(x) satisfiesaccuracy and conservation conditions and

1Ik(x) =

{1 if x ∈ Ik0 if x /∈ Ik.

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What is ENO?

Conservation enables us to recast as

ENO reconstruction =⇒ non-linear interpolation

So, define the primitive V (x) :=∫ x−∞ v(s)ds and perform the

interpolation for that!

ENO: L V (x) 7→ RL V (x) :=∑k

Fk(x)1Ik(x)

This interpolation satisfy both properties!

Note that Vj+1/2 :=

∫ xj+1/2

−∞v(s)ds =

j∑k=−∞

|Ik|vk

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ENO Reconstruction

Properties:

Use adaptive stencil, with data-dependent offset which isadapted to the smoothness of the data.

Measure smoothness by divided differences.

Use an iterative procedure to choose smoothest data for thestencil.

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Data Dependent Stencil based on divided difference

Algorithm 3.1. (ENO reconstruction: selection of ENO stencil)

Let point values of the primitive Vi−p+1/2, . . . , Vi+p−1/2 begiven.Set r1 = −1

2for j = 1, . . . , p− 1 do

if |V [xi+rj−1, . . . , xi+rj+j ]| < |V [xi+rj , . . . , xi+rj+j+1]|then

set rj+1 = rj − 1else

set rj+1 = rjend if

end forSet Fi(x) as the inrterpolant of V over the stencil {V }rp+pk=rp

Compute fi(x) := F ′i (x)

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A global, mesh-independent stability result, after 25 years!

Theorem 3.1. (The Sign Property)

Fix an integer p > 1. Given the cell averages {vi}, let RA v(x) be thepth-order ENO reconstruction of these averages.Let v+i+1/2 = RA v(x+i+1/2) and v−i+1/2 = RA v(x−i+1/2) denote left and

right reconstructed point values at the cell interface xi+1/2, Then foreach i {

if vi+1 − vi ≥ 0 then v+i+1/2 − v−i+1/2 ≥ 0,

if vi+1 − vi ≤ 0 then v+i+1/2 − v−i+1/2 ≤ 0.

Moreover, there is a constant Cp, depending only on p and on the meshratio of neighbouring grid cells s.t.

0 ≤v+i+1/2 − v

−i+1/2

vi+1 − vi≤ Cp.

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A global, mesh-independent stability result, after 25 years!

ENO reconstruction of randomly chosen cell averages [FMT, ’11]

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Notation

Divided difference for cell averages and primitive function

D[r,r+j] = V [xrj , . . . , xrj+j ], D[r,r+j] =D[r+1,r+j] −D[r,r+j−1]

xr+j − xr.

Also we have

D[−1/2,3/2] = V [x−1/2, x1/2, x3/2] =v1 − v0

x3/2 − x−1/2.

(we use it later!)

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Lemma 3.1.

v+1/2−v−1/2 =

sp∑r=rp

D[r,r+p+1] (xr+p+1 − xr)p−1∏

m=0, 6=l(x1/2 − xr+m+1)︸︷︷︸

has the same sign as (−1)r+p−1/2

remains to show that each non-zero summand has the same signas v1 − v0.

Lemma 3.2.

Let {rj}pj=1 and {sj}pj=1 be the signatures of the ENO stencilsassociated with cells I0 and, respectively, I1. Then the followingholds (for r = rp, . . . , sp):{

if D[−1/2,3/2] > 0 then (−1)r+p−1/2D[r,r+p+1] ≥ 0,

if D[−1/2,3/2] < 0 then (−1)r+p−1/2D[r,r+p+1] ≤ 0.37/̇ 53

Jumps are bounded!

Lemma 3.3.

Let rp, sp be the (half-integer) offsets of the ENO stencilsassociated with cell I0 and, respectively, I1. Then

D[r,r+p+1]

D[−1/2,3/2](−1)r+p−1/2 ≤ Cr,p, r = rp, . . . , sp,

where the constants Cr,p are defined recursively, starting withCr,1 = 1, and

Cr,p+1 =2

xr+p+2 − xrmax(Cr,p, Cr+1,p), ∀r,

and they only depend on grid sizes Ij .

38/̇ 53

. . . explicit expression for upper bound

Theorem 3.2.

v+1/2−v−

v1−v0 ≤ Cp :=

1x3/2−x−1/2

∑−1/2r=−p+1/2Cr,p

∣∣∣(xr+p+1 − xr)∏p−1m=0(x1/2 − xr+m+1)

∣∣∣ .if we use a uniform mesh, |Ii| , h we get

Cp = 2p−1

∑p−1k=0 k!(p− k− 1)!.

p Upper Bound (Cp)

1 12 23 3.3334 5.3335 8.5336 13.866

39/̇ 53

ENO interpolation, very short!

