Convection and total variation flow—erratum and improvement

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improvement

François Bouchut, David Doyen, Robert Eymard

To cite this version:

François Bouchut, David Doyen, Robert Eymard. Convection and total variation flow-erratum and

improvement. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2017, 37 (4),

pp.2139-2169. �10.1093/imanum/drw076�. �hal-03066497�

erratum and improvement

Fran¸cois Bouchut

, David Doyen

and Robert Eymard

Universit´ e Paris-Est

Abstract

This paper includes an erratum to “Convection and total variation flow” [3], that deals with a nonlinear hyperbolic scalar conservation law, regularised by the total variation flow operator (or 1-Laplacian), and in which a mistake has been written in the convergence proof of the numerical scheme to the continuous entropy solution. For correcting the proof, it is necessary to introduce an additional vanishing viscous term in the scheme. This modification imposes that the whole paper be cast in the framework of discrete and continuous solutions with unbounded support. This new version shows nevertheless a better result than the previous one, since the BV regularity and the compactness of the support of the initial data are no longer assumed.

Keywords: Hyperbolic scalar conservation law, total variation flow, 1-Laplacian, entropy formulation, finite volumes, finite elements

1 Introduction

Let us first briefly recall the problem studied in [3]. We consider a simplified model of unsteady Bingham flow with convection. This simplified model is scalar and consists in seekingu:R^d×(0, T)→Randλ:R^d×(0, T)→ R^d such that

∂tu+ divF(x, t, u)−divλ= 0 andλ∈Sgn(∇u), on QT :=R^d×(0, T), (1)

u(x,0) =u_ini(x), on R^d, (2)

where d∈N^?, T >0 is given, F :R^d×(0, T)×R→R^d is divergence-free with respect to the space variables anduini is a given function fromR^d to R. We denote by Sgn the vector sign function, which is the set-valued map fromR^d to P(R^d) defined by

λ∈Sgn(µ)⇔

(|λ| ≤1 ifµ= 0 λ= _|µ|^µ ifµ6= 0, where| · |denotes the euclidean norm inR^d.

Problem (1)-(2) is considered in this work under the following hypotheses, denoted by Hypotheses (HC) in this paper.

(HC1) The initial datumuini is assumed to belong toL¹(R^d)∩L^∞(R^d). The essential infimum and supremum ofuini are denoted bya0 andb0, respectively.

(HC2) The flux function F ∈ C¹(Q_T ×R,R^d) is assumed to be divergence-free with respect to the space variables, that is

d

X

i=1

∂Fi

∂x_i(x, t, u) = 0, ∀x= (xi)i=1,...,d∈R^d, ∀t∈[0, T], ∀u∈R. (3)

∗Email: francois.bouchut@u-pem.fr

†Email: david.doyen@u-pem.fr

‡Email: robert.eymard@u-pem.fr

1

Furthermore,^∂F_∂u is assumed to be locally Lipschitz continuous and such that, for all compact setK⊂R,

^∂F_∂u

≤CK a.e. onQT ×K, whereCK is a constant depending onK.

An entropy formulation for Problem (1)-(2) allows for the uniqueness of the entropy solution. A numerical approximation, based on a splitting scheme, is then proposed in the case where the initial data has a compact support. The hyperbolic flow is treated with finite volumes and the total variation flow is treated using P₁ finite elements. The finite volume mesh is built as a dual mesh of the finite element mesh, which makes simple the interpolation step between the two meshes. For the hyperbolic step (or finite volume step), we choose an explicit time discretisation. For the total variation flow step (or finite element step), we define an implicit scheme accounting for the non-regularity of the total variation flow operator. To guarantee the maximum principle, which is essential for the stability of the scheme, we use a non-obtuse finite element mesh.

Then the point is to prove the convergence of this numerical scheme to the unique entropy solution of Problem (1)-(2). Unfortunately, we made a mistake in the course of this proof. In [3, Proposition 4.2], the point was to derive an entropy inequality for the total variation approximation by the finite element step, holding for any regular convex entropy. This inequality, which holds at the continuous level, is also immediate at the discrete level if we restrict ourselves to the 1D case or to quadratic entropies. But we need here all the entropies in order to prove that the limit of the numerical approximation is solution to the entropy formulation. So we intended to prove such an inequality, using equation [3, (4.9)]. Unfortunately in this equation, the time and the space terms are not in agreement, the time test function is fixed and the space test function is let to be selected further, whereas it should in fact be the same function.

Turning to the correction of this wrong proof, we understood that the inequality that we tried to prove is in fact not true in the general multidimensional case (as written above, it holds in the 1D case or using quadratic entropies, but a two-dimensional counter-example is provided in Appendix A). We have therefore been led to modify the numerical scheme, by introducing a vanishing viscous term in the scheme (thanks to this term, the discrete solution is regular enough for passing to the limit in the above mentioned inequality). The compact support property, that was holding without this viscous term, is then no longer available. We have been therefore led to rewrite the whole numerical part of [3], accounting for solutions with non-compact support. Fortunately, this change does not imply to rewrite the uniqueness proof for the entropy solution of [3], since it does not account for the fact that the support of the solution is compact. Let us summarise in the next table the main differences between the original paper and this erratum.

[3] Erratum

support of the solution bounded bounded or unbounded

initial data inL^∞ with bounded variations

and bounded support inL^∞∩L¹

finite element mass lumping different from finite volume step identical to finite volume step finite element step without additional viscosity with additional viscosity estimates on discrete solution L^∞and bounded support L^∞∩L¹

inverted CFL needed for keeping bounded

support for discrete solution no longer needed

convergence proof uncorrect use of the viscous term

for passing to the limit

This erratum is organised as follows. In section 2, the concept of entropy solution for Problem (1)-(2) is redefined with respect to unbounded support for the solution. Section 3 describes the numerical approximation and the well-posedness and the maximum principle are proved. A priori estimates on the discrete solutions are provided in Section 4 and a discrete entropy formulation is established in Section 5. The convergence of the numerical approximation (and thus the existence of an entropy solution in the new framework) is finally proved in Section 6 using the results of the two previous sections. Appendix A provides a counter-example, justifying that we had to modify the scheme by addition of a vanishing viscous term.

