We study the convergence of the upwind Finite Volume scheme applied to the linear advection equation with a Lipschitz divergence-free speed in Rd

ADVECTION

BENOIT MERLET^∗

Abstract. We study the convergence of the upwind Finite Volume scheme applied to the linear advection equation with a Lipschitz divergence-free speed in R^d. We prove a h^1/2−ε-error estimate in the L^∞(R^d×[0,T ])-norm for Lipschitz initial data. The expected optimal result is a h^1/2-error estimate. In a second part, we also prove a h^1/2-error estimate in the L^∞(0,T ; L²(R^d))-norm for initial data in H¹(R^d).

Key words. scalar conservation laws, advection equation, Finite Volume method, error estimate AMS subject classifications. 35L65, 65M15

1. Introduction. We consider here the numerical approximation by the explicit upwind Finite Volume scheme of the following general non linear hyperbolic scalar problem in d- space dimension.

( u_t+div (V f (u)) = 0 on R^d_x×R_+t

u(x,0) = u0(x) on R^d, (1.1)

where the vector field V is a Lipschitz function of x and t with values in R^dand is divergence free.

Finite Volume schemes are widely used for the numerical approximation of such equa- tions and more generally of conservative hyperbolic systems. They have several good be- haviours: they can handle general meshes, they have a compact numerical stencil, they con- serve the quantityR

R^du and under some hypothesis (CFL condition) they have stability prop- erties. Moreover, in general, the solutions to (1.1) are not unique, and, again under some hypthesis concerning the time step and the regularity of the mesh, FV Schemes do converge towards the physical (entropic) solution when the mesh-size h tends to 0.

However, the convergence theory for these schemes is far from being complete. The first result is due to Kuztnetsov [7], who proves a h^1/2-error estimate in the L^∞(0,T ; L¹_x) norm for BV-data and a familly of structured cartesian meshes of mesh-size h tending to 0. The method of Kuznetsov has been extended to the case of non-regular cartesian meshes or almost cartesian meshes (see [2]). This method had also been extended to familly of unstructured uniformly regular meshes, but in this last case only a h^1/4-convergence rate is obtained (see Cockburn,Coquel, Lefloch [1], Vila [9] and Eymard, Gallou¨et, Herbin [6]).

We emphasize that numerical experiments show that the error behaves like h^1/2regardless of whether the familly of meshes is structured.

In this paper, we focus on the advection equation (i.e. the linear case f (u) := u) with W^1,pinitial data. Even in the linear case, it is more difficult to establish convergency results than in the setting of Finite Difference methods. Indeed, in general, FV schemes are not consistent in the FD classical sense and Lax’s theorem cannot be used. Provided that the Courant-Friedrich-Levy condition is satisfied and that the meshes are uniformely regular (see the condition (1.5) below), we expect a h^1/2convergence rate in the L^∞(0,T ; L^p_x)-norm.

In [8] the author and Vovelle prove this result for p =1 (BV-data) under a strict CFL condition in the general case and under a sharp CFL condition when the speed does not depend on the time variable.

∗Universit´e Paris Nord - Institut Galil´ee LAGA (Laboratoire d’Analyse, G´eom´etrie et Applications) Avenue J.B.

Cl´ement 93430 Villetaneuse ([email protected]).

1

In [4] Despr´es shows the result in the case p =2 under a strict CFL and for a constant speed. His proof is based on the study of the dissipation of the consistency error by the scheme and on a generalization of the Lax theorem. The lack of consistency of the scheme is compensated by the particular structure of the consistency error.

We also refer to Vila and Villedieu [10], who proved (under a strict CFL) a h^1/2-error estimate in the L²_locspace-time norm for the FV approximation of Friedrichs hyperbolic sys- tems with H¹ data — the linear advection equation appears to be a particular case of these systems. Their proof is based on energy estimates and on the weak consistency of the scheme in the dual of H_loc¹ .

The case p >2 has been attacked by Despr´es. In particular, in [5] a h^1/4−ε-estimate is obtained in the L^∞-norm for Lipschitz data.

Here, we study the case 2≤ p ≤ ∞. For p =2, we obtain the expected result under a strict CFL condition — Theorem 1.8. The ingredients of the proof are essentialy contained in [8]: the conservation of the L²-energy of the exact solution and energy estimates for the approximate solution coming from the dissipative properties of the scheme.

In the case p = +∞, we prove a h^1/2−ε-error estimate in the L^∞-norm for Lipschitz data under a strict CFL condition — Theorem 1.6. In the proof, the fact that the scheme is consistent of order 1 for the dual norm of the BV-seminorm is crucial — see Section 2.3 for a more precise description. This result is again the consequence of the structure of the consistency error: the size of the error isO(1) but the integral of this error on any finite union of cellsωis controlled by the measure of the boundary ofωtimes h — and not only by the measure ofω.