Algorithm 3.2. (ENO interpolation: selection of ENO stencil)

Let point values vi−p+1, . . . , vi+p−1 be given.Set l1 = 0for j = 1, . . . , p− 1 do

if |v[xi+lj−1, . . . , xi+lj+j−1]| < |v[xi+lj , . . . , xi+lj+j ]| thenset lj+1 = lj − 1

elseset lj+1 = lj

end ifend forSet fi(x) as the inrterpolant of v over the stencil {v}lp+p−1k=lp

fi(x) =

p−1∑j=0

v[xi+lj , . . . , xi+lj+j ]

j−1∏m=0

(x− xi+lj+m)

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Theorem 3.3., The Sign Property Revisited, ENO Interpolation

Fix an integer p > 1. Given the point values {vi}, let I v(x) be thepth-order ENO interpolant of these point values,

I v(x) =∑k

fk(x)1Ik(x), deg fk(x) ≤ p− 1.

Let v−i+1/2 := I v(x−i+1/2) and v+i+1/2 := I v(x+i+1/2) denote left and

right reconstructed point values at the cell interfaces xi+1/2. Then:{if vi+1 − vi ≥ 0 then v+i+1/2 − v

−i+1/2 ≥ 0,

if vi+1 − vi ≤ 0 then v+i+1/2 − v−i+1/2 ≤ 0.

Moreover, there is a constant Cp , depending only on p and on the meshratio of neighbouring grid cells, s.t.

0 ≤v+i+1/2 − v

−i+1/2

vi+1 − vi≤ Cp.

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Final results

If we use a uniform mesh with mesh width xj+1 − xj ≡ h then

v+1/2 − v−1/2

v1 − v0≤ Cp :=

2p−1

(p− 1)!

p−1∑l=0

∣∣∣ p−1∏m=1

(1/2− l −m)∣∣∣.

p Upper Bound (Cp)

1 12 23 3.54 65 10.3756 18.25

42/̇ 53

Sod shock tubeShock-entropy wave interaction

Numerical Experiments

We test the following schemes:

ENOk: k-th order accurate standard ENO scheme in theMUSCL formulation.

TeCNOk: k-th order accurate entropy stable scheme as (2) fork = 3, 4 and 5.

We consider the 1D Euler equations and diffusion matrix being ofthe Roe type. The eigenvalues and eigenvectors of the Jacobianare computed at the average of the left and right states. TheENO-MUSCL schemes use the standard Roe numerical flux.

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Sod shock tube

u(x, 0) =

{uL if x < 0

uR otherwise.

with ρLuLpL

,ρRuRpR

0.1250

44/̇ 53

Comparing ENO (blue circles) and TeCNO (red circles) with the referencesolution (black line) for the Sod shock tube. Density at T = 1.3 on a mesh of100 points is plotted. [FMT, ’12a]

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Shock-Entropy wave interaction [Shu, ’89]

u(x, 0) =

{uL if x < −4

uR otherwise.

with ρLuLpL

3.8571432.62936910.33333

,ρRuRpR

1 + εsin(5x)01

46/̇ 53

Comparing ENO (blue circles) and TeCNO (red circles) with a referencesolution (black line) on the Shu-Osher shock-entropy wave interaction problem.The plotted quantity is the density at time T = 1.8 on a mesh of 200 points.[FMT, ’12a] 47/̇ 53

References

[Arandiga, ’05] Arandiga, Francesc, et al.Interpolation and approximation of piecewise smooth functions,SIAM Journal on Numerical Analysis 43.1 (2005): 41-57.

[Barth, ’98] Barth, Timothy J.Numerical methods for gas dynamic systems on unstructuredmeshes,An introduction to recent developments in theory and numerics forconservation laws 5 (1998): 195-285.

[Harten, ’86] Harten, Ami, et al.Some results on uniformly high-order accurate essentiallynon-oscillatory schemes,Applied Numerical Mathematics 2.3 (1986): 347-377.

48/̇ 53

References

[Harten, ’87] Harten, Ami, et al.Uniformly high order accurate essentially non-oscillatory schemes,III,Journal of Computational Physics 71.2 (1987): 231-303.

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[Lefloch, ’00] LeFloch, Philippe G., and Christian Rohde.High-order schemes, entropy inequalities, and non-classical shocks,SIAM journal on numerical analysis 37.6 (2000): 2023-2060.

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[Shu, ’98] Shu, Chi-Wang.Essentially non-oscillatory and weighted essentially non-oscillatoryschemes for hyperbolic conservation laws,Advanced numerical approximation of non-linear hyperbolicequations (1998): 325-432.

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[Roe, ’09] Ismail, Farzad, and Philip L. Roe.Affordable, entropy-consistent Euler flux functions II: Entropyproduction at shocks,Journal of Computational Physics 228.15 (2009): 5410-5436.

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