Mensur Hrh

Omerbashich

Hyperresonance Unifying Theory and the resulting Law

Keywords: unifying theory, multiverse theory, string theory, relativity, Newtonian gravitational constant, geophysics

Hyperresonance Unifying Theory (HUT) is herein conceived based on theoretical and experimental geophysics, as that absolute extension of both Multiverse and String Theories, in which all universes (the Hyperverse) -of non-prescribed energies and scales -mutually orbit as well as oscillate in tune. The motivation for this is to explain oddities of "attraction at a distance" and physical unit(s) attached to the Newtonian gravitational constant G. In order to make sure HUT holds absolutely, we operate over non-temporal, unitless and quantities with derived units only. A HUT's harmonic geophysical localization (here for the Earth-Moon system; the Georesonator) is indeed achieved for mechanist and quantum scales, in form of the Moon's Equation of Levitation (of Anti-gravity). HUT holds true for our Solar system the same as its localized equation holds down to the precision of terrestrial G-experiments, regardless of the scale: to 10^-11 and 10^-39 for mechanist and quantum scales, respectively. Due to its absolute accuracy (within NIST experimental limits), the derived equation is regarded a law. HUT can indeed be demonstrated for our entire Solar system in various albeit empirical ways. In summary, HUT shows: (i) how classical gravity can be expressed in terms of scale and the speed of light; (ii) the tuning-forks principle is universal; (iii) the body's fundamental oscillation note is not a random number as previously believed; (iv) earthquakes of about M6 and stronger arise mainly due to Earth's alignments longer than three days to two celestial objects in our Solar system, whereas M7+ earthquakes occur mostly during two simultaneous such alignments or alignments longer than a week; etc. HUT indicates: (v) quantum physics is objectocentric, i.e. trivial in absolute terms so it cannot be generalized beyond classical mass-bodies; (vi) geophysics is largely due to the magnification of mass resonance; etc. HUT can be extended to multiverse (10^17) and string scales (10^-67) too, providing a constraint to String Theory. HUT is the unifying theory as it demotes classical forces to states of stringdom. The String Theory's paradigm on vibrational rather than particlegenic reality has thus been confirmed.

Introduction

Whilst theoretical physics was a tale-telling albeit lively trade, most efforts and resources in physics of the last century were spent on verifying only one of the two rival paradigms: on particlegenic reality. The alternative paradigm -on vibrational reality -favored by String Theory, has been largely ignored. Standard Model and Higgs Theory, the pinnacles of the former paradigm, might prove certain mass-transfers could have happened but not that they did happen, since their paradigm clearly could be verified using earthbound data only. However, a theory cannot be proven in theory but in practice instead: by using data from the same epoch as the starting physical hypothesis's, to derive equations that either prove or disprove that hypothesis. On the other hand, not nearly as much has been done in order to unlock G, the least understood classical constant and thus the most fundamental problem of science. Useless, yet exotic, theorizing took lead instead -coupled with particle-experiments that, trivially enough, all in the end find some particles. Rivaling such a fundamentally uncritical approach, while believing that theories must be worked out around both a concept and its null-epoch's data, we herein present an approach that is based on information critical for understanding G as the constant of earthbound physics only, i.e., without pretending that G is universal in value and units.

Then one must turn to global geophysical data as the only complete information on the Earth as a classical mass-body. Global geophysics too has been unable to explain its own largest problem: the mechanism that supplies most of the energy for tectonogenesis and seismicity [1]. Mechanical tectonogenesis seems to be the culprit, as our Solar system's moons, e.g., the Titan, were found recently to feature cryovolcanism. So tectonics is driven mechanically and not by heat dissipation as believed by some. As the Earth is seismically least active in the frigid zones, this preferentiality too cannot be explained by a mix of internal processes and Earth rotation, so certain external causes that cluster about the ecliptic are called for instead.

