The SMAI journal of computational mathematics

Mehdi Badsi

Christophe Berthon

Ludovic Martaud

A family of second-order dissipative finite volume schemes for hyperbolic systems of conservation laws

SMAI-JCM
SMAI Journal of
Computational Mathematics
Erratum to : “A family of
second-order dissipative finite
volume schemes for hyperbolic
systems of conservation laws”
Mehdi Badsi, Christophe Berthon & Ludovic Martaud
Volume 9 (2023), p. 311-318.
https://doi.org/10.5802/smai-jcm.103
© The authors, 2023.
The SMAI Journal of Computational Mathematics is a member
of the Centre Mersenne for Open Scientific Publishing
http://www.centre-mersenne.org/
Submissions at https://smai-jcm.centre-mersenne.org/ojs/submission
e-ISSN: 2426-8399
SMAI Journal of Computational Mathematics
Vol. 9, 311-318 (2023)
Erratum to : “A family of second-order dissipative finite volume
schemes for hyperbolic systems of conservation laws”
Mehdi Badsi 1
Christophe Berthon 2
Ludovic Martaud 3
1 Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue
de la Houssinière, BP 92208, 44322 Nantes, France
E-mail address: mehdi.badsi@univ-nantes.fr
2 Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue
de la Houssinière, BP 92208, 44322 Nantes, France
E-mail address: christophe.berthon@univ-nantes.fr
3 Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue
de la Houssinière, BP 92208, 44322 Nantes, France
E-mail address: ludovic.martaud@univ-nantes.fr.
Abstract. We, authors of the paper entitled “A family of second-order dissipative finite volume schemes for hyper-
bolic systems of conservations laws” present our apologies for many mistakes in the current version of the published
paper. We explain below the mathematical mistakes and how to modify the assumption of the Theorem (4.2) to
get the same conclusion (namely, the decay of the mass of the entropy). New material and a correct proof of the
Theorem (4.2) are given. A corrected version of the published paper has been uploaded on HAL server at the adress:
https://hal.science/hal-03564325.
2020 Mathematics Subject Classification. 65N08, 35L65, 35L67.
Keywords. Systems of conservation laws, Second-order finite Volume schemes, Explicit schemes, Global
entropy inequality.
1. Issues in the published version
The mathematical errors are located in Section 4 (Global entropy inequality in the general case). In
particular, Lemma 4.3 (Poincaré inequality) is wrong. It asserts a result of monotony of matrices (see
the second point in the section below) which is wrong for non diagonal matrices. Moreover, since in
Section 4 we work with a general convex set Ω of Rd, if 0 ̸∈ Ω, there is a mathematical contradiction
with the fact that we consider in the assumption of the Theorem 4.2 compactly supported state
(wni ) ⊂ Ω. What misleads us, is the linearized analysis of Section 3, which has to be understood as a
stability analysis around a constant state in Ω. Actually, in Section 3, (wni ) is rather understood as a
fluctuation and this is why the l2(Z) space is adapted.
2. The wrong arguments
The Lemma (4.3) which was intended to prove a Poincaré inequality is wrong. It relies on two points:
(1) It is supposed that the sequence (wni )i∈Z is compactly supported. But provided 0 /∈ Ω, it yields
a contradiction with the fact that we also ask (wni )i∈Z ⊂ Ω. The argument was intended to
mimick the case of the linearized equations around a constant state which misleads us and we
do apologize for this.
https://doi.org/10.5802/smai-jcm.103
© The authors, 2023
311
M. Badsi, et al.
(2) The last point but not the least: it uses at the beginning of the proof a fake argument which
is the following : let A ∈ Md(R) a symmetric positive definite matrix, then there exists C > 0
such that for all x, y ∈ Rd,
∑d
j=1 |(Ax)j ||yj | ≥ C
∑d
j=1 |xj ||yj |. It is wrong for non diagonal
matrices as soon as d > 1, as we prove here under.
Lemma 2.1. Let d > 1 and A = (aij) ∈ Md(R) a non diagonal symmetric positive definite matrix.
Then for all C > 0, there exists x, y ∈ Rd such that
∑d
j=1 |(Ax)j ||yj | < C
∑d
j=1 |xj ||yj |.