The proof of Theorem 1.6 may be sketched as follows. By duality, we are led to prove Theorem 1.7 concerning the behaviour of the FV approximation when the initial data is a probability measure concentrated on one cell. Let K0 be a cell, let u0 := 1/|K₀|1K₀ be the initial condition, u be the exact solution to the advection equation and u be the corresponding FV approximation. Roughly speaking, Theorem 1.7 says that the expectation of the distance kx−ykwith respect to the probability

dP(x,y) :=u(x,T )dλ(x)×u(y,T )dλ(y) in R^d_x×R^d_yis bounded by Ch^1/2−ε, namelyR

R^d×R^dkx−yku(x,T )u(y,T )dxdy≤Ch^1/2−ε. For simplicity, assume that the speed is constant: V(x,s) = V ; in this case the exact solution is given by u(·,t) = 1/|K₀|1_K0+Vt. Let x₀ ∈ K₀ and let B₀ be the ball centered at x0 with radius kσh^1/2where k is a positive integer andσis a parameter depending on V, T and the regularity of the mesh. We consider the solution to the exact advection equation with initial data: 1B₀. At time t this solution is still the characteristic function of a ball Bt=B0+Vt.

We have to prove that the integral of u on B^c_t remains small.

The evolution of the L²-energy of u on B^c_t, is drived by two phenomenons:

1) the dissipation of the scheme decreases the energy and provides energy estimates, 2) a source term created by the consistency error and located on the boundary of Bt

may increase the energy.

We use the energy estimates and the weak consistency of the scheme to prove (recursively on k) that the L²-energy of u on B^c_t is bounded by (t/T )^kh^−d/k! and we conclude by the Cauchy- Schwarz inequality. In fact, in the proof, the characteristic function of the set B^c_t is replaced by a smooth cut-offfunction in the neighborhood of∞.

The sequel is organized as follows. First we supplement this introduction by describing the linear advection problem (§1.1), the explicit upwind Finite Volume scheme (§1.2) and the main results of the paper (§1.3).

Section 2 is devoted to the proof of Theorems 1.6 and 1.7. In§2.1, we prove by duality that Theorem 1.6 ensues from Theorem 1.7. In§2.2, we establish an identity satisfied by the L²(Ψ²dλ(x))-energy of the numerical solution — in the applications the weightΨ²will be a cut-offfunction in the neighborhood of∞. In §2.3 we prove that the implicit in time downwind FV scheme is consistent of order 1 in a weak sense. Applying the results of

§§2.2-2.3, we prove in§2.4 that if two solutions of the advection equationΨ,Ψsatisfy 0≤ 1_|suppΨ≤Ψ, the growth of the L²(Ψ²dλ(x))-energy of the numerical solution between times 0 and T is controlled by its L²(_Ψ²dλ(x)dt)-energy. Finally, in§2.5, we build a decreasing sequence of cut-offfunctions, we apply recursively the result of§2.4 and we end the proof of Theorem 1.7.

Theorem 1.8 is proved in Section 3. In§3.1, we recall that the FV approximation satisfies the weak formulation of the advection equation up to an error term, we prove an energy in- equality and we give some technical results concerning some projections of the exact solution on the mesh. In§3.2 we derive successive estimates yielding Theorem 1.8.

1.1. The linear advection equation. We consider the linear advection problem in R^d: ( ut+div(Vu)=0, x∈R^d,t∈R₊,

u(x,0)=u0(x), x∈R^d, (1.2)

where we suppose that V∈W^1,∞(R^d×R₊,R^d) satisfies

div V(·,t)=0 ∀t∈R₊. (1.3)

The problem (1.2) is well-posed in L¹_loc(R^d). Let us recall the following classical result (see Theorem 1 in [8] for a sketch of proof.)

T1.1. For every u0 ∈L¹_loc(R^d), the problem (1.2) has a unique weak solution u, in the sense that u ∈ L¹_loc(R₊×R^d) such that: for every compactly supported test function φ∈W^1,∞(R^d×R₊),

Z

R₊

Z

R^d

u(φt+V· ∇φ)dxdt+ Z

R^d

u0(x)φ(x,0)dx=0. (1.4) This solution is given by the characteristic formula

u(x,t)=u0(X(x,t)), ∀(x,t)∈R^d×R₊,

with X defined by X(·,t) := Y(·,t)⁻¹, where then Y solves the Cauchy Problem∂tY(x,t) = V(Y(x,t),t) ; Y(x,0)=x.