The main theoretical problem with the Newton-Einstein setup is in its derivative concepts: of the peculiar "attraction at a distance" (over the void) and of the related "free fall"; and the main problem with G is in its meaningless unit(s) that Newton used only in order to close his theory physically as well as mathematically. In order to overcome these obscurities while avoiding absurd notions such as time, one is ideally led to the absolute blended concept of unitless G over the vibrating void, with temporal variations being ignored then. To detect the vibrating void, we use the most accurate global geophysical data -taken by the superconducting gravimeter -and look for at least one relationship that describes this confusing attraction between any two classical mass-bodies (here: the Earth; the Moon) by using harmonic quantities only, such as oscillatory and orbital information of those bodies. In order to find such a relationship, we compare generalizations of the Moon-forced geooscillator at two Earth-scales (mechanist, quantum), where physics is represented by (i.e., checked against) the value of G as estimated by NIST for each scale. No unit-bearing physical quantities can be used either, as it is unclear if those can exist as such in absolute terms, i.e., outside of the classical-mass bodies. Generalizations of such a relationship will also have to be freed from temporal variations, and include only the general part of G, i.e., its value, but not earthbound aspects of G, i.e., its unit(s). By building on the above information as it became available over the past decade, a blending view has indeed been achieved in the above way and termed Hyperresonance Unifying Theory (HUT).

Based on HUT, the String Theory's paradigm has been verified at last. It is shown how classical gravity can be expressed in terms of scale and the speed of light, and that the tuningforks principle is universal so that fundamental vibrational modes and orbital periods of heavenly bodies are tuned exactly, i.e., nothing is in so-called "free fall". The demonstration is in three steps (Sections 2-4): georesonator → hyperresonator → hyperresonance.

Georesonator

Georesonator is a concept in which the Earth is thought of as a mechanical oscillator forced mostly by the Moon's classical mass. The Earth's mass is taken totally (noise inclusive). It can be shown that the Earth's gravity correlates with seismicity, fig. 1, as well as oscillates with lunar synodic half-month period exactly, fig. 2, [2]. Then the Earth is a real georesonator.

, . Well-known equations of a viscous mechanical oscillator, here applied for the generally accepted values of Earth parameters including the mantle viscosity, give the commonly observed maximum displacement due to strongest earthquakes, of ~10 m, as well as the 3-days phase which is also seen in cross-correlation of gravity v. deep seismicity; fig. 1-b. For details on georesonator, see [2].

Hyperresonator

Hyperresonator is the absolute generalization of georesonator, such that georesonator applies to all quantities in all universes (of all energies; at all scales), all the time. This is achieved by operating over unitless and quantities with derived units only, while avoiding temporal considerations. On the other hand, a HUT's localization (for our space of choice: the Earth-Moon system), is achieved by comparing the relationship between Earth-Moon forced oscillators generalized along mechanist and quantum scales, with physics (represented by G) along those same scales. We thus tacitly assumed that all gravitational considerations in the vicinity of the Earth, such as G-experiments, had failed to account for resonant magnification of gravity oscillations due to heavenly objects from our Solar system primarily. This approach proved fruitful, resulting in the Moon's Equation of Levitation (of Anti-gravity): (2)

[3] NIST Planck 4 2 2 Moon Earth Earth Planck NIST Newton 2 1 Moon Earth Earth Newton G T T * G G T T * G = - ⋅ = ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = - ⋅ = ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = - - 39
Eqn.( 1) can be rewritten generally, for a scale s, as:

2 G e s ⋅ = , (3)
meaning that, what we have known so far as G, is just the scaled Euler's constant, i.e., the base of natural logarithm. Then nature itself is basically fractal in character, and therefore inherently anti-entropic. The above NIST, 2006 estimates of G do rectify the estimate of To Earth, Eqn.( 1), up from a coarse 3455 s (as measured by superconducting gravimeter -the world's most accurate instrument that is used also for estimating G), to 3454.8 ±0.1 s. While the NIST, 2006 estimates Eqn.(2) do close Eqn.(1) an order of magnitude better than the preceding NIST estimates used to, the NIST, 2010 updated estimates score against the NIST, 2006 estimates even better, albeit just under a half an order of magnitude so. Since the localization, Eqn.(1), of HUT, Eqn.(3), does successfully relate orbital periods with major eigenperiods of certain heavenly bodies in our Solar system, Eqn.( 3) is thereby understood as the Kepler's 4th law. That Eqn.( 3) is indeed a law can be seen from the fact that it holds true at least for our entire Solar system, i.e., to the precision of best terrestrial G-experiments, regardless the scale: to 10^-11 and 10^-39 for mechanist and quantum scales, respectively. Such a relationship requires a precisely tuned, vibrating medium, in order for orbital periods and eigenperiods of heavenly bodies to "communicate" with each other down to the absolute precision and in a closed form, such as the case with Eqn. (1). Hence the law, Eqn.(3), both completes and overturns the Newton's "action at a distance" unlikely concept, as well as its Einstein-Riemann mathematical enhancement for dynamic setups. At the same time, String Theory readily offers its own realm (classical aether; here: stringdom) as the natural setup for Hyperresonator as well. We note here that the Newton-Einstein concept had been fundamentally beset by the assumption that G was both a universal constant and a physical quantity with its own physical unit(s).

The law possesses many elemental properties, showing: (i) how classical gravity can be expressed in terms of scale and the speed of light; (ii) that the well known tuning-forks principle is universal (or, rather: hyperversal); (iii) that the body's fundamental oscillation note is not a random number as previously believed; and so on. Finally, manipulations in the law's power indices, reflecting the law's range of scales in both extents, enable HUT to be extended to both multiverse (10^17) and string scales (10^-67), thereby constraining String Theory: 67 10 6.74377 17 10 6.64095

- ⋅ = ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⋅ = ⋅ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = - - - T T * G T T * G
For details on the hyperresonator, see [3].

Hyperresonance

Hyperresonance is the hyperresonator's working state, i.e., it is the absolute vibration of stringdom. Then hyperresonance can be verified empirically only. By comparing astronomical with seismic data (primarily from the decade of 2000's, independent of the data used for considerations in Section 2), one easily obtains various empirical proofs of HUT for our Solar system, figs. 3-5. So hyperresonance as a phenomenon can indeed be seen at work, in multiple ways: as a Sun-centered hyperresonator; fig. 3, as a preferential rotation in strain-vortices of classical viscous mass, fig. 4; as the strengthening-peaking-weakening patterns in Earth's M6+ seismicity (an equivalent of 2/3+ of kinetic energy emitted by all earthquakes) during astronomical alignments that lasted for more than 3 days -the mantle-sensitive limiting phase of Georesonator, fig. 1-b) and fig. 5; and so on. Moreover, the peak magnitudes in such patterns, as seen from the alignments that lasted for weeks-to-months, also formed patterns of the same type. This renders the hyperresonance setting, i.e., the Hyperverse (the set of all the universes, where each tidally affects at least another) fractal in character and thereby anti-entropic, which corroborates the consequence of the law, Eqn. (3). The property of anti-entropy forbids singular events such as the alleged "Big Bang". For details on hyperresonance, see [4]. 5. An empirical proof of HUT via uncritically selected sample of the increasing-peaking-weakening patterns (blue callouts) which arise always in Earth's M6+ (an equivalent of 2/3+ of kinetic energy from earthquakes) seismicity during Earth's (-5°, +5°) alignments longer than 3 days to two Solar system's bodies. The same patterns occur in Earth's M7+ seismicity during two simultaneous such alignments, or such alignments that last for longer than a week. In those alignments that go on for weeks to months, the patterns' own peak-magnitudes themselves form patterns of the same type (green callouts), showing that the Hyperverse is fractal in character and thereby anti-entropic. For complete 6-months demonstration and other details, see [4]; as well as www.seismo.info.

Based on the above, our Solar system is then a real hyperresonator.

