A relaxation method via the Born-Infeld system

Michael Baudin∗†, Frederic Coquel‡, and Quang Huy Tran∗§

July 28, 2008

Abstract

The semilinear relaxation was introduced by Jin and Xin [Comm. Pure Appl.Math. 48, 235–276, 1995] in order to approximate the conservation law ∂tu+∂xf(u) =0 for any flux function f ∈ C 1(R;R). In this paper, we propose an alternate relaxationtechnique for scalar conservation laws of the form ∂tu + ∂x(u(1− u)g(u)) = 0, whereg ∈ C 1([0, 1];R) and 0 6∈ g(]0, 1[). We extend this alternate philosophy to an arbitraryflux function f whenever possible. Unlike the semilinear approach, the new relaxationstrategy does not involve any tuning parameter, but makes use of the Born-Infeldsystem. Another advantage of this method is that it enables us to achieve a maximumprinciple on the velocities w = (1−u)g and z = −ug, which turns out to be a physicallyinteresting and helpful feature in the context of some two-phase flow problems.

Keywords

Two-phase flow, relaxation methods,Born-Infeld system, Chapman-Enskog analysis, subcharacteristic condition

1 Introduction

We are concerned with the relaxation of the conservation law

∂tu+ ∂xf(u) = 0, x ∈ R, t > 0, (1.1)

where u is the unknown function taking on values in some convex domain D ⊂ R, andf ∈ C 1(D;R) is an arbitrary flux function. From the pioneering ideas of Whitham [24]and Liu [18], it has been suggested by Jin and Xin [15] that the solution u of (1.1) can beobtained as the limit of the family of solutions (Uλ)λ≥0 to the semilinear systems

∂tUλ+ ∂xF

λ = 0 (1.2a)

∂tFλ+ a2∂xU

λ = λ[f(Uλ)− F λ], (1.2b)

in which λ, representing the inverse of a relaxation time, is meant to go to +∞, and F λ isa fully-fledged variable which can be maintained close to f(Uλ) by choosing large values

∗Departement Mathematiques Appliquees, Institut Francais du Petrole, 1 et 4 avenue de Bois-Preau,

92852 Rueil-Malmaison Cedex, France†presently at: ALTRAN Aeronautics-Space & Defence, 2 rue Paul Vaillant Couturier, 92300 Levallois-

Perret, France‡Laboratoire Jacques-Louis Lions, CNRS et Universite Pierre et Marie Curie, B.C. 187, 75252 Paris

Cedex 05, France§Corresponding author: Q-Huy.Tran@ifp.fr, phone +33 1 47 52 74 22, fax +33 1 47 52 70 22

1

for λ. The parameter a is subject to an additional condition to ensure stability. Beforeelaborating on this condition, it is essential to point out that the advantage of Jin and Xin’srelaxation strategy lies in the fact that the left-hand side of (1.2) is linear, has constantcoefficients and characteristic speeds ±a. The theoretical behavior of this method for thescalar equation (1.1), as well as of its variant for systems, has been extensively investigatedby many authors such as Chen and Liu [9], Chen et al. [8], Natalini [19], Serre [23].For a thorough review of relaxation, the readers are also referred to the monographsby Bouchut [6], Hsiao [12], as well as the papers by Coquel and Perthame [10], Jin etSlemrod [14], Liu et al. [17].

The power of Jin-Xin’s semilinear relaxation (1.2) lies in the fact that it can be safelyapplied to any scalar conservation law (1.1), provided that the so-called subcharacteristiccondition (in Liu’s terminology [18])

a > maxu∈D

|f ′(u)| (1.3)

be satisfied for all values of u under consideration. In this respect, a acts as a tuningparameter that monitors the extra amount of viscosity due to the relaxation procedure.Under some practical circumstances, however, the semilinear approach may not be themost suitable scheme. For instance, when abruptly applied to a system, it is known togive rise to excessive dissipation, in which case a partial use of it would be preferable, asrecommended by Baudin et al. [2, 3] for some two-phase flow models. Another instanceof specific situations where Jin-Xin’s relaxation does not fit the purpose of the physicalproblem —and which motivates our present work— is the scalar equation

∂tu+ ∂x(u(1− u)g(u)) = 0, x ∈ R, t > 0, (1.4)

defined over the u-domain D = [0, 1] and g ∈ C 1(D;R), 0 6∈ g(D). Such a single conserva-tion law often arises in the context of simplified drift-flux models for two-phase flow usinga modified Zuber-Findlay hydrodynamic law (see Andrianov et al. [1, §2.2.2], Benzoni-Gavage [4], Zuber and Findlay [25], Seguin and Vovelle [21]). In (1.4), the quantities

w(u) = (1− u)g(u) and z(u) = −ug(u) (1.5)

play the role of convective velocities, insofar as equation (1.4) can be read as either

∂t(u) + ∂x(u · w(u)) = 0 or ∂t(1− u) + ∂x((1− u) · z(u)) = 0. (1.6)

The physics of the two-phase flow model requires that w and z be always well-defined. Now,if the semilinear relaxation (1.2) is used, the out-of-equilibrium values of these velocitieswill have to be computed as

W λ =F λ

Uλand Zλ = − F λ

1− Uλ. (1.7)

Since F λ is not equal to f(Uλ) = Uλ(1 − Uλ)g(Uλ) for λ < +∞, there is no guaranteethat the out-of-equilibrium velocities (1.7) remain bounded as uλ approaches 0 or 1.

In this paper, we would like to put forward a new relaxation technique for the scalarequation (1.4), the interest of which is to ensure a maximum principle on the values of theconvective velocities w and z. This technique is based upon the relaxation system

∂tUλ + ∂x(U

λ(1− Uλ)Gλ) = 0 (1.8a)

∂tGλ + (Gλ)2 ∂x Uλ = λ[g(Uλ)−Gλ], (1.8b)

2

or equivalently,

∂tWλ + Zλ∂xW

λ = λ[w(U(W λ, Zλ))−W λ] (1.9a)

∂tZλ +W λ∂xZ

λ = λ[z(U(W λ, Zλ)) − Zλ], (1.9b)

after a change of variables that will be made explicit in §2. To the astonishment ofourselves, the left-hand side of (1.9) is Serre’s version [22]) of the Born-Infeld system [5],an object that appeared about a century ago in electrodynamics and was recently revivedby Brenier [7], along with Neves and Serre [20]. This is why we also refer to the newrelaxation philosophy as the Born-Infeld relaxation for short. As for the original Born-Infeld system By construction, the Born-Infeld relaxation (1.8)–(1.9) allows us to controlthe out-of-equilibrium values of the convective velocities. Moreover, it is free from anytuning parameter and is stable under the condition

f ′(u) ∈ bw(u), z(u)e, (1.10)

where f(u) = u(1 − u)g(u) and bw(u), z(u)e denotes the convex interval of real numbersdetermined by the endpoints w(u) and z(u). This condition is still a subcharacteristicone, since it expresses that the characteristic speed f ′(u) of the original system must beenclosed by those of the relaxation system.

The core ideas of the Born-Infeld relaxation can be carried over to other flux functionsf(u), provided that we are given an appropriate notion of control variables w(u) and z(u).Two prototypes of such flux functions are:

• f(u) = −uh(u), where u ∈ R+ and h ∈ C 1(R+;R+), 0 6∈ h(R∗+), with

w(u) = −h(u) and z(u) =θ(u) +

√u2 + 4uh(u)

2; (1.11)

• f(u) = uh(u), where u ∈ R+ and h ∈ C 1(R+;R+), 0 6∈ h(R∗+), with

w(u) = h(u) and z(u) = negative root of z3 − uz + f(u). (1.12)

This paper is organized as follows. In §2, we start by presenting the Born-Infeldrelaxation for the scalar case (1.4), i.e., with flux functions of the form f(u) = u(1−u)g(u).Numerical results are shown and naturally compared to Jin-Xin’s relaxation. In §3, weset up a general abstract framework in order to attempt an extension of the Born-Infeldrelaxation to other flux functions f . Numerical simulations corresponding to the fluxfunctions f(u) = uh(u) are provided and commented on. Before concluding, we brieflyaddress some prospective extension of the Born-Infeld relaxation to systems in §4.