Moreover, for every t≥0, X(·,t) : R^d →R^d is one to one and onto, the map X belongs to W_loc^1,∞(R^d ×R₊,R^d). More precisely, for every T ≥ 0, X−Id_R^d_x and Y−Id_R^d_x belong to W^1,∞(R^d×[0,T ]) and there exists C0 ≥ 1 only depending on T ,kVk_W1,∞ and d, such that,

∀x∈R^d,∀t∈[0,T ],

kX(x,t)−xk ≤C0t, k∇X(x,t)k ≤C0, k∂_tX(x,t)k ≤C0, kY(x,t)−xk ≤C0t, k∇Y(x,t)k ≤C0, k∂tY(x,t)k ≤C0. Finally, for every t≥0, X(·,t) preserves the Lebesgue measureλon R^d, i.e:

λ(X(E,t))=λ(E), for every Borel subset E of R^d.

For t ≥0, the map u0 7→ u(·,t) acts on W^1,∞(R^d) as a continuous linear operator and if u0∈W^1,∞(R^d), then t7→u(·,t) is locally Lipschitz from R₊to L^∞(R^d). More precisely, the characteristic formula yields

C1.2. Let 0 ≤t ≤T , there exists a constant C0 depending onkVk_W1,∞ and T such that if u0∈W^1,p(R^d), for some 1≤p≤ ∞, then

ku(·,s)−u(·,t)k_Lp ≤C₀k∇u₀k_Lp|s−t|, 0≤s,t≤T, k∇u(·,t)k_Lp ≤C0k∇u₀k_Lp, 0≤t≤T.

1.2. Finite Volume scheme. The Finite Volume scheme which approximates (1.2) is defined on a meshT which is a family of closed connected polygonal subsets with disjoint interiors covering R^d; the time half-line is meshed by regular cells of sizeδt >0. We also assume that the partitionT satisfies the following properties: the common interface of two control volumes is included in an hyperplane of R^dand

there existsα >0 such that

( αh^d≤ |K|,

|∂K| ≤α⁻¹h^d−1, ∀K∈ T, (1.5) where h is the size of the mesh: h :=sup{diam(K),K∈ T },|K|is the d-dimensional Lebesgue measure of K and|∂K|is the (d−1)-dimensional Lebesgue measure of∂K. If K and L are two control volumes having a common edge, we say that L is a neighbour of K and denote (quite abusively) L∈∂K. We also denote KpL the common edge and n_KLthe unit normal to KpL pointing outward K. We set

V_KLⁿ := 1 δt

Z (n+1)δt nδt

Z

KpL

V·nKL

We denote by Kn := K×[nδt,(n+1)δt) a generic space-time cell, by M := T ×N the space-time mesh. The sets∂K_n⁻,∂K_n⁺are defined by

∂K_n⁻:={L∈∂K,V_KLⁿ <0}, ∂K_n⁺:={L∈∂K,Vⁿ_KL≥0}.

In the same way we set KpLn:=KpL×[nδt,(n+1)δt).

We will assume that a strict Courant-Friedrich-Levy condition is satisfied:

X

L∈∂K_n⁻

δt|V_KLⁿ | ≤(1−ξ)|K|, ∀Kn∈ M, for some 1≥ξ >0. (1.6) Moreover, in order to avoid the occurrence of terms with factorδt/h in our estimates, we add to (1.6) the following condition: there exists c0≥0 such that

δt≤c0h. (1.7)

The upwind Finite Volume method with explicit time-discretization is defined by the following set of equations:

u⁰_K := 1

|K|

Z

K

u₀(x) dx ∀K∈ T, (1.8)

uⁿ_K⁺¹:=uⁿ_K+ δt

|K|

X

L∈∂K_n⁻

−V_KLⁿ (uⁿ_L−uⁿ_K), ∀K_n∈ M. (1.9) We then denote by u the approximate solution of (1.2) defined by the FV scheme:

u(·,t) :=uⁿ_K a.e in K for every t∈[nδt,(n+1)δt) ; Kn ∈ M. (1.10)

Classical results. The divergence free assumption (1.3) leads to the following identity L1.3. Under the hypothesis (1.3) on the divergence of V, one has

X

L∈∂K

V_KLⁿ = X

L∈∂K_n⁻

V_KLⁿ + X

L∈∂K_n⁺

V_KLⁿ =0 ∀Kn∈ M.

Under the CFL condition (1.6) the Finite Volume scheme is order-preserving:

P1.4. Under condition (1.6), the linear applicationL: (uⁿ_K)7→(uⁿ_K⁺¹) defined by (1.9) is order-preserving and stable for the L^∞-norm.