Proof. Let C > 0. Since A is positive definite, one has for all i ∈ {1, . . . , d}, aii > 0 and since it is
non diagonal there exists a couple (i∗, j∗) ∈ {1, . . . , d}2 with i∗ ̸= j∗ such that ai∗j∗ ̸= 0. So consider
y = ei∗ the canonical basis vector of Rd. Let x = (x1, . . . , xi∗−1, 1, xi∗+1, . . . , xd)T ∈ Rd belonging to
the non-empty hyperplane
Pi∗ = {w ∈ Rd : ai∗ · w = 0},
where ai = (ai∗1, . . . , ai∗d)T ̸= 0. Then one has, one the one hand
∑d
j=1 |(Ax)j ||yj | = |(Ax)i∗ | =
0 because x ∈ Pi∗ and on the other hand
∑d
j=1 |xj ||yj | = |xi∗ | = 1. Hence,
∑d
j=1 |(Ax)j ||yj | <
C
∑d
j=1 |xj ||yj |.
3. Erratum
Mathematical mistakes are in Section 4. We provide here after the corrections to consider in order to
get the conclusion of the Theorem (4.2) still valid.
3.1. List of points to adress
(1) About the convergence of the various series:
(a) For the stability analysis, to justify that the various series are convergent, what is precisely
needed is the control in l2 of the discrete derivatives. So the adapted framework is the
homogeneous Sobolev space defined by
ḣ2(Z; Ω) :=
{
w ∈ ΩZ : (wi+1 − wi)i∈Z ∈ l2(Z), (wi+1 − 2wi + wi−1)i∈Z ∈ l2(Z)
}
(3.1)
which is no longer in contradiction with the fact that one may consider constante state
for (wni ) ⊂ Ω.
(b) The convergence in l2 of the discrete derivatives alone is not sufficient. One also needs
that the map s ∈ [0, 1] 7→ Qni (s), s ∈ [0, 1] 7→ F ni (s) and (u, s) ∈ [0, 1]2 7→ Ni(us) given
in (4.5), (4.7) and (4.11) are bounded uniformly for s ∈ [0, 1] and i ∈ Z. Noticing that
in the definition of these matrices the argument is a convex combination of two elements,
we have no choice that assuming that (wni ) ⊂ Kn where Kn ⊂ Ω is a convex compact
subset. We thus need the following lemma.
Lemma 3.1. Let Ω a non empty convex open set. Let K ⊂ Ω a convex compact subset.
Let a, b two elements in K. If A : Ω → Md(R) is a continuous map then for any two
points a, b ∈ K the map s ∈ [0, 1] 7→ A((1 − s)a + sb) is continuous on [0, 1]. In addition,
for any matrix norm ∥ · ∥ :
sup
s∈[0,1]
∥A((1 − s)a + sb)∥ ≤ max
w∈K
∥A(w)∥.
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Second-order dissipative schemes for hyperbolic systems
Proof. Since a, b ∈ K and K is a convex set one has (1 − s)a + sb ∈ K for all s ∈ [0, 1].
Therefore the map A : s ∈ [0, 1] 7→ ∥A((1 − s)a + sb)∥ is well-defined and continuous
on [0, 1] by composition of continuous maps. Since [0, 1] is a compact set in R, there
exists s∗ ∈ [0, 1] such that sups∈[0,1] ∥A((1 − s)a + sb) = ∥A((1 − s∗)a + s∗b)∥. Then since
(1 − s∗)a + s∗b ∈ K one has also ∥A((1 − s∗)a + s∗b)∥ ≤ sup
w∈K
∥A(w)∥ = max
w∈K
∥A(w)∥ where
the supremum is a maximum because K is a compact set.
(2) In Proposition (4.4) one has to select the parameter θ such that the inequality (4.4) holds.
Specifically, one needs to justify that the denominator in (4.11) cannot be zero. To remedy
this, we precisely eliminate this case. So it leads us to consider,
Sη(Z; Ω) :=
w ∈ ḣ2(Z; Ω) :
d∑
j=1
∑
i∈Z
|(∇η(wi+1) − ∇η(wi−1))j ||(wi+1 − 2wi + wi−1))j | ≠ 0
 . (3.2)
Note that Sη(Z; Ω) does not contain constante state. However it is not a restriction because
in this case, taking (Θni ) to be zero for example, the scheme (4.1) is exact (because it is
conservative).