2 Entropy formulation for nonlinear hyperbolic equation with total variation flow

In the usual entropy formulations of scalar conservation laws, the admissible entropies are the C¹ convex functions or the so-called Kruzhkov entropies. Let us recall that the Kruzhkov entropies are the functions| · −κ|

withκ∈R, the corresponding entropy fluxes being the functionsF(· ∨κ)−F(· ∧κ), wherea∨b denotes the maximum ofaandbanda∧bdenotes the minimum ofaandb.

The entropy formulation of the problem (1)-(2), owing to the term div Sgn(∇u), requires more regular entropies.

Definition 2.1: Under Hypotheses (HC), an admissible entropy is a convex Lipschitz continuous function η ∈ C^∞(R). The corresponding entropy flux is the locally Lipschitz continuous function Φ∈C⁰(QT ×R,R^d) such that

Φ(x, t, u) = 1 2

Z u a0

η⁰(s)∂F

∂u(x, t, s)ds+ 1 2

Z u b0

η⁰(s)∂F

∂u(x, t, s)ds+ 1

2(η⁰(a0)F(x, t, a0) +η⁰(b0)F(x, t, b0)). We then have ^∂Φ_∂u(x, t, u) =η⁰(u)^∂F_∂u(x, t, u), and we remark that, since the flux function F is divergence-free with respect to the space variables, the entropy flux is divergence-free in the same sense as well.

Remark 2.2 (Consistency with Kruzhkov entropy pairs): Definition 2.1 is such that, forη=|·−κ|witha0≤κ≤b0, then Φ(x, t,·) = F(x, t,· ∨κ)−F(x, t,· ∧κ). This consistency property is used in the course of the proof of the uniqueness theorem. Moreover, for any convex function η ∈C^∞(R), letting Φ be given by Definition 2.1, integrate by parts in [a₀, u] and [u, b₀] shows that the following relations hold for a.e. (x, t)∈Q_T:

η(u) = 1 2

Z b₀ a₀

η⁰⁰(s)|u−s|ds+1

2((η⁰(a0) +η⁰(b0))u+η(a0)−η⁰(a0)a0+η(b0)−η⁰(b0)b0), Φ(x, t, u) = 1

2 Z b₀

a₀

η⁰⁰(s) (F(x, t, u∨s)−F(x, t, u∧s))ds+η⁰(a0) +η⁰(b0)

2 F(x, t, u), ∀u∈[a0, b0]. (4) Regular entropy-entropy flux pairs in the sense of Definition 2.1 can now be used in the following definition of an entropy solution. This definition is classically resulting from the vanishing viscosity method (the computations are detailed in [3, Section 1.3]): assuming that, for all >0, there exists (u, λ) with

∂tu+ divF(x, t, u)−divλ−∆u= 0 and λ∈Sgn(∇u),

we multiply the preceding equation by η⁰(u)ϕfor a given admissible entropy-entropy flux pairs (η,Φ) and for a nonnegative test functionϕ∈C_c^∞(R^d×[0, T)), we integrate overQT and we let→0. Inequality (5) is then obtained, owing toη⁰⁰≥0 and to the semi-continuity of the total variation.

Definition 2.3 (Entropy solution): Under Hypotheses (HC), a function u∈L^∞(QT)∩L¹(0, T;BV(R^d)) is said to be an entropy solution of (1)-(2) if there existsλ∈L^∞(QT)^d, with|λ| ≤1 almost everywhere onQT, such that, for all admissible entropy-entropy flux pairs (η,Φ) in the sense of Definition 2.1 and all nonnegative test functionsϕ∈C_c^∞(R^d×[0, T)),

Z

Q_T

η(u)∂_tϕ+ Φ(x, t, u)−λη⁰(u)

· ∇ϕ dxdt−

Z

Q_T

ϕ

D[η⁰(u)]

dt+ Z

R^d

η(u_ini(x))ϕ(x,0) dx≥0. (5) Since η⁰ is in C¹(R), the function η⁰(u) is in L¹(0, T;BV(R^d)). Therefore, the term R

QTϕ

D[η⁰(u)]

dt is meaningful. The functionλ, which is not necessarily unique, is called a multiplier by analogy with a Lagrange multiplier. More details on functions with bounded variation are provided in [3, Section 1.1].

Theorem 2.4: Under Hypotheses (HC), there exists one and only one entropy solution of (1)-(2) in the sense of Definition 2.3.

The proof of the uniqueness part of the preceding theorem is not modified with respect to that of [3, Theorem 1.3]. The proof of the existence part is done by the proof that the numerical scheme used below is converging.

3 Numerical approximation 3.1 Notation and hypotheses

The finite element mesh, denoted byTh, is a conforming simplicial mesh of R^d of size h: Th is a (necessarily countable) set of disjoint open simplices such thatS

K∈ThK=R^d, andhis the maximum value of the diameter of allK∈ Th. The mesh is conforming in the sense that, for two distinct elementsK, LofTh,K∩Lis either

empty or a simplex included in an affine subset ofR^d with dimension strictly lower thand, whose vertices are simultaneously vertices of K and L. Therefore the set of the vertices of the mesh is countable as well, and denoted by {xp, p ∈ N}. In order to ensure the maximum principle, each element of Th is assumed to be nonobtuse; we recall that a simplex is said to be nonobtuse if the angles between any two facets are less than or equal toπ/2. For anyK∈ Th, we denote by VK ⊂Nthe set of thed+ 1 indices of the vertices ofK, and byEK the set of thed+ 1 faces ofK.