Proof. Eq. (1.9) gives uⁿ_K⁺¹=











1− X

L∈∂K⁻_n

−V_KLⁿ δt

|K|











uⁿ_K+ X

L∈∂K_n⁻

−V_KLⁿ δt

|K| uⁿ_L

i.e., under (1.6), uⁿ_K⁺¹ is a convex combination of uⁿ_K,(uⁿ_L)L∈∂K_n⁻. The stability result follows fromL(1)=1.

From Lemma 1.3 it is not difficult to see that the quantityP

K∈T|K|uⁿ_Kis conserved. By the Crandall-Tartar lemma [3], this fact and Proposition 1.4 imply

P1.5. Under condition (1.6), the schemeL : (uⁿ_K) 7→(uⁿ_K⁺¹) is stable for the L¹-norm:

X

K∈T

|K||uⁿ_K⁺¹| ≤X

K∈T

|K||uⁿ_K|, ∀n≥0.

Interpolating between the L¹- and L^∞-stability, the scheme is stable for the L^p-norm, 1<p<

∞.

1.3. Main results. We now assume that V ∈W^1,∞(R^d×R₊,R^d) satisfies (1.3), thatT is mesh of mesh-size h>0 satisfying conditions (1.5) and that the time stepδt>0 satisfies the CFL conditions (1.6)-(1.7). We fix a time T ≥δt and we assume that there exists an integer N≥0 such that T =(N+1)δt.

In the sequel, given an initial condition u0 ∈L¹_loc(R^d), the function u∈C(R₊,L¹_loc(R^d)) denotes the exact solution to (1.2) and u its numerical approximation obtained by the upwind FV method (1.8)-(1.9)-(1.10). In the results and in the proofs, C0 ≥1 is the constant intro- duced in Theorem 1.1 and Corollary 1.2; this constant only depends on T ,kVk_W1,∞and d. The letter C denotes various constants which are non decreasing functions ofα⁻¹, c0,kVk_W1,∞ and d but do not depend on h,δt,ξ, u0or T .

We have the following error estimate for Lipschitz data.

T1.6. Under a strict CFL condition (ξ >0), there exists h⋆ >0 depending only on C0,ξ, c₀and T such that if h≤h⋆, we have for every u0∈W^1,∞(R^d),

ku(·,T )−u(·,T )k_L∞ ≤CC²₀ξ^−1/2k∇u₀k_L∞T^1/2h^1/2Γ⁻¹(C²₀ξ⁻¹T h⁻¹),

whereΓ⁻¹ : [1,+∞)→ [2,+∞) is the inverse EulerΓfunction — in particularΓ⁻¹(n!)= n+1. This result will be the consequence of

T 1.7. Let K₀ ∈ T, x₀ ∈ K₀ and u₀ := 1/|K₀|1_K0, then under a strict CFL condition, for h≤h_⋆, as above, we have

Z

R^d

kx−Y(x₀,T )ku(x,T )dx≤CC²₀ξ^−1/2T^1/2h^1/2Γ⁻¹(C₀²ξ⁻¹T h⁻¹).

R1.1. The dependency in T in Theorem 1.6 and 1.7 is not interesting in the general case since C0depends on T . But in the case of a speed independent of x: V(x,t) :=V(t), we have X(x,t)=x−Rt

0 V(s)ds and we can choose C0=kVk_∞in Theorem 1.1. This is also valid for Theorem 1.8 below.

R1.2. The functionΓ⁻¹ is very slowly increasing (for exempleΓ⁻¹(R) =o(ln R) but we do not obtain a h^1/2-estimate. Proving this result (or exhibiting a counterexample) remains an open problem.

In this L² setting, we need an additionnal assumption on the mesh: there is no small interface:

αh^d−1≤ |KpL|, ∀K∈ T,L∈∂K. (1.11) T1.8. Assume that (1.5), (1.11), (1.6) and (1.7) hold. Then for every u0in H¹(R^d), we have

ku(·,T )−u(·,T )k²_L2+ku(·,0)k²_L2− ku(·,T )k²_L2≤C(1+C²₀)ξ⁻¹ku0k²_H1T h+Cku0k²_H1h², F(u,T )≤C(1+C²₀)ξ⁻²ku₀k²_H1T h+Cku₀k²_H1h², with

F(u,T ) :=

N

X

n=0

X

K∈T

|K||uⁿ_K⁺¹−uⁿ_K|²+

N

X

n=0

X

K∈T

X

L∈∂K_n⁻

δt|V_KLⁿ | |uⁿ_L−uⁿ_K|²,

R1.3. If the speed of advection V is constant, then assumption (1.11) is not nec- essary — see Remark 3.1.