2 Born-Infeld relaxation for a class of scalar conservation

laws

We consider the conservation law (1.1) for f(u) = u(1−u)g(u), with g ∈ C 1([0, 1];R∗) and0 6∈ g(]0, 1[). The latter assumption implies, that g keeps a constant sign over [0, 1].Theconservation law

∂tu+ ∂xf(u) = ∂tu+ ∂x(u(1− u)g(u)) = 0, (2.1)

3

defined for u ∈ [0, 1], is then hyperbolic. As mentioned earlier, equation (2.1) can bethought of as two nonlinear convective equations

∂t(u) + ∂x(u · w(u)) = 0 or ∂t(1− u) + ∂x((1− u) · z(u)) = 0 (2.2)

with the apparent velocities

w(u) = (1− u)g(u) =f(u)

uand z(u) = −ug(u) = − f(u)

1− u. (2.3)

The differenceg(u) = w(u)− z(u) (2.4)

then appears to be the slip velocity between the phase u and the phase 1−u. By inverting(2.3), we have

u =z(u)

z(u)− w(u)and f(u) =

w(u)z(u)

z(u)− w(u)(2.5)

provided that w(u) 6= z(u), which is the case, for w(u)z(u) ≤ 0 and g(u) 6= 0.

2.1 Change of variables

As we seek to ensure a monotonicity principle for the phase velocities, it is natural tochoose them as evolution variables. Let us introduce the pair of independent variables(W,Z), meant to be the relaxation counterparts of the equilibrium values (w(u), z(u)).The formulae (2.5) suggest to define the pair (U,F ), the relaxation counterparts of theequilibrium values (u, f(u)), as

U(W,Z) =Z

Z −Wand F (W,Z) =

WZ

Z −W(2.6)

for W 6= Z. The inverse transformation of (2.6) is

W (U,F ) =F

Uand Z(U,F ) = − F

1− U. (2.7)

In imitation of (2.4), we also define

G(W,Z) = W − Z (2.8)

to be the relaxation version of the slip velocity g(u). Then,

F = U(1− U)G, W = (1− U)G, Z = −UG. (2.9)

Our first task is to clarify the ranges of the variables involved so that the transforma-tions (2.6)–(2.7) make sense and so that we can make the identification U ≡ u ∈ [0, 1].Let us define two open sets Ω and 0 in the following way:

• if g(.) < 0,

Ω = (W,Z) |W ∈ R∗−, Z ∈ R∗+, 0 = (U,F ) | U ∈ ]0, 1[, F ∈ R∗− (2.10)

• if g(.) > 0,

Ω = (W,Z) |W ∈ R∗+, Z ∈ R∗−, 0 = (U,F ) | U ∈ ]0, 1[, F ∈ R∗+ (2.11)

4

Proposition 2.1. The (W,Z)-domain Ω and the (U,F )-domain 0 are diffeomorphic toeach other through the maps (2.6)–(2.7).The direct map (W,Z) → (U,F ), defined in (2.6), can be extended to Ω \ (0, 0) by

continuity. As for the F -component, it can be extended to Ω by setting

F (0, 0) = 0. (2.12)

The inverse map (U,F )→ (W,Z), defined in (2.7), can be extended to 0∪(0, 0)∪(1, 0)by continuity along the equilibrium manifold F = f(U) by setting

(W (0, 0), Z(0, 0)) = (w(0), z(0)) and (W (1, 0), Z(1, 0)) = (w(1), z(1)). (2.13)

Proof Let (W,Z) ∈ Ω. Then, Z − W 6= 0 and the formulae (2.6) are well-defined.Furthermore,

U(1− U) = − WZ

(Z −W )2> 0, (2.14)

which implies U ∈ ]0, 1[. On the other hand, since WZ < 0, F must have the same sign asW −Z, which is also that of W . As a result, (U,F ) ∈ 0. It is easy to check that if (W,Z)belongs to the boundary ∂Ω of Ω, but does not coincide with the origin (0, 0), then (U,F )remain well-defined. At the origin, F can be extended by (2.12) because

lim(W,Z)∈Ω→(0,0)

F (W,Z) = 0. (2.15)

It is not possible to do the same for U , but this does not cause any harm to the method.Conversely, let (U,F ) ∈ 0. Then, F 6= 0 and the formulae (2.7) are well-defined.

Furthermore, W must have the same sign as F and

WZ = − F 2

U(1− U)< 0, (2.16)

which implies (W,Z) ∈ Ω. If U = 0 or 1, the values (2.7) are undefined, except forF = f(U), in which case simplifications occur and enable us to set (2.13).

The key fact on which we wish to draw attention is that F can be defined continuouslyover Ω, so that the Born-Infeld flux is always well-defined (see Theorem 2.5) and thereforethe new relaxation method can be applied even for single-phase states U = 0 or 1.

2.2 Relaxation system

Definition 2.1. The Born-Infeld relaxation system for the scalar conservation law (2.1)is defined as

∂tWλ + Zλ∂xW

λ = λ[w(U(W λ, Zλ))−W λ] (2.17a)

∂tZλ +W λ∂xZ

λ = λ[z(U(W λ, Zλ)) − Zλ], (2.17b)

for (W λ, Zλ) ∈ Ω, where the superscript λ > 0 is the relaxation coefficient.

This name is justified by the fact that when λ = 0, system (2.17) coincides with Serre’sreduced form [22] of the Born-Infeld equations. The historical Born-Infeld system [5] stemsfrom classical electrodynamics and is actually more sophisticated. We also refer the readersto the works of Brenier [7], Neves and Serre [20] for the connection between Born-Infeld’s

5

original form and Serre’s reduced form. To our knowledge, this is the first time that itappears in the design of relaxation schemes, a context totally unrelated to field theory.

It seems to be the first time too that a nonlinear combination of the relaxation equationsmust be used in order to recover the “original” equation, as indicated in the followingLemma.

Lemma 2.1. For all λ > 0, smooth solutions of (2.17) are also solution of

∂tU(W λ, Zλ) + ∂xF (W λ, Zλ) = 0. (2.18)

Proof Since smooth solutions are differentiable, we multiply (2.17a) by UW = ZG−2,(2.17b) by UZ = −WG−2 and add them together. The superscript λ associated withvarious quantities is omitted for convenience. It is readily checked that the left-hand sideof the sum is equal to ∂tU + ∂xF . As for the right-hand side, it also vanishes thanks to

Z(w(U)−W )−W (z(U)− Z) = Zw(U)−Wz(U) = g(U)[(1− U)Z + UW ] (2.19)

and because of (2.6).For the moment, the property (2.18) holds true only for smooth solutions. In order to

extend it to weak solutions, we simply need to have a linear degeneracy of the characteristicfields of (2.17).

Proposition 2.2. The eigenvalues (Zλ,W λ) of (2.17), both linearly degenerate, are re-spectively associated with the strong Riemann invariants W λ and Zλ.

Proof See [22].This expected degeneracy is in accordance with the orthodoxy in the area of relaxation

systems [10, 13, 14]. The fact that all characteristic fields are degenerate by constructionenables us to state a more complete equivalent result.

Theorem 2.1. For all λ > 0, the Born-Infeld relaxation system (2.17) is equivalent tothe system

∂tUλ + ∂x(U

λ(1− Uλ)Gλ) = 0 (2.20a)

∂tGλ + (Gλ)2 ∂x Uλ = λ[g(Uλ)−Gλ]. (2.20b)

in the sense of weak solutions, where (Uλ, Gλ) ∈ [0, 1] × 0 is considered as a pair ofindependent variables.