The finite volume mesh, denoted by Dh, is a polyhedral mesh of R^d such that the interface between two cells is a finite union of faces. The meshDh is a dual mesh ofTh in the sense that each cell ofDh contains one and only one node ofT_h. For anyp∈N, the cell ofD_hcontaining the nodex_p is denoted byQ_p. We assume that

∀p∈N, Q_p⊂ [

K∈Ths.t. p∈VK

K.

Let us introduce some additional notation about Dh: Np is the set containing the indices of the neighbouring cells ofQ_p, Eh is the set of couples (p, q) such thatQ_p and Q_q are neighbours andp < q,σ_p,q is the interface between two neighbour cells Q_p and Q_q, ν_p,q is the unit normal vector to σ_p,q pointing toward Q_q, m_p is the measure ofQ_p,m_p,q is the measure ofσ_p,q.

In our scheme, the unknown function is simultaneously reconstructed from the valuesv_h= (v_p)_p∈Nat the vertices using a continuous piecewise affine reconstruction denoted bybvh, and using a piecewise constant reconstruction denoted byvh:

∀vh∈R^N,vbh∈C⁰(R^d) ; bv_h|K is affine for each K∈ Th, vbh(xp) =vp, ∀p∈N, v_h∈L¹_loc(R^d) ; v_h|Qp=v_p, ∀p∈N.

LettingK∈ Th, if we denote by{xp}_p∈VKthe vertices ofKand by{φp}_p∈VK the corresponding Lagrange basis function, we can write, for anyuh, vh∈R^N,

∇bv_h|K· ∇ub_h|K = X

p∈VK

X

q∈VK

vpuq∇φ_p|K· ∇φ_q|K.

Using the fact thatP

p∈VKφ_p|K= 1, and thus P

p∈VK∇φ_p|K = 0, the preceding equation can be rewritten as

∇bv_h|K· ∇ub_h|K = X

p∈V_K

X

q∈V_K

uq(vp−vq)∇φ_p|K· ∇φ_q|K. Exchanging the roles ofpandq, we get

∇bv_h|K· ∇ub_h|K = X

p∈VK

X

q∈VK

up(vq−vp)∇φ_p|K· ∇φ_q|K. Adding the two preceding relations provides

2∇bv_h|K· ∇bu_h|K=− X

p∈VK

X

q∈VK

(up−uq)(vp−vq)∇φ_p|K· ∇φ_q|K, and therefore

Z

K

∇bvh(x)· ∇buh(x)dx= X

p∈V_K

X

q∈V_K

T_pq^K(up−uq)(vp−vq) withT_pq^K =−1 2

Z

K

∇φp(x)· ∇φq(x)dx. (6) Since the simplexK is nonobtuse, we have the standard inequality (see for example [4])

∇φ_p|K· ∇φ_q|K ≤0 andT_pq^K≥0, ∀p, q∈ VK, p6=q. (7) Let us observe that, for anyuh∈R^N, we have, denoting for a.e. x∈R^d byp(x)∈Nsuch thatx∈Qp,

for a.e. x∈R^d, |ubh(x)−uh(x)|=|(x−xp(x))· ∇buh(x)| ≤h|∇ubh(x)|, (8) which implies, if∇buh∈L¹(R^d)^d,

kuh−ubhk_L1(R^d)≤hk∇ubhk_L1(R^d)^d. (9) For the finite volume step, we need numerical fluxes F_p,qⁿ (uⁿ_p, uⁿ_q) between two neighbouring cells p and q at timetⁿ, function ofuⁿ_p anduⁿ_q, the respective approximations of the unknown function inQ_p andQ_q at time tⁿ. We require that the family of numerical fluxes (F_p,qⁿ )_p,q,n∈Nis admissible and consistent with the fluxF in the sense of the two definitions below.

Definition 3.1: A family of numerical fluxes (F_p,qⁿ )_p,q,n∈Nis said to be admissible if

• F_p,qⁿ ∈C⁰(R²) and there existsL >0 such that, for allp∈Nandq∈ Np and for all n∈ {0, ..., N−1}, the function F_p,qⁿ is Lipschitz continuous with the constantmp,qLwith respect to each of its variables.

• F_p,qⁿ is monotone, in the sense that it is non-decreasing with respect to its first argument and non-increasing with respect to its second argument,

• F_p,qⁿ is conservative, i.e. F_p,qⁿ (u, v) =−F_q,pⁿ (v, u) for all (u, v)∈R².

Definition 3.2: LetF be a flux function. A family of numerical fluxes (F_p,qⁿ )_p,q,n∈Nis said to be consistent with F if

F_p,qⁿ (u, u) = 1 δt

Z tⁿ⁺¹ tⁿ

Z

σ_p,q

F(x, t, u)·νp,q dγ(x)dt, ∀u∈R². (10) Let us give two examples of consistent and admissible families of numerical fluxes:

• the Godunov numerical flux, defined by:

F_p,qⁿ (u, v) =











u≤s≤vmin 1 δt

Z tⁿ⁺¹ tⁿ

Z

σ_p,q

F(x, t, s)·νp,q dγ(x)dt ifu≤v

v≤s≤umax 1 δt

Z tⁿ⁺¹ tⁿ

Z

σ_p,q

F(x, t, s)·ν_p,q dγ(x)dt ifv≤u,

• and the Rusanov numerical flux, defined, denoting byLF a Lipschitz constant ofF, by:

F_p,qⁿ (u, v) = 1 δt

Z tⁿ⁺¹ tⁿ

Z

σp,q

F(x, t,u+v

2 )·νp,q dγ(x)dt+mp,qLF

2 (u−v).