R1.4. Using the techique of [8], we may prove that in the case of a constant speed V(x,t)=V, the result also holds under a sharp CFL condition.

R1.5. The conditions on the mesh (1.5), (1.11) are not stringent. They are satisfied in practice by every efficient mesh generator.

R1.6. We will see in the sequel that the L²-norm of the discrete solution is non increasing. Since the L²-energy of the exact solution is conserved, we deduce from the first estimate of Theorem 1.8 that 0 ≤ ku(·,T )k²

L²− ku(·,T )k²

L² =O(h). Sincek(u−u)(·,T )k_L2 = O(h^1/2), we may split the difference (u−u)(·,T ) into aO(h^1/2) component which is orthogonal to u(·,T ) in L²and aO(h) component.

Conventions and notations. If (X, µ) is a measurable set with finite (positive) measure andφ∈L¹(X), we denote the mean value ofφover X by

hφiX :=

?

X

φdµ:= 1 µ(X)

Z

X

φdµ.

The characteristic function of a set X is denoted by 1X.

For R>0, B(R) denotes the open ball centered at 0 with radius R in R^d.

In the sequel, we will use several functions in L¹_loc(R^d×R₊) which are almost everywhere constants on each space-time cell K_n. We will use the following convention, when (φ_Kn)_Kn∈M

is a family of real numbers, then the bold type letterφdenotes the functionφ:=P

K∈Mφⁿ_K1K_n. Whenφbelongs to C(R^d×R₊,R), then, for Kn∈ M,φⁿ_K :=hφ(·,nδtiKdenotes the mean value ofφon K at time nδt and in this caseφ= P

K∈Mhφ(·,nδtiK1K_n. This last convention holds for any letterϕbut u (u is always the exact solution of (1.2) and u its FV approximation).

For every Kn∈ M, we set V_Kⁿ :=−P

L∈∂K_n⁻V_KLⁿ =P

L∈∂K⁺_nV_KLⁿ and for every t≥0, M_t:={K_n∈ M : 0≤n<t/δt},

in particular, for T =(N+1)δt,MT ={Kn : K∈ T,0≤n≤N}.

Let (ϕⁿ_K)K_n∈Mbe a bounded family of real numbers. We define an energy associated to the weightϕand the FV solution u:

E(ϕ,u,t) := X

K_n∈M_t

ϕⁿ_K⁺¹eⁿ_K(u),

where eⁿ_K(u) := X

L∈∂K⁻_n

1−V_Kⁿδt

|K|

!

|V_KLⁿ |δt|uⁿ_L−uⁿ_K|²+1 2

X

L,M∈∂K_n⁻

V_KLⁿ V_{K M}ⁿ δt²

|K| |uⁿ_M−uⁿ_L|². In Section 3, we will writeE(u,t) forE(1,u,t).

We also define the local consistency error Qⁿ_K(Ψ) of the implicit in time downwind FV scheme associated to (1.2) applied to a functionΨ∈W^1,∞(R^d×R₊):

Qⁿ_K(Ψ) :=Ψⁿ_K⁺¹−Ψⁿ_K

δt + X

L∈∂K_n⁺

V_KLⁿ

|K|

Ψⁿ_L⁺¹−Ψⁿ_K⁺¹

, ∀K_n∈ M.

2. Proof of Theorems 1.6 and 1.7.

2.1. Proof of Theorem 1.7. In this section, u0 ∈ W^1,∞(R^d). We have to estimate the quantity

ku(·,T )−u(·,T )k∞= sup

K₀∈T

sup

x₀∈K₀

u^N_K0⁺¹−u(x0,T ) .

Let x₀ ∈ R^d and let K₀ ∈ T, such that x₀ ∈ K₀. Next, set v⁰ := 1/|K₀|1_K0 and define recursively for K∈ T and 0≤n≤N:

vⁿ_K⁺¹=vⁿ_K+ δt

|K|

X

L∈∂K⁺_N−n

V_KL^N−n(vⁿ_L−vⁿ_K). (2.1) Since u^N_K⁺¹

0 =P

K∈T|K|u^N_K⁺¹v⁰_K, (1.9) and repeated discrete integrations by part yield u^N_K⁺¹

0 =X

K∈T











|K|u^N_Kv⁰_K− X

L∈∂K⁻_N

δtV_KL^N (u^N_L−u^N_K)v⁰_K











=X

K∈T

|K|u^N_Kv¹_K =· · ·=X

K∈T

|K|u⁰_Kv^N_K⁺¹. Notice that the scheme (2.1) is the FV scheme with explicit time-discretization associated to the dual continuous problem vt(x,t)+div{−V(x,T−t)v(x,t)}=0, for (x,t)∈ R^d ×R₊; v(x,0)=v⁰(x) for x∈R^d) whose solution at time T is v(x,T )=v⁰(Y(x,T )).