Proof Thanks to linear degeneracy, we can restrict ourselves to smooth solutions. Equa-tion (2.20a) is none other than (2.18) of Lemma 2.1. As for (2.20b), it can obtained bysubtracting (2.17b) to (2.17a) and by arguing that

Z∂xW −W∂xZ = (ZWU |G −WZU |G)∂xU + (ZWG|U −WZG|U )∂xG (2.21)

in the left-hand side (again, we drop out the superscript λ in the proof).In preparation for the generalization of the Born-Infeld to an arbitrary flux function

(see §3), we point out the following property.

Corollary 2.1. The Born-Infeld relaxation system satisfies the additional equation

∂t((1− 2Uλ)Gλ)− ∂x(Uλ(1− Uλ)(Gλ)2) = λ(1− 2Uλ)[g(Uλ)−Gλ], (2.22)

the left-hand side of which is in conservation form.

6

Proof Let I = (1−2U)G and K = −U(1−U)G2. In terms of (W,Z), we have I = W +Zand K = WZ. Adding (2.17a) and (2.17b) together, we obtain the desired equality

∂tI + Z∂xW +W∂xZ = λ(1− U)− U[g(U)−G]. (2.23)

because Z∂xW +W∂xZ = ∂x(WZ) = ∂xK.

2.3 Chapman-Enskog analysis and subcharacteristic condition

We now show the extent to which the Born-Infeld relaxation system (2.17), or its equivalentform (2.20), is actually a “good” approximation to the original equation (2.1).

Theorem 2.2. At the first order approximation in λ−1, the component Uλ of the solutionto relaxation system (2.17) or (2.20) satisfies the equivalent equation

∂tUλ + ∂xf(U

λ) = λ−1∂x−[f ′(Uλ)− w(Uλ)][f ′(Uλ)− z(Uλ)]∂xUλ. (2.24)

Proof By omitting the superscript λ for clarity, we insert the standard Chapman-Enskogexpansion

G = g(U) + λ−1g1 +O(λ−2) (2.25)

into (2.20b). Keeping the leading terms only, we obtain

−g1 = ∂tg(U) + g2(U)∂xU. (2.26)

Arguing that ∂tU = −∂xf(U) is valid at the zeroth-order, we transform the right-handside into

−g1 = [g2(U)− g′(U)f ′(U)]∂xU, (2.27)

Now, plugging (2.25) into the first equation of (2.20), moving first-order terms to theright-hand side and using (2.27), we end up with

∂tU + ∂xf(U) = λ−1∂x−U(1− U)g1 (2.28a)

= λ−1∂xU(1− U)[g2(U)− g′(U)f ′(U)]∂xU. (2.28b)

Let us rearrange the kernel

D(U) = U(1− U)[g2(U)− g′(U)f ′(U)]. (2.29)

On one hand,U(1− U)g2(U) = −w(U)z(U). (2.30)

On the other hand, multiplying the equality f ′ = (1−2U)g+U(1−U)g′ by f ′ and isolatingthe product g′f ′, we have

U(1− U)g′(U)f ′(U) = [f ′(U)]2 − (1− 2U)g(U)f ′(U) (2.31a)

= [f ′(U)]2 − (w(U) + z(U))f ′(U). (2.31b)

Inserting (2.30) and (2.31) into (2.29) yields

D(U) = −w(U)z(U) + (w(U) + z(U))f ′(U)− [f ′(U)]2 (2.32a)

= −[f ′(U)− w(U)][f ′(U)− z(U)], (2.32b)

which completes the proof.

7

From now on, for (a, b) ∈ R2, we use the notation

ba, be = ra + (1− r)b, r ∈ [0, 1]. (2.33)

Theorem 2.3. A sufficient condition for the relaxation system (2.20) to be a dissipativeapproximation of the original equation (2.1) is that the subcharacteristic condition

f ′(u) ∈ bw(u), z(u)e, (2.34)

holds for all u in the range of the problem at hand.

Proof At the continuous level, the dissipative approximation means that the kernel D(u)in the equivalent equation (2.24) be a diffusive kernel, i.e., D(u) ≥ 0. This amounts to

[f ′(u)− w(u)][f ′(u)− z(u)] ≤ 0, (2.35)

hence (2.34).Rigorously speaking, we should have required D(u) > 0 at the PDE level. At the

discrete level, however, we will see in §2.5 that D(u) ≥ 0 is sufficient to guarantee themonotonicity of the numerical flux.

Unsurprisingly, the Whitham condition demands that the eigenvalue of the originalequation be enclosed by the two eigenvalues of the relaxation system [24]. The novelfeature is that, in the present method, condition (2.34) may appear to be quite severe:only a class of flux functions f are eligible for the Born-Infeld relaxation. Let us investigatefurther into this class of flux functions. The boundedness of g implies f(0) = f(1) = 0,and we can write

w(u) =f(u)− f(0)

u− 0and z(u) =

f(1)− f(u)

1− u. (2.36)

From (2.36), we infer the geometrical interpretation depicted in Fig. 1. On the curve Cgiven by the Cartesian equation f = f(u), let M = (u, f) be a running point, A = (0, 0)and B = (1, 0). Then, w is the slope of AM, while z is that of BM. Condition (2.34)amounts to saying that the tangent to C at M lies outside the angular sector ∠(MA,MB).

0 1 u

M

f

A Bw(u) z(u)

f(u)

u

Figure 1: Geometrical interpretation of the Whitham condition.

For u = 0 or 1, the subcharacteristic condition (2.34) holds with equality, since f ′(0) =w(0) and f ′(1) = z(1). We are going to characterize the subcharacteristic condition for alocal point u ∈ ]0, 1[ in two different ways. The first way is a condition in terms of w andz alone.

8

Lemma 2.2. Let u ∈ ]0, 1[. The subcharacteristic condition (2.34) is satisfied at u if andonly if the functions w and z are decreasing at u when f(u) > 0, increasing at u whenf(u) < 0, and critical when f(u) = 0.

Proof A straightforward computation shows that

w′ =uf ′ − f

u2and z′ = −(1− u)f ′ + f

(1− u)2. (2.37)

Suppose f(u) > 0. After (2.3), we have z(u) < 0 < w(u), because u ∈ ]0, 1[. Thus,condition (2.34) boils down to

z(u) = − f(u)

1− u≤ f ′(u) ≤ f(u)

u= w(u). (2.38)

This double inequality is equivalent to

−[(1− u)f ′(u) + f(u)] ≤ 0 and uf ′(u)− f(u) ≤ 0, (2.39)

whence z′(u) ≤ 0 and w′(u) ≤ 0. We proceed similarly in the case f(u) < 0 for whichw(u) < 0 < z(u). If f(u) = 0, necessarily w(u) = z(u) = 0 because u 6∈ 0, 1. By (2.34),we have f ′(u) = 0, from which we deduce that w′(u) = z′(u) = 0 via (2.37). To establishthe converse, we follow the same steps in reverse order.

It follows that a convex or concave function f with f(0) = f(1) = 0 is eligible functionsall over [0, 1]. There are eligible flux functions that are neither convexe nor concave. Asan example, consider

f(u) = u(1− u)

[1 +

sin(2πnu)

2πn

]for n ∈ N. (2.40)

It can be shown that w and z are strictly decreasing functions of u, but that f ′′ does nothave a constant sign over [0, 1] for n ≥ 2.

In Lemma 2.2, the function g is hardly visible. We now give it a prominent role in thesecond test of the subcharacteristic condition.

Lemma 2.3. Let u ∈ ]0, 1[. The subcharacteristic condition (2.34) is satisfied at u if andonly if

g′(u) ∈⌊−g(u)

u,g(u)

1− u

⌉. (2.41)

Proof In terms of g, we have

w′ = −g + (1− u)g′ and z′ = −g − ug′. (2.42)

We apply Lemma 2.2. If g(u) > 0, then f(u) > 0 and the decreasing conditions w′(u) ≤ 0and z′(u) ≤ 0 imply

g(u) ≥ max(1− u)g′(u),−ug′(u). (2.43)

Likewise, if g(u) < 0, we arrive at

g(u) ≤ min(1− u)g′(u),−ug′(u). (2.44)

From (2.43) and (2.44), we infer (2.41) for g(u) 6= 0. If g(u) = 0, by Lemma 2.2, we musthave w′(u) = z′(u) = 0, and using (2.42), we have g′(u) = 0. The converse follows thesame lines.