The following lemma, which is a discrete version of the divergence theorem, will be used below.

Lemma 3.3: Let (F_p,qⁿ )_p,q,n∈N be a family of numerical fluxes consistent with a flux functionF in the sense of Definition 3.2. IfF is divergence-free, then

∀u∈R, X

q∈Np

F_p,q(u, u) = 0. (11)

Proof Owing to (10) and to the divergence theorem, we have, for a givenu∈R, X

q∈Np

F_p,q(u, u) = 1 δt

Z tⁿ⁺¹ tⁿ

X

q∈Np

Z

σp,q

F(x, t, u)·ν_p,q dγ(x)dt= 1 δt

Z tⁿ⁺¹ tⁿ

Z

p d

X

i=1

∂Fi

∂x_i(x, t, u) dxdt, which vanishes owing to the assumption thatF is divergence-free.

3.2 Description of the numerical scheme and well-posedness

We consider a family of discretisations (Fh,δt)h,δt>0 – by discretisation, we mean a finite element mesh Th, a finite volume meshDh, a time stepδt, a family of numerical fluxes (F_p,qⁿ )_p,q,n∈N. We assume that the following hypotheses, denoted in the following by Hypotheses (HD), are satisfied uniformly by any element Fh,δt of the family.

(HD1) There existsα >0 such that, for allp∈N,mp≥α h^d,α|∂Qp| ≤h^d−1, and|K| ≥α h^d, for allK∈ Th. (HD2) There exists an admissible family of numerical fluxes (F_p,qⁿ )p,q,n∈Nin the sense of Definition 3.1, which is consistent with F in the sense of Definition 3.2. The constant L in Definition 3.1 is assumed to be independent of the discretisation.

(HD3) The time interval [0, T] is divided into N equal intervals of length δt, such that the following CFL condition holds

δt≤α²h

L . (12)

Note that, thanks to Hypothesis (HD1), the condition (12) implies that δt≤ 1

L

mp

P

q∈Npmp,q

, ∀p∈N. (13)

The scheme for approximating (1)-(2) is given by:

• Initialisation of (u⁰_p)_n∈Nsuch thatu⁰_h∈L^∞(R^d)∩L¹(R^d):

u⁰_p= 1 m_p

Z

Q_p

u_ini(x)dx, ∀p∈N. (14)

• Finite volume step. Letting (uⁿ_p)p∈Nsuch thatuⁿ_h∈L^∞(R^d)∩L¹(R^d), seek (uⁿ⁺

1

p 2)p∈Nsuch thatuⁿ⁺

1 2

h ∈

L^∞(R^d)∩L¹(R^d) and mp

uⁿ⁺p ¹² −uⁿ_p

δt + X

q∈Np

F_p,qⁿ (uⁿ_p, uⁿ_q) = 0, ∀p∈N. (15)

• Finite element step. Let θ∈C⁰((0,+∞)) be a positive function such that lim

h→0θ(h) = lim

h→0

h

θ(h) = 0, (16)

and let us define

Xh={vh∈R^N; ∇bvh∈L¹(R^d)∩L²(R^d), vh∈L²(R^d)}

Λh:={µh∈L^∞(R^d)^d;µ_h|K is constant for eachK∈ Th}.

Then the finite element step consists in seeking (uⁿ⁺¹_h , λⁿ⁺¹_h )∈Xh×Λhsuch that Z

R^d

uⁿ⁺¹_h −uⁿ⁺

1 2

h

δt v_hdx+ Z

R^d

(λⁿ⁺¹_h +θ(h)∇buⁿ⁺¹_h )· ∇bv_hdx= 0, ∀v_h∈X_h, (17)

λⁿ⁺¹_h ∈Sgn(∇buⁿ⁺¹_h ). (18)

An example of such a functionθisθ(h) =h^γ with 0< γ <1. Note that, ifd= 1, it is possible to letθ(h) = 0.

For each discretisation Fh,δt, we define the approximate solutions ubh,δt : QT → R, uh,δt : QT → R, and λh,δt:QT →Rsuch that

ub_h,δt(·, t) :=ubⁿ⁺¹_h ift∈(tⁿ, tⁿ⁺¹], uh,δt(·, t) :=uⁿ⁺¹_h ift∈(tⁿ, tⁿ⁺¹], λ_h,δt(·, t) :=λⁿ⁺¹_h ift∈(tⁿ, tⁿ⁺¹].

The proposition below proves that the scheme has at least one solution, which is unique with respect to the unknownu_h.

Proposition 3.4: Let us assume Hypotheses (HC) and Hypotheses (HD). Then there exists at least one solution to Scheme (14)-(18) such that u⁰_h ∈L^∞(R^d)∩L¹(R^d) and, for alln∈N, uⁿ⁺

1 2

h , uⁿ⁺¹_h ∈L^∞(R^d)∩L¹(R^d) and (uⁿ⁺¹_h , λⁿ⁺¹_h ) ∈ X_h×Λ_h. Moreover, u⁰_h and, for all n ∈ N, uⁿ⁺_h ¹² and uⁿ⁺¹_h are unique and, for all p ∈ N, a0≤uⁿ⁺

1

p 2 ≤b0 anda0≤uⁿ_p ≤b0.

Proof Thanks to (14), we get from Hypothesis (HC1) that

ku⁰_hk_L1(R^d)≤ kuinik_L1(R^d),

anda₀≤u⁰_p≤b₀ for allp∈N, which completes the proof thatu⁰_h∈L^∞(R^d)∩L¹(R^d).