From the monotony and the L¹-stability of the scheme (2.1), we have v^N_K⁺¹ ≥0, for every K∈ T andP

K∈T|K|v^N_K⁺¹=1, thus for every x0∈K0, we have

u^N_K⁺¹

0 −u(x₀,T ) ≤X

K∈T

|K| |u⁰_K−u(x₀,T )|v^N_K⁺¹=X

K∈T

|K| |u⁰_K−u₀(X(x₀,T ))|v^N_K⁺¹

≤ k∇u₀k∞

Z

R^d

kx−X(x₀,T )kv(x,T )dx+h

! ,

and Theorem 1.6 is the consequence of Theorem 1.7 applied to the dual problem.

R2.1. In general, the dual problem associated to (1.2) is ( v_t+V(·,T− ·)· ∇v=0, x∈R^d,t∈[0,T ],

v(x,0)=v₀(x), x∈R^d,

but since div V=0, we have V(·,T− ·)· ∇v=div(V(·,T− ·)v) and the dual problem is similar to the original problem.

2.2. Energy Estimates. We now turn to the proof of Theorem 1.7. We begin by proving an energy identity.

L2.1. Let u0∈L²(R^d), and (ϕⁿ_K)K_n∈Mbe a family of real numbers. Then, for every t≥0,

Z

R^d

u²(x,t)ϕ(x,t)dx− Z

R^d

u²(x,0)ϕ(x,0)dx+E(ϕ,u,t)=− X

K_n∈M_t

|K|δt(uⁿ_K)²Qⁿ_K(ϕ).

Proof. We assume without loss of generality that t=(n0+1)δt for some n0∈N.

Let n≥0, we consider the quantity eⁿ:=X

K∈T

|K|ϕⁿ_K⁺¹(uⁿ_K⁺¹)²−X

K∈T

|K|ϕⁿ_K(uⁿ_K)², Using (1.9), the first sum is equal to: P

K∈T|K|ϕⁿ_K⁺¹P

L∈V⁻_n(K)aⁿ_KLuⁿ_L2

, where, for Kn ∈ M, we have setV⁻_n(K) :={L∈∂K_n⁻} ∪ {K},

aⁿ_KK:=1−V_Kⁿδt/|K|, and for L∈∂K_n⁻, aⁿ_KL:=−V_KLⁿ δt/|K|.

Next, using Lemma 1.3, we rewrite the second sum:

X

K∈T

|K|ϕⁿ_K(uⁿ_K)²=X

K∈T

|K| −V_Kⁿδtϕⁿ_K(uⁿ_K)²−X

K∈T

X

L∈∂K⁺_n

Vⁿ_LKδtϕⁿ_K(uⁿ_K)². In the last sum, we permute the indexes K and L, to get:

X

K∈T

|K|ϕⁿ_K(uⁿ_K)²=X

K∈T

|K| −V_Kⁿδtϕⁿ_K(uⁿ_K)²+X

K∈T

X

L∈∂K_n⁻

−V_KLⁿ δtϕⁿ_L(uⁿ_L)²

=X

K∈T

|K| X

L∈V⁻_n(K)

aⁿ_KLϕⁿ_L(uⁿ_L)²

=X

K∈T

|K|ϕⁿ_K⁺¹ X

L∈V⁻_n(K)

aⁿ_KL(uⁿ_L)²+X

K∈T

X

L∈V⁻_n(K)

|K|aⁿ_KL(ϕⁿ_L−ϕⁿ_K⁺¹)(uⁿ_L)². Thus, we have, eⁿ =I+II, with

I :=X

K∈T

|K|ϕⁿ_K⁺¹



















 X

L∈V⁻_n(K)

aⁿ_KLuⁿ_L











2

− X

L∈V⁻_n(K)

aⁿ_KL(uⁿ_L)²









 ,

II :=X

K∈T

X

L∈V⁻_n(K)

|K|aⁿ_KL(ϕⁿ_K⁺¹−ϕⁿ_L)(uⁿ_L)². For every Kn∈ M, using the identityP

M∈V⁻_n(K)aⁿ_{K M}=1, an easy computation yields









 X

L∈V⁻_n(K)

aⁿ_KLuⁿ_L











2

− X

L∈V⁻_n(K)

aⁿ_KL(uⁿ_L)²=−1 2

X

L,M∈V⁻_n(K)

aⁿ_KLaⁿ_{K M}(uⁿ_L−uⁿ_M)². Thus, replacing the terms aⁿ_KLby their definitions and summing on K ∈ T, we get

I=−X

K∈T

ϕⁿ_K⁺¹eⁿ_K(u).