9

2.4 Riemann problem

As was explained in [2, 3], we need to solve the Riemann problem corresponding to thehomogeneous relaxation system in order to know the value of the numerical flux.

A generic point in Ω is designated by v = (W,Z). The symbols v− and v+ stand forthe negative and positive parts of any real number v ∈ R. We also denote by

. thecharacteristic function.

W W

ZZ

vL

v vR

vLv

vR

1a 1b

Figure 2: Solution to the Riemann problem for the homogeneous Born-Infeld system.

Theorem 2.4. Let vL = (WL, ZL) ∈ Ω and vR = (WR, ZR) ∈ Ω be two arbitrarily givenstates. Define the two speeds

sL = min(WL, ZL) ≤ 0 and sR = max(WR, ZR) ≥ 0. (2.45)

When λ = 0, the solution to (2.17) for (t, x) ∈ R∗+ × R with the initial data

v(t = 0;x) = vL

x<0 + vR

x>0 (2.46)

is given byv(t, x) = vL

x<tsL + v

tsL<x<tsR + vR

x>tsR (2.47)

withv = (W , Z) = (W−

L +W+R , Z

−L + Z+R ). (2.48)

The intermediate state v satisfies the maximum principle

W ∈ bWL,WRe, Z ∈ bZL, ZRe. (2.49)

Proof In the (W,Z)-plane, W -curves are vertical lines, while Z-curves are horizontallines. Riemann problems can thus be solved “by hand,” depending on the locations of thetwo initial states, as depicted in Fig. 2. The claims about the maximum principle are easyto check.

The benefit of the maximum principle (2.49)) is obvious: should we need the apparentvelocities at some intermediate state, they are always well-defined and bounded. We nowconcentrate on the flux that is to be plugged in the numerical scheme. In this respect, itis well known [11,16] that in the context of Godunov-like methods, we have to extract theinterface values

W ∗ = W (x/t = 0+), Z∗ = Z(x/t = 0+) (2.50)

from the full Riemann solution. In the special cases sL = 0 or sR = 0, although W ∗, Z∗

may take different values at x/t = 0+ and at x/t = 0−, the relaxed flux

F ∗ = F (W ∗, Z∗) =W ∗Z∗

Z∗ −W ∗(2.51)

10

is continuous across x/t = 0, as detailed by the following procedure.

Theorem 2.5. Let vL = (WL, ZL) ∈ Ω and vR = (WR, ZR) ∈ Ω be two arbitrarily given

states. Then, the intermediate state (W , Z) = (W−L +W+

R , Z−L +Z+R ), defined in Theorem

2.4, necessarily belongs to Ω, and

F ∗ = F (W , Z). (2.52)

The numerical flux satisfies the sign property

F ∗ · F (WL, ZL) ≥ 0, F ∗ · F (WR, ZR) ≥ 0. (2.53)

Proof The proof is done with the help of Theorem 2.4, the extension (2.12) and byscrutinizing all the situations that can occur. We leave it to the reader.

The sign property (2.53) is another asset of the Born-Infeld relaxation: in some appli-cations, it might be essential that the sign of the numerical flux does not contradict thephysical fluxes of the left and right states. To emphasize that F ∗ depends on the left statevL and the right state vR, we write

F ∗ = F (vL,vR). (2.54)

Definition 2.2. Let uL ∈ [0, 1] and uR ∈ [0, 1] be two arbitrary states. Consider v(uL) =(w(uL), z(uL)) and v(uR) = (w(uR), z(uR)). Then,

HBI(uL, uR) = F (v(uL),v(uR)) (2.55)

is said to be the Born-Infeld numerical flux associated to (uL, uR).

Thanks to Theorem 2.5, the Born-Infeld numerical flux is well-defined for all (uL, uR) ∈[0, 1]2. It is a trivial matter to check that this is a consistant flux, that is, HBI(u, u) = f(u)for all u ∈ [0, 1].

2.5 Stability at the discrete level

In §2.3, the Whitham condition was derived and discussed at the continuous level from theChapman-Enskog analysis. We are going to evidence its connection with the monotonicityof the numerical flux at the discrete level. Consider the first-order explicit scheme

un+1i = uni −∆t

∆x[H(uni , u

ni+1)−H(uni−1, u

ni )]. (2.56)

We recall [11] that the numerical flux H is said to be monotone if it is increasing withrespect to its first argument and decreasing with respect to its second argument. If wespecify H(uL, uR) for the arguments, then monotonicity is expressed as

HuL≥ 0 and HuR

≤ 0. (2.57)

provided that the partial derivatives exist. We wish to investigate about the monotonicityof the Born-Infeld numerical flux HBI(uL, uR) defined in (2.55).

Theorem 2.6. The Born-Infeld numerical flux HBI(uL, uR) is monotonous for all (uL, uR) ∈]0, 1[2 if and only if the subcharacteristic condition (2.34) is satisfied for all u ∈ ]0, 1[.

11

Proof We recall the assumption 0 6∈ g(]0, 1[), which means that the sign of g does notchange over [0, 1]. Let us assume g < 0. Then

vL = (WL, ZL) = (w(uL), z(uL)) ∈ R∗− × R∗+ (2.58a)

vR = (WR, ZR) = (w(uR), z(uR)) ∈ R∗− × R∗+. (2.58b)

We are in Case 1a of Fig. 2, which yields v∗ = v = (WL, ZR). It follows that

HBI(uL, uR) =w(uL)z(uR)

z(uR)− w(uL)(2.59)

Thus, the Born-Infeld numerical flux is differentiable and one has

HBIuL

= w′(uL)

[z(uR)

z(uR)− w(uL)

]2(2.60a)

HBIuR

= −z′(uR)[

w(uL)

z(uR)− w(uL)

]2. (2.60b)

The monotonicity conditions (2.57) take place if and only if w′(uL) ≥ 0 and z′(uR) ≥ 0for all (uL, uR) ∈ ]0, 1[2. This is equivalent to saying that w′(u) ≥ 0 and z′(u) ≥ 0 forall u ∈ ]0, 1[. Since f(u) < 0, this is still equivalent to the statement that the Whithamcondition (2.34) is satisfied at all u ∈ ]0, 1[ by virtue of Lemma 2.2. The case g > 0 worksalike.

As a matter of fact, it is fair to acknowledge that for the Jin-Xin relaxation scheme,the monotonicity of the numerical flux

HJX(uL, uR) =f(uL) + f(uR)

2− a

uR − uL2

for all (uL, uR) (2.61)

is also equivalent to the non-strict Whitham condition

a ≥ |f ′(u)| for all u. (2.62)

2.6 Numerical results

Comparer Born-Infeld a Jin-Xin 1 (a global) et Jin-Xin 2 (a local) pour

f(u) = u(1− u)(1 + u) (2.63)

Choc, detente. Courbes de convergence. Beau programme :)

3 Born-Infeld relaxation for a general scalar conservation

law

In this section, we provide an abstract setting that enables us to extend the Born-Infeldrelaxation to a scalar conservation law

∂tu+ ∂xf(u) = 0, u ∈ D, (3.1)

where the flux f(u), defined over a domain D ⊂ R, is any function subject to somecompatibility conditions. The first of these conditions is that 0 6∈ f(D), i.e.,f keeps thesame sign in the interior domain D.

12

In the case f(u) = u(1−u)g(u) of §2, the control variables w(u) and z(u) were dictatedto us by the physics of the problem via (2.3). Now, given an arbitrary function f , andwithout any further knowledge about the physics, we need to define the control velocitiesourselves. Ideally, the relaxation strategy would be the same as in §2, that is:

1. Define the equilibrium values of the control variables (w(u), z(u)) from (u, f(u));

2. Extend this definition to compute the relaxation variables (W,Z) from (U,F );

3. Introduce a Born-Infeld system similar to (2.17) and declare it as the relaxationmodel for the original conservation law.