Discussion and conclusions

The fact that we could arrive at the experimental value of G at various scales and with absolute precision just by operating over quantities without real units means that no physical units are real to begin with. Their very appearance is always a local phenomenon then. Physical units refer to some pre-selected space of interest, here the Earth-Moon system, the Earth's surface primarily. So we demonstrated absolutely (independently of the concepts of time and units; at various scales) that the so-called Newtonian gravitational constant, G, is actually a parameter whose value (but not the units) is independent of space selection or any imaginary mathematical constructs such as time, space-time, etc. Imaginary "gravitational pull" from the classical view is just a vibrational push along the stringdom. Mathematical singularities (errors, really) from Einstein's General Relativity Theory, referred to by some as real "black holes" are multilateralities -the points of locally highest oscillatory excitement of stringdom. In case of galaxies, the excitement originates mostly from within the galaxy, giving the appearance of a "black hole" around galactic centres. The "holes" however, are not phenomena but purely geometric, Keplerian foci. The resonant magnification amounts then, all through the inter-galactic Space -to what is oddly regarded by some as "dark matter/energy". Galaxies themselves are just stringdom vortices -"pinwheel toys" -which by their preferential rotation indicate that our known universe orbits about as well as oscillates in tune with outer universe(s), thus supporting our starting physical hypothesis -that HUT is indeed a correct generalization of both Multiverse Theory and String Theory. The here demonstrated anti-entropy property of HUT renders alternative grand unifying theories forbidden by virtue of scale non-preferentiality. Brane Theory for instance, requires parallel universes to belong to rather same scales.

It is inadequate to express classical mass of an object that is positioned outside of the Earth in units of weight such as kilogram; "weightlessness" being the most obvious hint at such inadequacy. Since the concept of classical mass cannot be extended universally and uniquely, no alternative description of mass of an object positioned outside the Earth can be real either. In other words, such a description, as well as its units and constants, are always only local (here: earthbound), i.e., they are objectocentric and therefore highly unreliable in terms of prediction for Space. Objectocentricity is what makes Newton's Theory and its detailed form the so-called Standard Model such a success story in the vicinity of heavenly objects (based on Newton's mechanism of "central gravity fields"), and at the same time such a failure for everywhere else in the universe(s). Then the solution of a famous multi-body problem of Newton Theory is simple in HUT, where it should be searched in Dermott's rule -an empirical formula for orbital period of satellites that orbit planets in our Solar system. Basically, thanks to HUT, near ratios of mean motion now have physical meaning.

As we herein provided the first mathematical-physical constraint to String Theory, the classical gravity (tidal) force between the extreme realms of the Hyperverse and strings, where strings are enmeshed into orbiting-oscillating multiverses, is neither the Newtonian "attraction at a distance" apparent at inertial mechanist scale, nor the Riemannian "distortion of space-time" apparent at dynamic mechanist scales. Rather, we speak of an incessant ubiquitous influx of forward-propagating string vibration on to the point of interest. This spreading leaves background energy signatures which can forge the "Big Bang" type of events. Finally, HUT also indicates that "strange quantum phenomena", like quantum entanglement, wave-particle duality, etc., are all trivial as quantum physics too is objectocentric, i.e., appears only in the vicinity of classical mass-bodies which are all now non-independent, mathematically completely describable realms of georesonators.

Geophysics itself seems largely due to classical-mass resonance magnification. Alfred Wegener's main proposal -that tectonics arises externally -is basically correct as Solar system's hyperresonance shapes continents and topography, fig. 5. It turns out that, as Einstein once remarked, geophysics was the "missing link" of theoretical physics. As classical gravity now can be expressed in terms of scale and the speed of light, it is clear that true gravity (i.e., as a phenomenon) is not a classical force. Instead, true gravity is the totality of all magnified oscillations throughout stringdom. True gravity thus consists of two main components: the objectocentric (apparent) and the hyperresonant (hidden) component that was, out of ignorance, misnamed the "dark energy/matter".

The fork-tuning principle is universal. Systemic uniformity (rotation-and size-wise) of stringdom's strain-vortices confirms that hyperresonance is the observable working state of hyperresonator as the system of all universes -mutually orbiting and oscillating at all scales and energies, all the time. HUT is the unifying theory as it capably demotes classical forces to states of stringdom. At last, the String Theory's paradigm on vibrational and not particlegenic reality has been verified.