Now, rearranging the terms in II, we compute II=X

K∈T

|K|δt(uⁿ_K)²











ϕⁿ_K⁺¹−ϕⁿ_K

δt + X

L∈∂K_n⁺

V_KLⁿ δt

|K|

ϕⁿ_L⁺¹−ϕⁿ_K⁺¹









 . Summing these two equations and then summing on 0≤n≤n₀, we get the lemma.

2.3. Weak consistency. In this section, the integer 0≤n≤N is fixed. Let (vK)K∈T be a family of real numbers and set v :=P

KvK1K, then let us define the weak total variation of v with respect to the familly (V_KLⁿ )K,L∈T by:

kvk_{Vn−WT V} :=X

K∈T

X

L∈∂K_n⁻

|V_KLⁿ ||vK−vL|.

Now forη∈R, we defineωη(v) to be the union of cells K such that v_K < η <0 or 0< η <v_K and we consider the level set decomposition v(x)=R

Rsgn(η) 1_{ω(v, η)}dη. Whenω ⊂R^dis a finite union of elements ofT, we havek1_ωk_{Vn−WT V} = ¹₂P

KpL⊂∂ω|V_KLⁿ |, and it is not difficult to see that

kvk_{Vn−WT V} = Z

R

k1_{ω(v, η)}k_{Vn−WT V}dη.

This suggests the following definition. Let (w_K)K∈T ⊂R and w :=P

K∈Tw_K1_K, we set:

kwk^′

Vn−WT V :=sup

ω

R

ωw(x)dx

k1_ωk_{Vn−WT V} ,

where the supremum is taken over all subsetsω⊂R^dsuch thatωis a finite union of elements ofT. This norm is the dual norm associeted to the seminormk · k_{Vn−WT V}. We do not prove it, we we will only need the following inequality, which is a direct consequence of the two last identities.

L2.2. Let v, w as above such thatkvk_{Vn−WT V} <∞andkwk^′

Vn−WT V <∞. Then Z

R^d

v(x)w(x)dx=X

K∈T

|K|vKwK ≤ kvk_{Vn−WT V}kwk^′

Vn−WT V. We now prove a consistency result for the implicit downwind FV scheme.

L2.3. LetΨ0 ∈ L^∞(R^d) such that∇Ψ0 is bounded. For (x,t) ∈ R^d ×[0,T ], set Ψ(x,t) := Ψ0(X(x,t)) and Qⁿ(Ψ) :=P

K∈TQⁿ_K(Ψ)1K_n. There exists R^m_K(Ψ), Km∈ M, such that for n≤N, Rⁿ(Ψ) :=P

K∈TRⁿ_K(Ψ)1K_n, and Sⁿ(Ψ) :=Qⁿ(Ψ)−Rⁿ(Ψ) satisfy kRⁿ(Ψ)k∞≤CC0(k∇Vk∞+k∂_tVk∞)k∇Ψ0k∞h, kSⁿ(Ψ)k^′

Vn−WT V ≤CC0kVk∞k∇Ψ0k∞h.

Proof. For Km ∈ M, let us noteΨ^m_K the mean value ofΨ on K at time mδt: Ψ^m_K = hΨ(·,mδti_K. Let K∈ T, sinceΨsolves∂tΨ +div(VΨ)=0 on R^d×R₊, we have

Ψⁿ_K⁺¹−Ψⁿ_K

δt = 1

|K|δt Z

K_n

∂tΨ(x,s)dxds =− 1

|K|δt X

L∈∂K

Z

KpL_n

Ψ(x,s)V(x,s)·nKLdxds.

For K_m∈ M, we set R^m_K(Ψ) := 1

|K|δt X

L∈∂K

Z

KpLm

Ψ(x,s)n

hV·n_KLi_KpLm −V(x,s)·n_KLo dxds.

Since, for every space-time edge KpLm, we haveR

KpLm{hV·nKLi_KpLn−V(x,s)·nKL}dxds=0, we may replaceΨ(x,s) byΨ(x,s)− hΨiKpL_m in the above equation. Then, for every K∈ T,

|Rⁿ_K(Ψ)| ≤ 1

|K|δt X

L∈∂K

Z

KpLn

nΨ(x,s)− hΨi_KpLnon

V(x,s)·n_KL− hV·n_KLi_KpLno dxds

≤|∂K|

|K|

k∇Ψk_L∞(R^d×[0,T ])h+k∂tΨk_L∞(R^d×[0,T ])δt

(k∇Vk∞h+k∂tVk∞δt).