In practice, although the essence of the strategy is the same, we are compelled to swapthe order of the steps.

3.1 Admissible changes of variables

Indeed, the success of a Born-Infeld relaxation strategy relies on whether or not we areable to recover the conservative equation

∂tU(W λ, Zλ) + ∂xF (W λ, Zλ) = 0 (3.2)

by a (possibly nonlinear) combination of the equations in the Born-Infeld relaxation sys-tem for (W λ, Zλ), as was the case for (2.18). In other words, (U(W,Z), F (W,Z)) mustnecessarily be an entropy-entropy flux pair1 for the homogeneous Born-Infeld system

∂tW + Z∂xW = 0 (3.3a)

∂tZ +W∂xZ = 0. (3.3b)

In this respect, we recall the following technical result.

Lemma 3.1. A necessary and sufficient condition for (U,F ) to be an entropy-flux pairfor the homogeneous Born-Infeld system is that there exist two arbitrary smooth functionsA(W ) and B(Z) such that

U(W,Z) =A(W ) +B(Z)

Z −Wand F (W,Z) =

ZA(W ) +WB(Z)

Z −W. (3.4)

Proof See [22].For convenience, such a pair (U,F ) is said to be a Goursat pair, due to the fact that

U must obey the Goursat equation [22]

UW − UZ + (Z −W )UWZ = 0. (3.5)

To quote a few examples, the pair (U,F ) introduced in (2.6) is a Goursat pair with(A,B) = (0, Z). The pair (I,K) = (W + Z,WZ), which shortly appeared in the proof ofCorollary 2.1, is also a Goursat pair with (A,B) = (−W 2, Z2).

The two functions A(W ) and B(Z) in (3.4) are at our disposal to pick the correspondingGoursat pair. However, to set the stage for the Born-Infeld relaxation, we have to consider

1Note that despite the label “entropy,” no convexity property is required.

13

a Goursat pair with a few additional properties. To begin with, since f(u) is assumed tokeep the same strict sign for u ∈ D, we consider the open sets

Ω = (W,Z) |W ∈ R∗−, Z ∈ R∗+, 0 = (U,F ) | U ∈ D, F ∈ R∗− (3.6)

orΩ = (W,Z) |W ∈ R∗+, Z ∈ R∗−, 0 = (U,F ) | U ∈ D, F ∈ R∗+, (3.7)

depending on f < 0 or f > 0.

Definition 3.1. The Goursat pair (U,F ) is said to be an admissible first pair if

1. It is a diffeomorphism between the (W,Z)-domain Ω and the (U,F )-domain 0. Thedirect map (W,Z) ∈ Ω→ (U,F ) ∈ 0 are extendable by continuity to Ω \ (0, 0), whilethe F -component is extendable to Ω;

2. The inverse map (U,F ) ∈ 0 → (W,Z) ∈ Ω is extendable to the equilibrium points ofthe boundary ∂0. In other words, the equilibrium control velocities

w(u) = W (u, f(u)) and z(u) = Z(u, f(u)) (3.8)

are well-defined and locally bounded for u ∈ D;

3. Both components (W,Z) are strictly monotone with respect to F , that is,

WF (U,F ) 6= 0 and ZF (U,F ) 6= 0 (3.9)

for all (U,F ) ∈ 0.

3.2 Relaxation system

Our departure point is of course a Goursat admissible first pair (U,F ), representing therelaxation values of (u, f(u)), from which we deduce (W,Z) by inverse mapping.

Definition 3.2. For a given Goursat admissible first pair, the generalized Born-Infeldrelaxation of equation (3.1) is the system

∂tWλ + Zλ∂xW

λ = λWF [f(U(W λ, Zλ))− F (W λ, Zλ)] (3.10a)

∂tZλ +W λ∂xZ

λ = λ ZF [f(U(W λ, Zλ))− F (W λ, Zλ)], (3.10b)

for (W λ, Zλ) ∈ Ω, where λ > 0 is the relaxation coefficient.

In (3.10), the quantities (WF , ZF ) have to be understood as functions of the velocity-like variables (W λ, Zλ) via

WF = WF (U(W λ, Zλ), F (W λ, Zλ)), ZF = ZF (U(W λ, Zλ), F (W λ, Zλ)). (3.11)

The monotonicity condition (3.9) ensures that that the right-hand sides of the relaxationsystem (3.10) does not vanish for λ → +∞. Note that the right-hand sides of (3.10) arenot written in the same form as those of (2.17). However, they do have the same valueswhen f = u(1− u)g and for the change of variables (2.6)–(2.7).

As in Proposition 2.2, the characteristic fields of the Born-Infeld system (3.10) arelinearly degenerate, which enables us to extend results valid for smooth solutions to weaksolutions.

14

We wish to express the relaxation system (3.10) in another set of variables (U,J ), ina fashion similar to (2.20), since this will make the Chapman-Enskog analysis easier. Tothis end, we consider a second Goursat pair (I,K), which is intended to play the role of((1− 2U)G,−U(1− U)G2) in (2.22) of Corollary 2.1. According to Lemma 3.1, this pairis given by

I(W,Z) =Q(W ) +R(Z)

Z −W, and K(W,Z) =

ZQ(W ) +WR(Z)

Z −W(3.12)

for (W,Z) ∈ Ω, where Q(W ) and R(Z) are two functions. We chain the second pair withthe inverse of the first pair in order to obtain

I (U,F ) = I(W (U,F ), Z(U,F )), and K (U,F ) = K(W (U,F ), Z(U,F )) (3.13)

as functions of the variables (U,F ) ∈ 0. Since our goal is to switch to the set of variables(U,I ), we have to require some technical properties in order to legitimate this plan.

Definition 3.3. The Goursat pair (I,K) is said to be an admissible second pair if

1. (U,I ), where I is defined by (3.13), can replace (U,F ) as independent variables.In other words, for all (U,F ) ∈ 0 under consideration, we must have

IF (U,F ) 6= 0; (3.14)

2. The equilibrium values I (u, f(u)) and K (u, f(u)) remain bounded for u ∈ D.

Note that we do not postulate invertibility for the second Goursat pair.We are now in a position to state the following equivalence result, in which the condition

(3.14) guarantees that the right-hand side does not vanish for λ→ +∞.

Theorem 3.1. The generalized Born-Infeld relaxation system (3.10) is equivalent to theconservative form

∂tUλ + ∂xF

λ = 0 (3.15a)

∂tI (Uλ, F λ) + ∂xK (Uλ, F λ) = λIF (Uλ, F λ)[f(Uλ)− F λ] (3.15b)

for all (I ,K ) defined from an admissible second pair (I,K).

Proof To prove (3.15a), multiply (3.10a) by UW , (3.10b) by UZ , add them together andmake use of (3.17a). The right-hand side vanishes because UWWF + UZZF = UF = 0.Proceed similarly for (3.15b) and invoke the chain rule IWWF + IZZF = IF .

3.3 Chapman-Enskog analysis and subcharacteristic condition

The good news is that Theorem 2.2 remains still valid, although its proof involves heaviercalculations.

Theorem 3.2. At the first order approximation in λ−1, the component Uλ of the solutionto relaxation system (3.15) satisfies the equivalent equation

∂tUλ + ∂xf(U

λ) = λ−1∂x−[f ′(Uλ)− w(Uλ)][f ′(Uλ)− z(Uλ)]∂xUλ, (3.16)

where the equilibrium velocities w(.) and z(.) have been defined in (3.8).

15

Before entering the proof, we need to state a few preliminary equalities.