Independent confirmations of Hyperresonance Theory up to April 2013

Positive Verifications of Hyperresonance Theory, which is thus deemed confirmed conclusively:

The alignments-seismicity physical relationship, as the core of the georesonance mechanism and by extent of the Hyperresonance Theory, have been confirmed by Professor Boucouvalas's research group at U of Peloponnese, Greece: "Virtually all M7+ earthquakes from 2004-2011 occurred on the day of Earth's astronomical alignment." [1].

Hyperresonance Theory's realization in the Earth-Moon system, Georesonator mechanism in particular, results in seismic magnitudes' strengthening-peaking-weakening pattern discovered here to hold for M5.7+ earthquakes. Geophysicists at USGS & U of California in Berkeley have confirmed that mega-quakes "trigger" strong (M5.6+) earthquakes globally. They independently stumbled upon the mechanism of the externally forced Georesonator. While clueless about the mechanism (they try Earthbound approaches only!), such approaches to global-scale phenomena obviously necessitate too many ifs, as well as coincidences which occur virtually simultaneously. So there cannot exist an Earthbound mechanism that actually does trigger strong earthquakes globally, yet so successfully (so fast and in such a perfectly ordered manner magnitude-wise) as was the case with the 11 April 2012, Indonesia mega-quake, which they call "the strangest ever". Of course, there are no strange natural events; we are supposed to decipher nature, not vice versa. One mega-quake is thus forcing all global geophysics theories to be rewritten. Cf. [2] Hyperresonance Theory predicts that our universe moves relative to another universe(s). This was confirmed by astronomers at U of California Berkeley, Cal Tech et al., who found that all galaxies tend to go whirlpool-like. This is a direct consequence of hyperresonance as our universe is oscillatory-orbitally engaged with at least another universe. Just like a pin-wheel toy spins when moved frontally against the air, galaxies too spin as our known (their common) universe moves through the omnipresent oscillatory-orbitally tuned environment -stringdom. The same applies to all scales including georesonance scales, for example the Jupiter's Great Red Spot, the Earth's Ring of Fire, etc. Cf. [3] Hyperresonance Theory predicts that our universe is gravitationally engaged with another universe(s). This was confirmed by physicists at Large Hadron Collider (CERN), who have disproved Theory of Supersymmetry -thus disproving the "dark matter" concept as a name for "we don't know how to explain (around 23% of) our universe". [4] Hyperresonance Theory predicts that our universe is gravitationally engaged with another universe(s). This was confirmed by astronomers at the Berkeley National Lab, who have mapped Space in BOSS Survey, showing when exactly the alleged "dark energy" took over gravity in our universe. Since "dark energy" is just another name for "we don't know how to explain (around 73% of) our universe", the instance that they pinpointed demonstrates by its sheer existence that, indeed, at least another universe is gravitationally affecting ours. This, by elimination alone: what else could pull/push an entire universe, in such a sudden yet uniform fashion? The most reasonable answer seems to be: another universe. So "dark matter" does not exist, and there is nothing dark about "dark energy" either. The so-called "Big Bang" events are overstated, and the universes continue ad infinitum. Cf. [START_REF]Baryon Acoustic Oscillations in the Ly-α forest of BOSS quasars[END_REF] European Space Agency (ESA) has released a long-awaited map of known universe, as based on the 2009-2012 Planck Space Mission. The new map represents also the conclusive confirmation of the Hyperresonance Theory, namely its main consequence: our known universe must be gravitationally engaged with another universe(s). The mission was named after Max Planck as it was expected to confirm the cosmological model built on Big Bang theories and quantum theories first started by Planck. But instead, the new map showed significant disagreement with the two groups of theories, revealing drastic anomalies. Due to these absolutely astonishing anomalies, the map cannot be called "almost perfect". Thus the only perfectly clear conclusion is that the observed anomalies are so huge (on universe scales), and span an incredible 1/2 of the map (with a huge contrast-spot and a forbiddingly nonhomogeneous distribution for inflationary scenarios to work) -that the anomalies can only be explained by tidal effects of another universe(s) exerted on our known universe. Cf. [6] Fundamentally, the here reported discovery shows that quantum physics is a byproduct of secondary, resonant effects of the Moon (tides) against the Earth. This confirms Einstein's claims that quantum mechanics is locally real. Locality becomes self-evident beyond celestial resonant systems such as the Earth-Moon that hinder locality and impose common-resonance settings for quantum phenomena to trivially apply even "at a distance" but only within the system.