For any n∈N, we get from Propositions 3.5 and 3.7, assuminguⁿ_h ∈L^∞(R^d)∩L¹(R^d) witha0≤uⁿ_p ≤b0 for allp∈N, thatuⁿ⁺

1 2

h ∈L^∞(R^d)∩L¹(R^d) witha0≤uⁿ⁺

1

p 2 ≤b0for allp∈N. Using Proposition 3.8, we get the existence of (uⁿ⁺¹_h , λⁿ⁺¹_h )∈Xh×Λh such that (18) holds, and we get that uⁿ⁺¹_h is unique. It now suffices to apply Proposition 3.9, for obtaining thatuⁿ⁺¹_h ∈L^∞(R^d)∩L¹(R^d) witha₀≤uⁿ⁺¹_p ≤b₀ for allp∈N. Let us now state and prove the propositions used in the proof of the preceding result. We have first the following result.

Proposition 3.5: Let us assume Hypotheses (HC) and Hypotheses (HD). Letn∈N,κ∈Rand a family (uⁿ_p)_p∈N be given such thata0≤uⁿ_p ≤b0 for allp∈N. Let (uⁿ⁺

1

p 2)p∈Nbe given by (15). Then there holds uⁿ⁺

1

p 2 ∨κ≤uⁿ_p∨κ− δt mp

X

q∈Np

F_p,qⁿ (uⁿ_p ∨κ, uⁿ_q ∨κ), ∀p∈N, (19) uⁿ⁺

1

p 2 ∧κ≥uⁿ_p∧κ− δt mp

X

q∈Np

F_p,qⁿ (uⁿ_p ∧κ, uⁿ_q ∧κ), ∀p∈N, (20)

mp

|uⁿ⁺p ¹² −κ| − |uⁿ_p−κ|

δt + X

q∈N_p

F_p,qⁿ (uⁿ_p∨κ, uⁿ_q ∨κ)−F_p,qⁿ (uⁿ_p∧κ, uⁿ_q ∧κ)

≤0, ∀p∈N. (21)

Consequentlya0≤uⁿ⁺

1

p 2 ≤b0 for allp∈N.

Proof The proof of this proposition is done in [5, Lemma 3] or [7, Lemma 27.1]. We recall it since it is very brief. We consider the functionH_pⁿ : R^1+#Np→R, defined by

H_pⁿ(a,(b_q)_q∈Np) =a− δt mp

X

q∈Np

F_p,qⁿ (a, b_q).

We observe that, for anya⁰> a, there holds

H_pⁿ(a⁰,(b_q)_q∈Np)−H_pⁿ(a,(b_q)_q∈Np)≥(a⁰−a)(1−δtLP

q∈Npmp,q

mp

)≥0,

thanks to Definition 3.1 of admissible fluxes and to condition (13) implied by (12). Therefore the function H_pⁿ is non-decreasing with respect to all its arguments. Noticing that κ = H_pⁿ(κ,(κ)_q∈Np) and uⁿ⁺

1

p 2 = H_pⁿ(uⁿ_p,(uⁿ_q)q∈N_p), we get that κ ≤H_pⁿ(uⁿ_p ∨κ,(uⁿ_q ∨κ)q∈N_p) and uⁿ⁺

1

p 2 ≤H_pⁿ(uⁿ_p ∨κ,(uⁿ_q ∨κ)q∈N_p), which implies (19). The proof of (20) is similar, and (21) is obtained by the difference between (19) and (20).

Lettingκ=b0 in (19) and using (11) on one hand, lettingκ=a0in (20) on the other hand complete the proof thata₀≤uⁿ⁺p ¹² ≤b₀for allp∈N.

We then deduce the following result.

Proposition 3.6: Let us assume Hypotheses (HC) and Hypotheses (HD). Let (η,Φ) be an entropy-entropy flux pair in the sense of Definition 2.1, and letn∈Nbe given. Then, the family (Φⁿ_p,q)p,q,n∈Nof admissible numerical fluxes defined by

Φⁿ_p,q(x, y) := 1 2

Z b0

a₀

η⁰⁰(κ) F_p,qⁿ (x∨κ, y∨κ)−F_p,qⁿ (x∧κ, y∧κ)

dκ+η⁰(a₀) +η⁰(b₀)

2 F_p,qⁿ (x, y), (22) is consistent with Φ in the sense of Definition 3.2, and is such that, ifuⁿ⁺_h ¹² ∈R^Nis obtained fromuⁿ_h ∈R^Nby (15) witha0≤uⁿ_p ≤b0, for allp∈N, then

mp

η(uⁿ⁺p ¹²)−η(uⁿ_p)

δt + X

q∈Np

Φⁿ_p,q(uⁿ_p, uⁿ_q)≤0, ∀n∈ {0, ..., N−1}, ∀p∈N. (23) Furthermore, there is a constantL⁰, depending only onL, η, a₀ andb₀, such that, for all (p, q)∈ Eh and for all n∈ {0, ..., N−1}, the function Φⁿ_p,q is Lipschitz continuous with respect of each of its variables with the constantm_p,qL⁰.

Proof Thanks to Proposition 3.5, we have that all valuesuⁿ_p anduⁿ⁺p ¹² belong to [a₀, b₀], which enables to use relations (4). The consistency of Φⁿ_p,q with Φ is a consequence of (4) and of the consistency of F_p,qⁿ with F. Multiplying (21) byη⁰⁰(κ) and integrating onκ∈[a₀, b₀] implies (23), owing to (4) and (22).

From the above result, we get the following one.