Lemma 3.2. The two admissible Goursat pairs (U,F ) and (I,K) satisfy

FW = ZUW , FZ = WUZ (3.17a)

KW = ZIW , KZ = WIZ . (3.17b)

As far as the inverse mapping (U,F )→ (W,Z) is concerned, we have

WU = −WWF and ZU = −ZZF . (3.18)

Proof The equalities (3.17) are none other than the compatibility conditions betweenan entropy and its entropy-flux for the homogeneous Born-Infeld system [22]. As for therelations (3.18), they can be obtained by arguing that

(WU WF

ZU ZF

)=

(UW UZFW FZ

)−1(3.19)

and by using (3.17a) for a direct checking.Proof of Theorem 3.2 We insert the standard Chapman-Enskog expansion

F = f(U) + λ−1f1 +O(λ−2) (3.20)

into (3.15b) and drop the superscript λ for clarity. Keeping the leading terms only, whileremembering that all functions of (U,F ) must be evaluated at (U, f(U)), yields

−IF (U, f(U))f1 = ∂tI (U, f(U)) + ∂xK (U, f(U)) (3.21a)

= I ′(U, f(U))∂tU + K ′(U, f(U))∂xU (3.21b)

= [K ′(U, f(U))− f ′(U)I ′(U, f(U))]∂xU (3.21c)

where (.)′ designates the total derivative

(.)′ = ∂U (.) + f ′(U)∂F (.). (3.22)

Inserting (3.22) into (3.21c), and with the help of (3.17b), we obtain

K ′ − f ′I ′ = [KWW ′ +KZZ′]− f ′[IWW ′ + IZZ

′] (3.23a)

= [ZIWW ′ +WI ′Z ]− f ′[IWW ′ + IZZ′] (3.23b)

= (z(U)− f ′(U))W ′IW + (w(U)− f ′(U))Z ′IZ . (3.23c)

Applying (3.22) again to transform (W ′, Z ′) and invoking (3.18), we end up with

−IF (U, f(U))f1 = −IF (f′(U)− w(U))(f ′(U)− z(U))∂xU. (3.24)

After simplification by IF 6= 0, plug f1 into (3.15a) and deduce the desired result.

Theorem 3.3. A sufficient condition for the relaxation system (3.15) to be a dissipativeapproximation of the original equation (3.1) is that the subcharacteristic condition

f ′(u) ∈ bw(u), z(u)e (3.25)

holds for all u in the range of the problem at hand.

Proof Identical to that of Theorem 2.3.

16

3.4 Numerical flux and monotonicity

Like the Born-Infeld relaxation of §2, the Riemann problem associated with (3.10) can besolved for the initial left and right data

vL = (w(uL), z(uL)), vR = (w(uR), z(uR)). (3.26)

The intermediate states obey a maximum principle componentwise, as in Theorem 2.4.As in Theorem 2.5, the flux F ∗ at x = 0 always makes sense, and satisfies a sign property.This allows us to consider the numerical flux

HBI(uL, uR) = F (vL,vR) = F ∗ (3.27)

and to ask question about the equivalence between the monotonicity of HBI and thesubcharacteristic condition (3.25), as was the case in Theorem 2.6.

Theorem 3.4. A necessary condition for the Born-Infeld numerical flux HBI(uL, uR) tobe monotone for all (uL, uR) ∈ D2 is that the subcharacteristic condition (3.25) is satisfiedfor all u ∈ D.This condition is also sufficient if each component of (UW (W,Z), UZ(W,Z)) keeps a

constant sign for all (W,Z) ∈ Ω.

Proof Without loss of generality, suppose we are in Case 1a of Fig. 2, that is, Ω =R∗− × R∗+. Then the intermediate state is v∗ = v = (w(uL), z(uR)), and

HBI(uL, uR) = F (w(uL), z(uR)). (3.28)

By differentiation,

HBIuL

= FW (w(uL), z(uR))w′(uL) = z(uL)UW (w(uL), z(uR))w

′(uL), (3.29)

where we have used (3.17a). Let

b(uL, uR) = UW (w(uL), z(uR))w′(uL). (3.30)

Then,HBIuL

= b(uL, uR)z(uL). (3.31)

Similarly, we haveHBIuR

= a(uL, uR)w(uR). (3.32)

witha(uL, uR) = UZ(w(uL), z(uR))z

′(uR). (3.33)

Therefore, the monotonicity conditions HBIuL≥ 0 and HBI

uR≤ 0 are tantamount to

a(uL, uR) ≥ 0, b(uL, uR) ≥ 0. (3.34)

Taking the derivatives of

u = U(w(u), z(u)), f(u) = F (w(u), z(u)) (3.35)

with respect to u, we obtain

1 = UW (w(u), z(u))w′(u)+ UZ(w(u), z(u))z′(u) (3.36a)

f ′(u) = FW (w(u), z(u))w′(u)+ FZ(w(u), z(u))z′(u) (3.36b)

= z(u)UW (w(u), z(u))w′(u)+ w(u)UZ(w(u), z(u))z′(u), (3.36c)

17

again because of (3.17a). With the notations (3.30), (3.33), this system can be cast underthe form

1 = a(u, u)+ b(u, u) (3.37a)

f ′(u) = w(u)a(u, u)+ z(u)b(u, u). (3.37b)

The Whitham condition (3.25) says that f ′(u) must be a convex combination of w(u) andz(u). In view of (3.37), this is equivalent to

a(u, u) ≥ 0, b(u, u) ≥ 0. (3.38)

It is plain that

(3.34) for all (uL, uR) ∈ D2 =⇒ (3.38) for all u ∈ D.

Hence, monotonicity of the numerical flux implies the subcharacteristic condition. Further-more, from the definitions (3.30), (3.33), we clearly see that if (UW , UZ) keeps a constantsign componentwise, the monotonicity condition (3.34) splits into two separated and de-coupled conditions, one on z′(uL), the other on w′(uR). The subcharacteristic condition(3.38) also splits into two conditions, one on z ′(u), the other on w′(u). In this case, wehave the converse

(3.38) for all u ∈ D =⇒ (3.34) for all (uL, uR) ∈ D2,

which completes the proof.

3.5 Concrete examples

In order to better grasp the Born-Infeld relaxation in the general case, let us look atsome prominent examples for which we have closed-form formulae. For a given choice ofthe functions A and B in the Goursat transform (3.4), we determine the family of fluxfunctions f for which the Born-Infeld relaxation is feasible.

For A(W ) = 0 and B(Z) = Z, we recover the Goursat pair (2.6), namely

U(W,Z) =Z

Z −Wand F (W,Z) =

WZ

Z −W, (3.39)

the inverse of which is (2.7), i.e.,

W (U,F ) =F

Uand Z(U,F ) = − F

1− U. (3.40)

Obviously, WZ ≤ 0 if and only if U(1−U) ≥ 0, which reduces the u-domain to D = [0, 1].From the requirements that (i) f must keep a constant strict sign over D and that (ii) theequilibrium values

w(u) =f(u)

uand z(u) = − f(u)

1− u(3.41)

must remain bounded for u ∈ D, it follows that f(u) = u(1 − u)g(u), with g ∈ C 1(D;R)and 0 6∈ g(D). According to Lemma 2.3, the subcharacteristic condition is satisfied if andonly if g is subject to (2.41).

For A(W ) = 0 and B(Z) = Z2, we have

U(W,Z) =Z2

Z −Wand F (W,Z) =

WZ2

Z −W, (3.42)

18

from which we infer that

W (U,F ) =F

Uand Z(U,F ) = root of Z2 − UZ + F. (3.43)

To make sure that a real root Z(U,F ) exists and that WZ ≤ 0, we restrict ourselves tothe situation where U ≥ 0 and F ≤ 0, and we select

Z(U,F ) =U +

√U2 − 4F

2. (3.44)

Hence, the u-domain is D = R+ and

(W,Z) ∈ Ω =: R∗− × R∗+ and (U,F ) ∈ 0 =: D+ × R∗+. (3.45)

Because f must keep a constant sign over D and that the equilibrium value w(u) = f(u)/umust remain locally bounded for u ∈ D, we necessarily have f(u) = −uh(u), whereh ∈ C 1(D;R+) and 0 6∈ h(D).