3.2. Correction of the statements
• In the introduction, the sentence at the end of page 2 must be completed with the word convex.
It should be written: “Namely, it is required to belong to a convex compact subset of Ω.”
• In the Definition 2.3, the third sentence must be completed with the word strictly. It should
be written : Let η ∈ C1(Ω,R) be a strictly convex entropy function.
• In Section 3, the second sentence should be replaced by: We therefore consider momentarily
a linearization of a non linear hyperbolic scalar equation around a constant state
which here takes the form{
∂tw + a∂xw = 0, t > 0, x ∈ R,
w(x, t = 0) = w0(x)
(3.3)
where a ̸= 0.
• The correct statement of Theorem 4.3 is the following:
Theorem 3.2 (Global entropy inequality). Let Ω be a non-empty convex open subset of Rd.
Consider (η, G) ∈ C2(Ω,R) × C1(Ω,R) a pair of strictly convex entropy and entropy-flux
which satisfies (1.2). Let (wni )i∈Z ∈ Sη(Z; Ω) such that
∑
i∈Z η(wni )∆x is finite. Let (Θni )i∈Z, a
sequence of bounded matrices such that
Sn :=
∫ 1
0
∑
i∈Z
Nni (s)P ni (s)Dni · Dni ds > 0, (3.4)
313
M. Badsi, et al.
where the block matrices (Dni , Nni , P ni ) ∈ R2d × (M2d(R))2 are respectively defined by
Dni =
δni− 12
δn
i+ 12
 ,
Nni (s) =
∇
2η
(
wni − sδni− 12
)
0
0 ∇2η
(
wni + sδni+ 12
)
 ,
P ni (s) =
(1 − 2s)I + Θni −Θni
Θni − I 2(1 − s)I − Θni
 ,
(3.5)
for all s ∈ [0, 1]. Also assume there exists a convex compact set Kn ⊂ Ω such that (wni )i∈Z ⊂
Kn. Let the numerical diffusion λ be such that
λ > λn, (3.6)
where
λn =
2 max
0, ∑
i∈Z
∫ 1
0
s
(∫ 1
0
Nni (us)du
)
F ni (s)Dni · Dni ds

∑
i∈Z
∫ 1
0
Nni (s)P ni (s)Dni · Dni ds
≥ 0,
F ni (s) =
−∇f
(
wni − sδni− 12
)
0
0 ∇f
(
wni + sδni+ 12
)
 , ∀s ∈ [0, 1],
(3.7)
and ∆t∆x be such that
0 < ∆t∆x ≤
−
∑
i∈Z
∇η(wni ) · Rni∫ 1
0
(1 − s)
∑
i∈Z
∇2η
(
wni + s
∆t
∆xR
n
i
)
Rni · Rni ds
. (3.8)
If Ω = Rd, then one has the global entropy inequality,∑
i∈Z
η(wn+1i )∆x ≤
∑
i∈Z
η(wni )∆x. (3.9)
If Ω ̸= Rd then the global entropy inequality (3.9) still holds if moreover 0 < ∆t∆x ≤ c
n where
cn is a positive non explicit constant given by the Lemma (4.1).
• Lemma 4.3 (Poincaré inequality) should be removed.
• The conclusion of Proposition 4.3 (Existence of dissipative corrections) is still valid but it
needs supplementary material (see the Section 3.3 below).
• In the Section (4.1) entitled “Reformulation of the global dissipation”: at the begining of the
section, it should be mentionned that owing to Propositions 3.4 and 3.5 (below), it is fully
understood, in this section, that the various series are convergent so that we only concentrate
on algebraic computations.
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Second-order dissipative schemes for hyperbolic systems
• In accordance with the previous point, in the Section (4.1), for the series to be convergent, one
must replace, in Lemma (4.5) and (4.6), the assumptions by the following ones: Let the sequence
(wni )i∈Z ∈ Sη(Z; Ω) and let (η, G) ∈ C2(Ω,R) × C1(Ω,R) a pair of strictly convex entropy-
entropy-flux which satisfies (1.2). Assume in addition that there exists a convex compact subset
Kn ⊂ Ω such that (wni ) ⊂ Kn.