Proposition 3.7: Let us assume Hypotheses (HC) and Hypotheses (HD). Assume that, for a given n ∈ N, a0 ≤uⁿ_p ≤b0 for all p∈N anduⁿ_h ∈ L¹(R^d). Then for all entropy η in the sense of Definition 2.1 such that η(0) = 0 andη⁰(0) = 0,

Z

R^d

η(uⁿ⁺

1 2

h (x))dx≤ Z

R^d

η(uⁿ_h(x))dx, (24)

and consequently

kuⁿ⁺_h ¹²k_L1(R^d)≤ kuⁿ_hk_L1(R^d)andkuⁿ⁺_h ¹²k_L2(R^d)≤ kuⁿ_hk_L2(R^d). (25) Proof We get, from (23) and applying Lemma 3.3,

m_pη(uⁿ⁺

1

p 2)−η(uⁿ_p)

δt + X

q∈Np

(Φⁿ_p,q(uⁿ_p, uⁿ_q)−Φⁿ_p,q(0,0))≤0, ∀n∈ {0, ..., N−1}, ∀p∈N,

Forε >0, we multiply the preceding inequality byδtexp(−ε|xp|) and we sum the result onp∈N. We get X

p∈N

m_p(η(uⁿ⁺p ¹²)−η(uⁿ_p)) exp(−ε|xp|) +X

p∈N

X

q∈Np

Φⁿ_p,q(uⁿ_p, uⁿ_q)−Φⁿ_p,q(0,0)

exp(−ε|xp|)≤0.

Defining, for any (p, q)∈ Eh, the pointxpq= ¹₂(xp+xq) and using the property Φⁿ_p,q(u, v) =−Φⁿ_q,p(v, u) for all (u, v)∈[a0, b0]²(which therefore implies Φⁿ_p,q(uⁿ_p, uⁿ_q)−Φⁿ_p,q(0,0) + Φⁿ_q,p(uⁿ_q, uⁿ_p)−Φⁿ_q,p(0,0) = 0), we get

Φⁿ_p,q(uⁿ_p, uⁿ_q)−Φⁿ_p,q(0,0)

exp(−ε|xp|) + Φⁿ_q,p(uⁿ_q, uⁿ_p)−Φⁿ_q,p(0,0)

exp(−ε|xq|) =Aⁿ_pq+Aⁿ_qp, with

Aⁿ_pq= Φⁿ_p,q(uⁿ_p, uⁿ_q)−Φⁿ_p,q(0,0)

(exp(−ε|xp|)−exp(−ε|xpq|)).

Hence the two preceding relations imply X

p∈N

m_p(η(uⁿ⁺p ¹²)−η(uⁿ_p)) exp(−ε|x_p|) +X

p∈N

X

q∈Np

Aⁿ_pq≤0, Observing that Proposition 3.6 implies

|Φⁿ_p,q(uⁿ_p, uⁿ_q)−Φⁿ_p,q(0,0)| ≤L⁰m_pq(|uⁿ_p|+|uⁿ_q|), we get, using|exp(−ε|xp|)−exp(−ε|xpq|)| ≤ε||xp| − |xpq|| ≤ε¹₂|xp−xq|, that

Aⁿ_pq≥ −L⁰

2mpq(|uⁿ_p|+|uⁿ_q|)εh.

This leads to X

p∈N

mpη(uⁿ⁺

1

p 2) exp(−ε|xp|)≤X

p∈N

mpη(uⁿ_p) exp(−ε|xp|) +εL⁰ 2

X

p∈N

X

q∈Np

mpqh(|uⁿ_p|+|uⁿ_q|).

Since we have

1 2

X

p∈N

X

q∈Np

m_pqh(|uⁿ_p|+|uⁿ_q|) =X

p∈N

X

q∈Np

m_pqh|uⁿ_p|=X

p∈N

|uⁿ_p| |∂Q_p|h,

we can write, using Hypothesis (HD1), X

p∈N

mpη(uⁿ⁺

1

p 2) exp(−ε|xp|)≤X

p∈N

mpη(uⁿ_p) +εL⁰ α²

X

p∈N

mp|uⁿ_p|.

Lettingε→0, we get by monotonous convergence (recall that thanks to the hypothesis uⁿ_h ∈L¹(R^d), we also haveη(uⁿ_h)∈L¹(R^d) andη(s)≥η(0) = 0),

X

p∈N

mpη(uⁿ⁺

1

p 2)≤X

p∈N

mpη(uⁿ_p), which concludes the proof of (24).

Lettingηtend to η(s) =|s| and lettingη(s) =s² allow for concluding (25).

The proposition below proves that the finite element step is well-posed, provided thatuⁿ⁺_h ¹² ∈L²(R^d), which holds ifuⁿ⁺

1 2

h ∈L^∞(R^d)∩L¹(R^d), and gives a variational characterisation ofuⁿ⁺¹_h .

Proposition 3.8: Let us assume Hypotheses (HC) and Hypotheses (HD). Letn∈Nbe given and let us assume that uⁿ⁺

1 2

h ∈L²(R^d). Then, equations (17)-(18) admit a solution (uⁿ⁺¹_h , λⁿ⁺¹_h )∈Xh×Λh. Furthermore, uⁿ⁺¹_h is unique and is the minimiser of the functionalJ_hⁿ⁺¹:X_h→Rdefined by

J_hⁿ⁺¹(vh) := 1 2δt

Z

R^d

vh−uⁿ⁺

1 2

h

² dx+

Z

R^d

(|∇bvh|+1

2θ(h)|∇vbh|²) dx. (26) Proof We remark that, defining the following norm on the Banach spaceXh,

kvhkX_h=kvhk_L2(R^d)+k∇bvhk_L1(R^d)+k∇vbhk_L2(R^d), we obtain that the stricly convex functionJ_hⁿ⁺¹is such that

kv_hklim_Xh→∞J_hⁿ⁺¹(vh) = +∞.