Figure 1 :

 1 Figure 1: Cross-correlation of 1990-ies mean-diurnal mass (gravity) oscillations from Canadian superconducting gravimeter's 1Hz, 220 billion data, with global M6.3+ seismicity (in seismic energies) along low-frequency poloidal modes from a geophysical Earth-model (left panel); same, but with global M6.3+ deep seismicity along low-frequency toroidal modes from another geophysical Earth-model (right panel).

Figure 2 :

 2 Figure 2: Spectrum of spectral magnitudes of total-Earth (all masses) mean-diurnal gravity is absolutely (to sampling resolution) periodic with lunar synodic half-month. Measured using Gauss-Vaníček spectral analysis, in variance-spectrum of variancespectra of Canadian superconducting gravimeter's 10+ billion, 1Hz gravity data from 1990-ies. Data span 10.3 years; band 2 day -5 yr. Solar tide removed.

  -Moon= 14.7655 day, fig.2 and T0 Earth=3455 s [2], which agrees with the NIST 2006 estimates:

Figure

  Figure 3. A Earth's long earthquakes is currently in both seri orbital inclin same as oth non-gradual

Acknowledgment

We thank Dr. Petr Vaníček, Dr. Barbara Romanowitz, and Dr. Jeffrey Park.


Jérôme Droniou
email: droniou@math.univ-montp2.fr

A numerical method for fractal conservation laws

Keywords: Mathematics Subject Classification: 65M12, 35L65, 35S10, 45K05 conservation laws, Lévy operator, fractal operator, integral operator, numerical scheme, proof of convergence, numerical results

We consider a fractal scalar conservation law, that is to say a conservation law modified by a fractional power of the Laplace operator, and we propose a numerical method to approximate its solutions. We make a theoretical study of the method, proving in the case of an initial data belonging to L ∞ ∩ BV that the approximate solutions converge in L ∞ weak- * and in L p strong for p < ∞, and we give numerical results showing the efficiency of the scheme and illustrating qualitative properties of the solution to the fractal conservation law.

Introduction

Partial differential equations involving non-local operators are used in several models, from mathematical finance [START_REF] Soner | Optimal control with state-space constraint. II[END_REF] to dislocation dynamics [START_REF] Alvarez | Dislocation dynamics: short time existence and uniqueness of the solution[END_REF] to gas detonation [START_REF] Clavin | Instabilities and nonlinear patterns of overdriven detonations in gases[END_REF] and anomalous diffusion in semiconductor growth [START_REF] Woyczyński | Lévy processes in the physical sciences[END_REF]. We consider in this paper the following model of non-local scalar conservation law, which appears in particular in the last two references:

∂ t u(t, x) + ∂ x (f (u(t, x))) + g[u(t, •)](x) = 0 t > 0 , x ∈ R , u(0, x) = u 0 (x) x ∈ R (1.1)
where f : R → R is locally Lipschitz-continuous, u 0 ∈ L ∞ (R)∩BV (R) and g is a fractional power of order λ/2 of the Laplacian, with λ ∈]0, 2[. The natural definition of g can be written via Fourier transform g[φ] = F -1 (| • | λ F(φ)), but it will be more useful in the sequel to consider the following formula (see [START_REF] Droniou | Fractal first order partial differential equations[END_REF]): for all r > 0 and all φ ∈ S(R),

g[φ](x) = -c(λ) |z|≤r φ(x + z) -φ(x) -φ (x)z |z| 1+λ dz -c(λ) |z|>r φ(x + z) -φ(x) |z| 1+λ dz = g λ,r [φ](x) + g 0,r [φ](x) (1.2)
where c(λ)