Theorem 3.5. The scalar conservation law (3.1), defined for u ∈ D = R+ with the fluxfunction

f(u) = −uh(u), h ∈ C 1(D;R+), 0 6∈ h(D), (3.46)

can be approximated by the Born-Infeld relaxation associated to the Goursat pair (3.42),in order to ensure a maximum principle on the control velocities

w(u) = −h(u) and z(u) =u+

√u2 + 4uh(u)

2. (3.47)

This relaxation is stable if and only if

h′(u) ∈[−h(u)

u− 1 +

√1 + 4h(u)/u

2, 0

]. (3.48)

for all u ∈ D. In such a case, the Born-Infeld numerical scheme is also monotone.

Proof To demonstrate the feasibility of the Born-Infeld relaxation, we must prove that(U,F ) given by (3.42) is an admissible first pair in the sense of Definition 3.1. It is readilychecked that the open sets Ω and 0, defined in 3.45, are diffeomorphic to each otherthrough the direct map (3.42) and the inverse map (3.43)–(3.44). The partial derivatives

WF =1

Uand ZF =

1√U2 − 4F

(3.49)

do not vanish for (U,F ) ∈ 0. Inserting (U,F ) = (u, f(u)) into (3.43)–(3.44), we end upwith the equilibrium values (3.47). These values are locally bounded for finite values ofu ∈ D.

The claim (3.48) regarding the subcharacteristic condition

−h(u) = w(u) ≤ f ′(u) = −h(u)− uh′(u) ≤ z(u) (3.50)

follows from straightforward calculations. As for the monotonicity of the numerical flux,it is a consequence of Theorem 3.4 and of the observation that

UW =Z2

(Z −W )2and UZ =

Z2 − 2ZW

(Z −W )2(3.51)

are both strictly positive for (W,Z) ∈ Ω.

19

An example of functions h satisfying (3.48) is

h(u) =1

(1 + u)α, 0 ≤ α ≤ 1. (3.52)

For A(W ) = 0 and B(Z) = Z3, we have

U(W,Z) =Z3

Z −Wand F (W,Z) =

WZ3

Z −W, (3.53)

from which we infer that

W (U,F ) =F

Uand Z(U,F ) = root of Z3 − UZ + F = 0. (3.54)

To make sure that a negative root exists and that WZ ≤ 0, we restrict ourselves to thesituation where U ≥ 0 and F ≥ 0. From (3.53), it is plain that

W (U,F ) =F

U≥ 0. (3.55)

This strongly suggests to work with the u-domain is D = R+ and to see to it that

(W,Z) ∈ Ω =: R∗+ × R∗− and (U,F ) ∈ 0 =: D+ × R∗−. (3.56)

The existence of a unique Z(U,F ) ≤ 0 is clarified below.

Lemma 3.3. For all (U,F ) ∈ 0, there exists a unique pair (W,Z) ∈ Ω such that theequalities (3.53) are satisfied.

Proof Let (U,F ) ∈ 0. We immediately have W = F/U ∈ R∗+. Instead of focusing onZ3 − UZ + F = 0, we remark that, as a consequence of (3.53),

F 2

U3=W 2(Z −W )

Z3=

(−WZ

)2(1− W

Z

), (3.57)

in which ξ = −W/Z is a positive number. Now, the polynomial ξ → ξ2(1 + ξ) is strictlyincreasing for ξ ≥ 0, and takes values from 0 to +∞. Since F 2/U3 is positive, there existsa unique ξ(F 2/U3) that solves ξ2(1 + ξ) = F 2/U3. Once ξ(F 2/U3) is known, we obtain

Z(U,F ) = − W

ξ(F 2/U3)= − F

Uξ(F 2/U3). (3.58)

This completes the proof.Similarly to the previous example, requiring that f keeps a constant sign over D

and that the equilibrium value w(u) = f(u)/u remains locally bounded for u ≥ 0 yieldsf(u) = uh(u), where h ∈ C 1(D;R+) and 0 6∈ h(D).

Theorem 3.6. The scalar conservation law (3.1), defined for u ∈ D = R+ with the fluxfunction

f(u) = uh(u), h ∈ C 1(D;R+), 0 6∈ h(D), (3.59)

can be approximated by the Born-Infeld relaxation associated to the Goursat pair (3.53),in order to ensure a maximum principle on the control velocities

w(u) = h(u) and z(u) = − h(u)

ξ([h(u)]2/u), (3.60)

20

where ξ(y) is the unique positive root of the equation ξ2(1 + ξ) = y. This relaxation isstable if and only if

h′(u) ∈[−h(u)

u

(1 +

1

ξ([h(u)]2/u)

), 0

]. (3.61)

for all u ∈ D. In such a case, the Born-Infeld numerical scheme is also monotone.

Proof Again, we must prove that (U,F ) given by (3.53) is an admissible first pair in thesense of Definition 3.1. It is readily checked that the open sets Ω and 0, defined in 3.56,are diffeomorphic to each other through the direct map (3.53) and the inverse map (3.55),(3.58). On one hand, WF = 1

U6= 0. On the other hand, by differentiating Z3−UZ+F = 0

with respect to F , we have(U − 3Z2)ZF = 1. (3.62)

This equality implies ZF 6= 0 for finite values of U and Z.Inserting (U,F ) = (u, f(u)) into (3.55), (3.58), we end up with the equilibrium values

(3.60). These values are clearly locally bounded for u > 0. When u → 0, we have[h(u)]2/u→ +∞. Therefore ξ([h(u)]2/u)→ +∞ and finally z(u)→ 0. In summary, z(u)and w(u) = h(u) are well-defined and bounded for u = 0.

The claim (3.61) regarding the subcharacteristic condition

z(u) ≤ f ′(u) = h(u) + uh′(u) ≤ w(u) = h(u) (3.63)

follows from straightforward calculations. As for the monotonicity of the numerical flux,it is a consequence of Theorem 3.4 and of the observation that

UW =Z3

(Z −W )2and UZ =

2Z3 − 3Z2W

(Z −W )2(3.64)

are both strictly negative for (W,Z) ∈ Ω.An example of functions h satisfying (3.61) is

h(u) =1

(1 + u)α, 0 ≤ α ≤ 1. (3.65)

3.6 Numerical results

Comparer Born-Infeld a Jin-Xin 1 (a global) et Jin-Xin 2 (a local) pour

• f(u) = − u√1 + u

avec la transformation (A,B) = (0, Z2)

• f(u) = +u√

1 + uavec la transformation (A,B) = (0, Z3)

Ca va saigner!

4 Prospective extensions

We wish to demonstrate the versatility of the Born-Infeld relaxation by sketching out twopotential applications to systems. Since the scope is purely prospective, we do not dwellinto a details and do not show numerical results.

21

4.1 A porous media model

The Seguin-Vovelle system [21]

∂tu+ ∂x(u(1− u)k) = 0 (4.1a)

∂tk = 0, (4.1b)

with u ∈ [0, 1] and k > 0, is a simplistic model for porous media. Setting the equilibriumphase velocities

w(u, k) = (1− u)k and z(u, k) = −uk, (4.2)

and introducing the relaxation variables U,K,W,Z, related to each other via

W = (1− U)K and Z = −UK, (4.3)

we approximate (4.1) by the relaxation model

∂tWλ+Zλ ∂xW

λ = λ[w(U(W λ, Zλ), k)−W λ] (4.4a)

∂tZλ +W λ∂xZ

λ = λ[ z(U(W λ, Zλ), k)− Zλ] (4.4b)

∂tk = 0, (4.4c)

with U(W,Z) = Z/(Z −W ). After combination, we show that the linearly degeneratesystem (4.4) is equivalent to

∂tUλ + ∂x(U

λ(1− Uλ)Kλ) = 0 (4.5a)

∂tKλ + (Kλ)2 ∂x Uλ = λ[k −Kλ] (4.5b)

∂tk = 0. (4.5c)

By a Chapman-Enskog analysis, we can show that the U -component of this system obeys,at the first order in λ−1, to the equivalent equation

∂tUλ + ∂x(U

λ(1− Uλ)k) = λ−1∂xUλ(1− Uλ)k2∂xUλ, (4.6)

the right-hand side of which always correspond to a diffusive kernel. Thus, there is noneed for imposing a subcharacteristic condition.