Hence J_hⁿ⁺¹ reaches its unique minimum value at some point uⁿ⁺¹_h ∈ Xh. Let us prove that this point is characterised by (17)-(18).

Let us first assume that there exists (uⁿ⁺¹_h , λⁿ⁺¹_h )∈Xh×Λh, solution to (17)-(18). Writing, for anywh∈Xh, J_hⁿ⁺¹(wh) =J_hⁿ⁺¹(uⁿ⁺¹_h +vh) withvh=wh−uⁿ⁺¹_h , we have using (17)

J_hⁿ⁺¹(wh)−J_hⁿ⁺¹(uⁿ⁺¹_h ) = 1 2δt

Z

R^d

vh(x)²dx+A+ Z

R^d

1

2θ(h)|∇bvh(x)|²dx, with

A= Z

R^d

(|∇(ubⁿ⁺¹_h +bvh)(x)| − |∇ubh(x)| −λⁿ⁺¹_h (x)· ∇bvh(x))dx

= X

K∈Th

|K|(|∇(ubⁿ⁺¹_h +bv_h)_|K| − |(∇buⁿ⁺¹_h )_|K| −λⁿ⁺¹_K ·(∇bv_h)_|K).

Recall that, if (∇buⁿ⁺¹_h )_|K6= 0, then λⁿ⁺¹_K = ^(∇bu

n+1 h )_|K

|(∇buⁿ⁺¹_h )_|K| and otherwise that|λⁿ⁺¹_K | ≤1.

Using that for anya, b∈R^dwitha6= 0, we have |a+b| − |a| −^a·b_|a| =|a||a+b|−a·(a+b)

|a| ≥0, and|b| −λⁿ⁺¹_K ·b≥0, we getA ≥0, which proves that J_hⁿ⁺¹(wh)−J_hⁿ⁺¹(uⁿ⁺¹_h )≥0, and therefore that J_hⁿ⁺¹ reaches its minimum value atuⁿ⁺¹_h .

Reciprocally, let us assume thatJ_hⁿ⁺¹ reaches its minimum value at some point uⁿ⁺¹_h ∈X_h. Let us prove that there exists λⁿ⁺¹_h ∈Λ_h such that (17)-(18) holds. We denote byT_h,0ⁿ⁺¹ ={K ∈ Th, (∇buⁿ⁺¹_h )_|K = 0}, and we define the linear formLⁿ⁺¹_h : Xh→Rby

Lⁿ⁺¹_h (vh) := 1 δt

Z

R^d

uⁿ⁺¹_h −uⁿ⁺

1 2

h

vhdx+

Z

{∇buⁿ⁺¹_h 6=0}

∇ubⁿ⁺¹_h

|∇ubⁿ⁺¹_h |· ∇bvhdx+θ(h) Z

R^d

∇ubⁿ⁺¹_h · ∇bvhdx.

For a givenε∈Rwithε6= 0 and for any vh∈Xh, we write that

J_hⁿ⁺¹(uⁿ⁺¹_h +εv_h)−J_hⁿ⁺¹(uⁿ⁺¹_h )≥0.

Assumingε >0, dividing the above equation byεand lettingε→0, we get Lⁿ⁺¹_h (v_h) +

Z

{∇buⁿ⁺¹_h =0}

|∇bv_h|dx≥0.

Assumingε <0, dividing the above equation byεand lettingε→0, we get Lⁿ⁺¹_h (v_h)−

Z

{∇buⁿ⁺¹_h =0}

|∇bv_h|dx≤0.

Hence we get that

∀vh∈Xh, |Lⁿ⁺¹_h (vh)| ≤ Z

{∇ubⁿ⁺¹_h =0}

|∇bvh|dx= X

K∈T_h,0ⁿ⁺¹

|K| |∇bv_h|K|. (27)

We consider the setE of all functionsf fromT_h,0ⁿ⁺¹→R^d which are bounded for the norm kfkE= X

K∈T_h,0ⁿ⁺¹

|K| |fK|.

We observe that, defining F = {f ∈ E, ∃vh ∈ Xh, f = ∇bvh}, the linear form B : F → R, such that B(f) =Lⁿ⁺¹_h (vh) is well defined (since if∇bvh=∇wbh on allK∈ T_h,0ⁿ⁺¹, then we get from (27) thatLⁿ⁺¹_h (vh) = Lⁿ⁺¹_h (wh)) and continuous (again from (27), which proves thatkBkF⁰ := sup_f∈F\{0}kfk^B(f)

E ≤1). Applying the Hahn-Banach theorem, this linear form can be extended onE by a linear form, again denoted byB, with the same norm (hence lower or equal to 1). Hence there exists (λⁿ⁺¹_K )_K∈Tn+1

h,0 with

∀f ∈E, B(f) =− X

K∈T_h,0ⁿ⁺¹

|K|λⁿ⁺¹_K ·fK.

andkBkE⁰ = sup_K∈Tn+1

h,0 |λⁿ⁺¹_K | ≤1. Therefore we have

∀vh∈Xh, Lⁿ⁺¹_h (vh) + X

K∈T_h,0ⁿ⁺¹

|K|λⁿ⁺¹_K ·(∇bvh)_|K = 0,

which concludes, denotingλⁿ⁺¹_K = ^(∇ub

n+1 h )_|K

|(∇ubⁿ⁺¹_h )|K| if (∇buⁿ⁺¹_h )_|K 6= 0, the proof thatλⁿ⁺¹_h ∈Λhis such that (17)-(18) holds.

Proposition 3.9: Let us assume Hypotheses (HC) and Hypotheses (HD). Let n∈N. Letη ∈C²(R) such that η(0) = 0,η⁰(0) = 0 and there existsM ∈R⁺ withη⁰⁰(s)∈[0, M] for alls∈R. Assume thatuⁿ⁺

1 2

h ∈R^Nis such that

X

p∈N

mpη(uⁿ⁺

1

p 2)<∞.