= λΓ( 1+λ 2 ) 2 √ ππ λ Γ(1-λ 2 )
with Γ the Euler function (this value of c(λ) corresponds to the convention F(φ)(ξ) = R e -2iπxξ φ(x) dx, and gives in fact g = (2π) -λ (-∆) λ/2 ); the notations g λ,r and g 0,r refer to the order of each term: the first term is of order λ (the singularity of the weight in the integral sign necessitates some regularity on φ, of the kind φ(x + z) -φ(x) -φ (x)z = o(|z| λ )), whereas the second term can be applied to any bounded non-regular φ and is therefore of order 0.

There are several theoretical studies and results regarding such equations. To our knowledge, [START_REF] Biler | Fractal Burgers Equations[END_REF] presents some of the first results on these problems, mainly with f (s) = s 2 (or other powers) and a H s or Morrey framework, studying in particular traveling wave or self-similar solutions; more on self-similar solutions, as well as time decay estimates, can be found in [START_REF] Biler | Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws[END_REF]. In the framework of bounded solutions, classical for pure scalar conservation laws, existence and uniqueness of a regular solution if λ > 1 has been proved in [START_REF] Droniou | Global solution and smoothing effect for a non-local regularization of an hyperbolic equation[END_REF]. If λ ≤ 1, the solution is not smooth in general (see [START_REF] Alibaud | Occurrence and non-appearance of shocks in fractal Burgers equations[END_REF]) and obtaining general existence and uniqueness results for (1.1) requires to use an appropriate notion of entropy solution, introduced and studied in [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF]; this notion, which is constructed from the classical notion for scalar conservation laws [START_REF] Krushkov | First order quasilinear equations with several independent variables[END_REF], relies on the formula (1.2). Numerical studies of non-local operators in first-order PDE seem more scarce. A scheme for a Hamilton-Jacobi equation modelling dislocation dynamics and involving a non-local zero-order velocity is studied in [START_REF] Ghorbel | Well-posedness and numerical analysis of a one-dimensional nonlocal transport equation modelling dislocations dynamics[END_REF] (see also the references therein). Closer to the framework of scalar conservation laws, [START_REF] Alibaud | A non-motone nonlocal conservation law for dune morphodynamics[END_REF] studies an equation modelling the formation and movement of dunes, which is (1.1) with an additional term -∆u and g given by the opposite of (1.2) with λ = 4/3 and an integral sign only on R -(the non-local operator is therefore a lower order term in the PDE); besides theoretical results on the solution to this non-monotone equation, numerical results are obtained using a simple finite difference scheme (explicit and with centered discretizations), the study of which remains to be done. Regarding numerical approximations of (1.1) itself, to our best knowledge the only existing results are those based on the probabilistic interpretation of this equation (fractal conservation laws can be, as the classical heat equation, linked with a stochastic differential equation): [START_REF] Jourdain | Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws[END_REF] and [START_REF] Stanescu | Numerical study of interacting particles approximation for integro-differential equations[END_REF] use this interpretation to construct and study, in the case λ > 1, a numerical method for (1.1); however, in order to avoid having a too noisy approximation of the solution, the probabilistic method must be applied on the equation on ∂ x u obtained by derivating (1.1) and expressing u as the integral sum of its derivative (the local non-linearity in (1.1) is thus transformed into a non-local non-linearity); this technique is easy to implement in dimension 1, but its adaptation to the multi-dimensional case is less straightforward (the derived equation becomes a system in which, in order to reconstruct u from its derivatives, one has to introduce a convolution product with the derivative of the fundamental solution to the Laplace equation, see [START_REF] Jourdain | Probabilistic Interpretation via Spatial Derivation of Some Nonlinear Parabolic Evolution Equations[END_REF] for g = -∆; this derivative is however a singular function and therefore does not seem easy to use, in a numerical method, without introducing additional errors).