3.3. New material to complete the Proof of Theorem 3.2
We give below the new material to complete the proof of the Theorem 3.2.
3.3.1. Preparatory material
Before proving the main theorem, we need preparatory material. It is needed to justify the convergence
of the various series and specifically to justify the finiteness of the numerical diffusion (3.7).
Lemma 3.3 (Relative uniform upper bound). Let Ω a non empty convex open set. Let K ⊂ Ω a
convex compact set. Let a, b two elements in K. If A : Ω → Md(R) is a continuous map then for any
two points a, b ∈ K the map s ∈ [0, 1] 7→ A((1 − s)a + sb) is continuous on [0, 1]. In addition, for any
matrix norm ∥ · ∥ :
sup
s∈[0,1]
∥A((1 − s)a + sb)∥ ≤ max
w∈K
∥A(w)∥.
Proof. Since a, b ∈ K and K is a convex set one has (1 − s)a + sb ∈ K ⊂ Ω for all s ∈ [0, 1].
Therefore the map A : s ∈ [0, 1] 7→ ∥A((1 − s)a + sb)∥ is well-defined and continuous on [0, 1] by
composition of continuous maps. Since [0, 1] is a compact set in R, there exists s∗ ∈ [0, 1] such
that sups∈[0,1] ∥A((1 − s)a + sb) = ∥A((1 − s∗)a + s∗b)∥. Then since (1 − s∗)a + s∗b ∈ K one has
also ∥A((1 − s∗)a + s∗b)∥ ≤ supw∈K ∥A(w)∥ = maxw∈K ∥A(w)∥ where the supremum is a maximum
because K is a compact set.
Proposition 3.4 (Existence of dissipative corrections). Let (wni )i∈Z ∈ Sη(Z; Ω) and let η ∈ C2(Ω,R)
a strictly convex entropy. Assume in addition that there exists a convex compact subset Kn ⊂ Ω such
that (wni ) ⊂ Kn. Then if (Θni )i∈Z verifies
Θni = −θ diag1≤j≤d
(
sign
((
∇η(wni+1) − ∇η(wni−1)
)
j
(δn
i+ 12
− δn
i− 12
)j
))
, (3.10)
with
θ >
− min
0, ∫ 1
0
∑
i∈Z
Qni (s)Dni · Dni ds

∑
i∈Z
j∈{1,...,d}
|
(
∇η(wni+1) − ∇η(wni−1))
)
j
(δn
i+ 12
− δn
i− 12
)j |
,
Qni (s) =
(1 − 2s)∇2η(wni − sδni− 12 ) −∇2η(wni + sδni+ 12 )
0 2(1 − s)∇2η(wni + sδni+ 12 )
 ,
(3.11)
for all s ∈ [0, 1], then the dissipative inequality (3.4) holds.
Proof. We firstly prove that the right hand side in (3.11) is finite. By assumption the denominator
in (3.11) cannot be zero. Moreover, one has using a Cauchy–Schwarz inequality∣∣∣ ∫ 1
0
∑
i∈Z
Qni (s)Dni · Dni ds
∣∣∣ ≤ ∑
i∈Z
∫ 1
0
∥Qni (s)∥2ds∥Dni ∥22 ≤
∑
i∈Z
sup
s∈[0,1]
∥Qi(s)∥2∥Dni ∥22.
315
M. Badsi, et al.
where ∥ · ∥2 denotes both the Euclidean norm on R2d and the induced norm on M2d(R). Using the
equivalence of norm in a finite dimensional vector space one has in particular that there exists a
constant c(d) which only depends on the dimension d such that
sup
s∈[0,1]
∥Qi(s)∥2 ≤ c(d) sup
s∈[0,1]
∥Qi(s)∥∞.
From now, observe that thanks to Lemma 3.3 applied with A = ∇2η and the matrix norm ∥ · ∥∞, we
obtain for all i ∈ Z
sup
s∈[0,1]
∥∇2η((1 − s)wi + swi−1)∥∞ ≤ max
w∈Kn
∥∇2η(w)∥∞.