Then

X

p∈N

m_pη(uⁿ⁺¹_p )≤X

p∈N

m_pη(uⁿ⁺p ¹²). (28)

As a consequence, if a0 ≤uⁿ⁺

1

p 2 ≤b0 for all p∈N, thena0 ≤uⁿ⁺¹_p ≤b0 for allp∈N, and ifuⁿ⁺

1 2

h ∈L¹(R^d) thenuⁿ⁺¹_h ∈L¹(R^d).

Proof We remark thatvh, defined byvp=η⁰(uⁿ⁺¹_p ), is such thatvh∈Xh. Indeed, there holds|vp| ≤M|uⁿ⁺¹_p |, and, using (6)-(7), we have

k∇bv_hk²_L2(R)^d= X

K∈Th

X

p∈VK

X

q∈VK

T_pq^K(η⁰(uⁿ⁺¹_p )−η⁰(uⁿ⁺¹_q ))²

≤M² X

K∈Th

X

p∈VK

X

q∈VK

T_pq^K(uⁿ⁺¹_p −uⁿ⁺¹_q )²=M²k∇ubⁿ⁺¹_h k²_L2(R)^d,

which implies vh ∈ Xh. We now remark that, for any K ∈ Th, if ∇ubⁿ⁺¹_h|K = 0, then uⁿ⁺¹_p = uⁿ⁺¹_q for any p, q∈ VK, and therefore∇bv_h|K= 0. So one can write

λⁿ⁺¹_h|K · ∇bv_h|K = 0 =∇ubⁿ⁺¹_h|K · ∇vb_h|K. If∇ubⁿ⁺¹_h|K 6= 0, we then have

λⁿ⁺¹_h|K · ∇bv_h|K= 1

|∇ubⁿ⁺¹_h|K|∇buⁿ⁺¹_h|K · ∇bv_h|K. Hence, definingαⁿ⁺¹_K byαⁿ⁺¹_K = 1 if∇ubⁿ⁺¹_h|K = 0, andαⁿ⁺¹_K = ¹

|∇buⁿ⁺¹_h|K| otherwise, we get Z

R^d

(λⁿ⁺¹_h +θ(h)∇buⁿ⁺¹_h )· ∇vb_hdx= X

K∈Th

(αⁿ⁺¹_K +θ(h)) X

p∈VK

X

q∈VK

T_pq^K(uⁿ⁺¹_p −uⁿ⁺¹_q )(η⁰(uⁿ⁺¹_p )−η⁰(uⁿ⁺¹_q ))≥0.

Hence we can write from (17)

X

p∈N

m_p(uⁿ⁺¹_p −uⁿ⁺p ¹²)η⁰(uⁿ⁺¹_p )≤0.

Applyingη(b)−η(a) =η⁰(a)(b−a) +η⁰⁰(c)^(b−a)₂ ² forcbetweenaandb, we get X

p∈N

mp(η(uⁿ⁺¹_p )−η(uⁿ⁺

1

p 2))≤X

p∈N

mp(uⁿ⁺¹_p −uⁿ⁺

1

p 2)η⁰(uⁿ⁺¹_p )≤0, which concludes the proof of (28).

Then, assuminguⁿ⁺p ¹² ≤b₀ for allp∈Nand lettingη(s) tend tos∨b₀−b₀ (this is possible sinceu_ini∈L¹(R^d) implies a₀ ≤ 0 ≤ b₀), we get that η(uⁿ⁺¹_p ) = 0, which shows that uⁿ⁺¹_p ≤ b₀. The same reasoning holds, lettingη(s) tend toa₀−s∧a₀, which shows thatuⁿ⁺¹_p ≥a₀. Finally, lettingη(s) tend to|s|, we conclude that uⁿ⁺¹_h ∈L¹(R^d).

4 A priori estimates on the approximate solutions

AL^∞(QT) estimate on the family of approximate velocities (uh,δt)h,δt>0has already been proved in Proposition 3.4. The aim of this section is to establish additional estimates on (buh,δt)h,δt>0, namely a L¹(0, T;BV(R^d)) estimate, and L¹_loc(QT) estimates on the space and time translates. The estimates on the space and time translates are deduced from theL¹(0, T;BV(R^d)) estimate.

Remark 4.1: Hypothesis (HD1) implies that each cell ofDhhas a finite number of neighbours (and this number is bounded independently ofFh,δt).

Remark 4.2: Throughout this section and the next one, C denotes a generic constant independent of the dis- cretisationF_h,δt.

4.1 L

(0, T ; BV ( R

)) estimate

Proposition 4.3: Let us assume Hypotheses (HC) and Hypotheses (HD). Letu⁰_h∈L^∞(R^d)∩L¹(R^d) and, for all n∈N, uⁿ⁺

1 2

h , uⁿ⁺¹_h ∈L^∞(R^d)∩L¹(R^d) and (uⁿ⁺¹_h , λⁿ⁺¹_h )∈Xh×Λh be a solution to Scheme (14)-(18). Then there holds

1

2kuh,δtk²_L∞(0,T;L²(R^d))+

N

X

n=1

δt Z

R^d

|∇buⁿ_h|dx≤ 1

2kuinik²_L2(R^d), (29) and

N−1

X

n=0

δt Z

R^d

θ(h)|∇buⁿ⁺¹_h |² dx=θ(h)

N−1

X

n=0

δt X

K∈T_h

X

p∈V_K

X

q∈V_K

T_pq^K(uⁿ⁺¹_p −uⁿ⁺¹_q )²≤ 1

2kuinik²_L2(R^d). (30)