4.2 A two-phase drift-flux model

The drift-flux model in Eulerian coordinates reads [1–3]

∂t(ρ) + ∂x(ρv) = 0 (4.7a)

∂t(ρv) + ∂x(ρv2 +P (q)) = 0 (4.7b)

∂t(ρY ) + ∂x(ρYv+ σ(q)) = 0, (4.7c)

with the vector notation

q ∈ E := (ρ, ρv, ρY ) | ρ > 0, v ∈ R, Y ∈ [0, 1]. (4.8)

Here, ρ is the total density of the mixture, v is the mass-averaged velocity, and Y is thegas mass-fraction. The pressure-like function P (u) is highly nonlinear, as well as the slipmomentum

σ(q) = Y (1− Y )ρφ(q). (4.9)

22

By analogy with (2.3), we introduce the (Lagragian) phase velocities

w(q) =σ(q)

Y= (1− Y )ρφ(q), z(q) = − σ(q)

1− Y= −Yρφ(q) (4.10)

The differenceg(q) = w(q)− z(q) = −ρφ(q) (4.11)

then appears to be the Lagrangian slip velocity between the gas (Y ) and the liquid (1−Y ).The transformations (4.10) can be inverted to yield

Y =z(u)

z(u)− w(u), σ =

w(u)z(u)

z(u)− w(u). (4.12)

Consider the relaxation variables W,Z,G and define

Υ(W,Z) =Z

Z −W, Σ(W,Z) =

WZ

Z −W. (4.13)

The relaxation counterpart of q is denoted by

Q = (ρ, ρv, ρΥ(W,Z)). (4.14)

We decide that the relaxation system for (4.7) is

∂t(ρ)λ + ∂x(ρv)

λ = 0 (4.15a)

∂t(ρv)λ + ∂x(ρv

2 +Π)λ = 0 (4.15b)

∂t(ρΠ)λ + ∂x(ρΠv + a2v)λ = λρ(P (Qλ)−Πλ) (4.15c)

∂t(ρW )λ+ ∂x(ρWv)λ + Zλ∂xWλ = λρ(w(Qλ)−W λ) (4.15d)

∂t(ρZ)λ + ∂x(ρZvλ) +W λ∂xZ

λ = λρ( z(Qλ)− Zλ) (4.15e)

In (4.15), the new variable Π is meant to act as the relaxation variable for the pressure-like quantity P . The first three equations (4.15a)–(4.15c) formally appear to be a partialJin-Xin relaxation of the acoustic block (4.7a)–(4.7b), using the tuning parameter a. Asfor the last two equations (4.15d)–(4.15e), they can be assimilated to a partial Born-Infeldrelaxation of the kinematic block (4.7c).

It is easy to check that the relaxation system (4.15) is always hyperbolic. Its (un-ordered) eigenvalues

vλ − a

ρλ, vλ +

W λ

ρλ, vλ, vλ +

Zλ

ρλ, vλ +

a

ρλ, (4.16)

are all linearly degenerate. This allows us to derive the equivalent form

∂t(ρ)λ + ∂x(ρv)

λ = 0 (4.17a)

∂t(ρv)λ + ∂x(ρv

2 +Π)λ = 0 (4.17b)

∂t(ρΠ)λ+ ∂x(ρΠv + a2v)λ = λρ(P (qλ)−Πλ) (4.17c)

∂t(ρY )λ+ ∂x(ρY v + Y (1− Y )G)λ = 0 (4.17d)

∂t(ρG)λ+ ∂x(ρGv)λ + (Gλ)2∂xY

λ = λρ( g(qλ)−Gλ) (4.17e)

by standard calculations.

23

5 Conclusion

As evidenced throughout this contribution, there is a close connection between some scalarconservation laws —and especially those of the type (2.1)— and the Born-Infeld equations(2.17).

Insofar as the relaxation method we propose takes advantage of this specific connection,it looks somehow more natural than the generic Jin-Xin relaxation. For one, there is noneed for tuning an extra parameter in order to ensure stability. For another, the eigenvaluesof the relaxation system coincide with the physically meaningful phase velocities associatedto the original. In terms of numerical quality, it does compete well with the Jin-Xin methodfor the class of flux functions that satisfy the subcharacteristic condition (1.10).

References

[1] N. Andrianov, F. Coquel, M. Postel, and Q. H. Tran, A relaxation multiresolutionscheme for accelerating realistic two-phase flows calculations in pipelines, Int. J. Numer. Meth.Fluids, 54 (2007), pp. 207–236.

[2] M. Baudin, C. Berthon, F. Coquel, R. Masson, and Q. H. Tran, A relaxation methodfor two-phase flow models with hydrodynamic closure law, Numer. Math., 99 (2005), pp. 411–440.

[3] M. Baudin, F. Coquel, and Q. H. Tran, A semi-implicit relaxation scheme for modelingtwo-phase flow in a pipeline, SIAM J. Sci. Comput., 27 (2005), pp. 914–936.

[4] S. Benzoni-Gavage, Analyse Numerique des Modeles Hydrodynamiques d’Ecoulements

Diphasiques Instationnaires dans les Reseaux de Production Peroliere, PhD dissertation, EcoleNormale Superieure de Lyon, 1991.

[5] M. Born and L. Infeld, Foundations of a new field theory, Proc. Roy. Soc., (1934), pp. 425–451.

[6] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Lawsand Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhauser, 2004.

[7] Y. Brenier, Hydrodynamic structure of the augmented Born-Infeld equations, Arch. RationalMech. Anal., 172 (2004), pp. 65–91.

[8] G. Q. Chen, C. D. Levermore, and T. P. Liu, Hyperbolic conservation laws with stiffrelaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), pp. 787–830.

[9] G. Q. Chen and T. P. Liu, Zero relaxation and dissipation limits for hyperbolic conservationlaws, Comm. Pure Appl. Math., 46 (1993), pp. 755–781.

[10] F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers forgeneral pressure laws in fluid dynamics, SIAM J. Numer. Anal., 35 (1998), pp. 2223–2249.

[11] E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Mathematiqueset Applications, SMAI, Ellipses, 1991.

[12] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanism, World Scientific, Sin-gapour, 1997.

[13] A. In, Numerical evaluation of an energy relaxation method for inviscid real fluids, SIAM J.Sci. Comput., 21 (1999), pp. 340–365.

[14] S. Jin and M. Slemrod, Regularization of the Burnett equations for rapid granular flowsvia relaxation, Physica D, 150 (2001), pp. 207–218.

24

[15] S. Jin and Z. P. Xin, The relaxation schemes for systems of conservation laws in arbitraryspace dimension, Comm. Pure Appl. Math., 48 (1995), pp. 235–276.

[16] R. J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics, ETHZurich, Birkhauser Verlag, Berlin, 1992.

[17] H. L. Liu, J. Wang, and T. Yang, Stability for a relaxation model with a nonconvex flux,SIAM J. Math. Anal., 29 (1998), pp. 18–29.

[18] T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987),pp. 153–175.

[19] R. Natalini, Convergence to equilibrium for the relaxation approximation of conservationlaws, Comm. Pure Appl. Math., 49 (1996), pp. 795–823.

[20] W. Neves and D. Serre, Ill-posedness of the Cauchy problem for totally degenerate systemof conservation laws, Elec. J. Diff. Eq., 124 (2005), pp. 1–25.

[21] N. Seguin and J. Vovelle, Analysis and approximation of a scalar conservation law witha flux function with discontinuous coefficients, M3AS, 13 (2003), pp. 221–257.

[22] D. Serre, Systemes de Lois de Conservation I & II, Collection Fondations, Diderot EditeurArts et Sciences, Paris, 1996.

[23] D. Serre, Relaxations semi-lineaires et cinetique des systemes de lois de conservation, Ann.Inst. Henri Poincare, Analyse non lineaire, 17 (2000), pp. 169–192.

[24] J. Whitham, Linear and Nonlinear Waves, New-York, 1974.

[25] N. Zuber and J. Findlay, Average volumetric concentration in two-phase flow systems, J.Heat Transfer, C87 (1965), pp. 453–458.

25