We therefore have that for all i ∈ Z, sup
s∈[0,1]
∥Qi(s)∥∞ ≤ 2 max
w∈Kn
∥∇2η(w)∥∞. It eventually yields the
upper bound ∑
i∈Z
sup
s∈[0,1]
∥Qi(s)∥2∥Dni ∥22 ≤ 2c(d) max
w∈Kn
∥∇2η(w)∥∞
∑
i∈Z
∥Dni ∥22 < +∞.
The last series being convergent because (wni ) ∈ ḣ2(Z; Ω). So the ratio in (3.11) is finite. Next, since the
matrix (Θni )i∈Z defined by (3.10) are symmetric (because they are diagonal) we have Θni a · b = a · Θni b,
for all vectors (a, b) ∈ (Rd)2. As a consequence, from the definition of the matrices P ni , Nni given
by (3.5) and the definition of Sn given in (3.4), we have
Sn =
∫ 1
0
∑
i∈Z
Qni (s)Dni · Dni ds −
∑
i∈Z
Θni
(
∇η(wni+1) − ∇η(wni−1)
)
·
(
δn
i+ 12
− δn
i− 12
)
.
Using the (Θni )i∈Z formula (3.10), we eventually obtain
Sn =
∫ 1
0
∑
i∈Z
Qni (s)Dni · Dni ds + θ
∑
i∈Z
j∈{1,...,d}
|
(
(∇η(wni+1) − (∇η(wni−1))
)
j
(δn
i+ 12
− δn
i− 12
)j |,
which is positive with θ verifying the inequality (3.11).
Proposition 3.5 (Finiteness of the numerical diffusion). Let (wni )i∈Z ∈ Sη(Z; Ω) and let η ∈ C2(Ω,R)
a strictly convex entropy. Assume in addition that there exists a convex compact subset Kn ⊂ Ω such
that (wni ) ⊂ Kn. Then the numerical diffusion λn defined by (3.7) is positive and finite.
Proof. We prove that the numerator in (3.7) is finite. The denominator was proven to be positive by
construction of the matrix parameter (Θni )i∈Z in Proposition 3.4. So one has, using a Cauchy–Schwarz
inequality and the fact that a matrix norm is submultiplicative∣∣∣ ∑
i∈Z
∫ 1
0
s
(∫ 1
0
Nni (us)du
)
F ni (s)Dni · Dni ds
∣∣∣ ≤ ∑
i∈Z
sup
s∈[0,1]
∥Nni (s)∥2 sup
s∈[0,1]
∥F ni (s)∥2∥Dni ∥22
where ∥ · ∥2 denotes both the Euclidean norm on R2d and the induced norm on M2d(R). Using the
equivalence of norm in a finite dimensional vector space one has in particular that there exist a constant
c(d) which only depends on the dimension d such that
sup
s∈[0,1]
∥Nni (s)∥2 ≤ c(d) sup
s∈[0,1]
∥Nni (s)∥∞, sup
s∈[0,1]
∥F ni (s)∥2 ≤ c(d) sup
s∈[0,1]
∥F ni (s)∥∞.
Observe that thanks to Lemma 3.3 applied respectively with A = ∇2η and A = ∇f and the matrix
norm ∥ · ∥∞, we obtain for all i ∈ Z
sup
s∈[0,1]
∥∇2η((1 − s)wi + swi−1)∥∞ ≤ max
w∈Kn
∥∇2η(w)∥∞,
sup
s∈[0,1]
∥∇f((1 − s)wi + swi−1)∥∞ ≤ max
w∈Kn
∥∇f(w)∥∞.
316
Second-order dissipative schemes for hyperbolic systems
It therefore yields that for all i ∈ Z,
sup
s∈[0,1]
∥Nni (s)∥∞ ≤ max
w∈Kn
∥∇2η(w)∥∞, sup
s∈[0,1]
∥F ni (s)∥∞ ≤ max
w∈Kn
∥∇f(w)∥∞.