The regularity of the mesh (1.5), leads to|∂K|/|K| ≤α⁻²h⁻¹, and using the bounds onk∇Xk andk∂tXkof Theorem 1.1 and the CFL condition (1.7), we get the first part of the lemma.

Next, since V_KLⁿ =|KpL|hV·nKLiKpL_n, we have ΨⁿK⁺¹−ΨⁿK

δt −Rⁿ_K=−X

L∈∂K

Vⁿ_KL

|K|hΨi_KpLn,

and using the identity of Lemma 1.3, Qⁿ_K(Ψ)−Rⁿ_K(Ψ)=−X

L∈∂K

Vⁿ_KL

|K|hΨi_KpLn+ X

L∈∂K_n⁺

V_KLⁿ

|K|

ΨⁿL⁺¹−ΨⁿK⁺¹

= 1

|K|









 X

L∈∂K⁺_n

V_KLⁿ (ΨⁿL⁺¹− hΨi_KpLn)+ X

L∈∂K_n⁻

V_KLⁿ (ΨⁿK⁺¹− hΨi_KpLn)









 .

Letω=∪K∈I_ωK be a finite union of elements ofT, we have Z

ω

Sⁿ(Ψ)= X

K∈I_ω

|K|(Qⁿ_K(Ψ)−Rⁿ_K(Ψ)).

If the cells K, L with L ∈ ∂K_n⁺are such that K and L belong to Iω, then the edge KpL has two contributions in the sum above: V_KLⁿ (Ψⁿ_L⁺¹ − hΨi_KpLn) (due to K ∈ I_ω, L ∈ ∂K_n⁺) and V_LKⁿ (Ψⁿ_L⁺¹− hΨi_LpKn) (due to L∈I_ω, K ∈∂L⁻_n). These contributions cancel each other. Thus, if∂ω⁺(resp. ∂ω⁻) denotes the set{KpL,K ∈ I_ω,L ∈ ∂K_n⁺\I_ω} (resp. {KpL,K ∈ I_ω,L ∈

∂K_n⁻\Iω}), we have:

Z

ω

Sⁿ(Ψ)= X

KpL∈∂ω⁺

V_KLⁿ (ΨⁿL⁺¹− hΨi_KpLn)+ X

KpL∈∂ω⁻

V_KLⁿ (ΨⁿK⁺¹− hΨi_KpLn), and from the definition ofk · k_Vn−WT V, we get

Z

ω

Sⁿ(Ψ)

≤







 1 2

X

KpL⊂∂ω

|V_KL|









k∇Ψk_L∞(R^d×[0,T ])h+k∂_tΨk_L∞(R^d×[0,T ])δt

≤ k1ωkVⁿ−WT VC0(1+c0)k∇Ψ0k∞h, which is the desired estimate.

2.4. Key estimate. We use the results of Sections 2.2 and 2.3 to estimate the growth of Rd

Ru²(x,t)Ψ²(x,t)dx, whenψis a cut-offfunction.

L2.4. LetΨ0,Ψ0 ∈ W_∞¹(R^d,[0,1]), such that∇Ψ0is bounded. For (x,t) ∈ R^d× R₊, let us defineΨ(x,t) := Ψ0(X(x,t)) andΨ(x,t) := Ψ0(X(x,t)). Assume thatΨ0 ≡1 on suppΨ0+B(0,C0(2+c0)h)={x∈R^d : Ψ0|B(x,C0(1+c0)h).1}. Then, for every 0≤t≤T ,

Z

R^d

u²(·,t)Ψ²(·,t)− Z

R^d

u²₀Ψ²(·,0)≤n

C1,⋆k∇Ψ0k²_∞h+C2,⋆k∇Ψ0k_∞hoZ t 0

Z

R^dΨ²u², with C1,⋆=CC²₀ξ⁻¹and C2,⋆=CC0.

Proof. Let 0≤t≤T , if t< δt, the left hand side vanishes and there is nothing to prove.

So we assume that there exists n0 ≥0 such that (n0+1)δt ≤t <(n0+2)δt, since the left hand side does not depend on t for (n₀+1)δt≤t<(n₀+2)δt and the right hand side is non- decreasing, we may assume that t=(n₀+1)δt. In this caseM_t={K_n∈ M : 0≤n≤n₀}.