So we eventually obtain,∑
i∈Z
sup
s∈[0,1]
∥Nni (s)∥2 sup
s∈[0,1]
∥F ni (s)∥2∥Dni ∥2 ≤ c(d)2 max
w∈Kn
∥∇2η(w)∥∞ max
w∈Kn
∥∇f(w)∥∞
∑
i∈Z
∥Dni ∥22 < +∞.
The last series is finite because (wni ) ∈ ḣ2(Z; Ω).
3.3.2. Proof of the Theorem 3.2 (related to the Section (4.2) “Proof of the main result”)
Let (η, G) ∈ C2(Ω,R) × C1(Ω,R) a pair of strictly convex entropy, entropy-flux which satisfies (1.2).
Consider the sequence (wni )i∈Z verifying the assumptions of the Theorem 3.2. Consider the CFL
condition ∆t∆x given in the Theorem 3.2. Therefore the sequence (w
n+1
i )i∈Z is contained in Ω. Since
η ∈ C2(Ω,R), using a Taylor expansion, in the above equations, we deduce
η(wn+1i ) = η(w
n
i ) +
∆t
∆x∇η(w
n
i ) · Rni +
( ∆t
∆x
)2 ∫ 1
0
(1 − s)∇2η
(
wni + s
∆t
∆xR
n
i
)
Rni · Rni ds.
It implies that,∑
i∈Z
η(wn+1i )∆x =
∑
i∈Z
η(wni )∆x +
∆t
∆x
∑
i∈Z
∇η(wni ) · Rni ∆x
+
( ∆t
∆x
)2 ∫ 1
0
(1 − s)
∑
i∈Z
∇2η
(
wni + s
∆t
∆xR
n
i
)
Rni · Rni ds∆x.
The second sum after the equality was proven being convergent thanks to Propositions 3.4 and 3.5
(of this document). We now justify that the third series inherited from the second order term in the
Taylor expansion is finite. Owing to the Lemma (4.1), since 0 < ∆t∆x < c
n, one has for all s ∈ [0, 1], for
all i ∈ Z,
wni + s
∆t
∆xR
n
i ∈ Kn+1 := Kn + B(0,
∆t
∆xR
n) ⊂ Ω
where B(0, ∆t∆xR
n) is the closed ball (for a norm ∥ · ∥ on Rd) centered at 0, of radius ∆t∆xR
n and where
Rn > 0 is a non explicit constant. Since Kn+1 is the sum of two compact sets, then Kn+1 is also a
compact set. So repeating the same arguments as in Propositions 3.4 and 3.5 (of this document), one
infers that there exists a constant c(d) that only depends on the dimension d such that∣∣∣ ∫ 1
0
(1 − s)
∑
i∈Z
∇2η
(
wni + s
∆t
∆xR
n
i
)
Rni · Rni ds
∣∣∣ ≤ c(d) max
w∈Kn+1
∥∇2η(w)∥∞
∑
i∈Z
∥Rni ∥22 < +∞.
Using again a similar argument as in Propositions 3.4 and 3.5 (that we do not detail), one again shows
that the above series is finite because (wni ) ∈ ḣ2(Z; Ω), the matrix parameter (Θni )i∈Z is bounded and
(wni ) ⊂ Kn. To establish the global entropy inequality (3.9), it is now sufficient to prove the following
inequality ∑
i∈Z
∇η(wni ) · Rni +
∆t
∆x
∫ 1
0
(1 − s)
∑
i∈Z
∇2η
(
wni + s
∆t
∆xR
n
i
)
Rni · Rni ds ≤ 0. (3.12)
Thanks to the Lemmas (4.6) and (4.7), we deduce that the dissipation reformulates as∑
i∈Z
∇η(wni ) · Rni =
∑
i∈Z
∫ 1
0
(
s
2
(∫ 1
0
Nni (us)du
)
F ni (s) −
λ
4 N
n
i (s)P ni (s)
)
Dni · Dni ds. (3.13)
317
M. Badsi, et al.
But, as the sequence of matrices (Θni )i∈Z is selected in order to satisfy the inequality (3.4), and λ is
such that the inequality (3.6) is verified, we have∑
i∈Z
∇η(wni ) · Rni < 0,
which is the definition of a dissipative flux given by (2.3). Using the CFL condition (3.8) concludes
the proof.
318


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