In this paper, we propose and study a numerical method to directly approximate the solution to (1.1) for any λ ∈]0, 2[. The scheme is based on classical techniques of numerical approximation of scalar conservation laws and diffusion equations (monotone fluxes, semi-implicit scheme, etc.) and therefore, though we present it on (1.1) for the sake of legibility, its adaptation to multi-dimensional equations with heterogeneous fluxes and source terms (such as ∂ t u + div(f (t, x, u)) + g[u(t, •)](x) = h(t, x, u)) is straightforward. This approach also allows us to obtain a stable and robust method, valid for any λ ∈]0, 2[ and which preserves the qualitative properties of the solution, such as the symmetry, the maximum principle (the solution takes its values between the upper and lower bounds of the initial datum) or the smoothing or non-smoothing effects (depending on the position of λ with respect to 1). The plan is as follows. In the next section, we present the numerical method, using only general properties on the discretizations of ∂ x (f (u)) and g [u] and covering therefore a wide range of possible schemes. The study of this method is done in Section 3, where we prove, thanks to usual techniques associated with monotone numerical fluxes for conservation laws, the existence of an approximate solution and its convergence toward the (entropy) solution to (1.1). An example of discretization of g, satisfying the properties used in the theoretical study of the scheme, is presented in Section 4, along with considerations on the practical implementation; some numerical results are also provided and show the efficiency of the scheme in catching known qualitative properties of the solution to (1.1) (such as the presence of shocks, speed of diffusion, or the asymptotic behavior). A few technical lemmas are gathered in an appendix (Section 5) which closes the article.

Definition of the scheme and main result

Let δt > 0 and δx > 0 be time and space steps. The scheme consists in computing approximate values u n i of the solution to (1.1) on [nδt, (n + 1)δt[×[iδx, (i + 1)δx[ for n ∈ N and i ∈ Z, thanks to the following relations:

∀i ∈ Z : u 0 i = 1 δx (i+1)δx iδx u 0 (x) dx , (2.1
)

∀n ≥ 0 , ∀i ∈ Z : δx δt (u n+1 i -u n i ) + F (u n i , u n i+1 ) -F (u n i-1 , u n i ) + δxg δx [u n+1 ] i = 0 (2.2)
where F is a numerical flux corresponding to the continuous flux f and g δx is a discretization of the nonlocal term g. Notice that the hyperbolic term of the equation is discretized using an explicit method; this imposes a CFL condition on the time and space steps (see (2.4)), but this condition is not very binding and, more importantly, the explicit discretization has the double advantage to avoid the solving of a non-linear equation at each time step and to allow us to consider as easily more complicated numerical fluxes (see Remark 2.2; higher order fluxes are not really adapted to an implicit discretization [START_REF] Eymard | Finite Volume Methods, Handbook of Numerical Analysis[END_REF]). On the contrary, since the non-local operator is linear and has diffusive properties (similar to the ones of -∆), we use an implicit discretization for g [u] in order not to have to impose, during the proof of a priori estimates on the approximate solution, a more restrictive condition than (2.4) on the time and space steps (see however Section 4.3).

Remark 2.1 Non-uniform time and space steps can as easily be considered but, for the sake of legibility, we only take here uniform δt and δx.

The numerical fluxes we consider are classical 2-points finite volume monotone fluxes (see [START_REF] Eymard | Finite Volume Methods, Handbook of Numerical Analysis[END_REF]):

F : R 2 → R is Lipschitz-continuous on [inf R u 0 , sup R u 0 ] 2 ,
non-decreasing with respect to its first variable, non-increasing with respect to its second variable, and

F (a, a) = f (a) for all a ∈ [inf R u 0 , sup R u 0 ]. (2.3)
Defining Lip 1,u0 (F ) and Lip 2,u0 (F ) as the Lipschitz constants of F with respect to its first and second variable on [inf R u 0 , sup R u 0 ] 2 , it is know that the following CFL condition is required to ensure the stability of explicit schemes involving such monotone fluxes:

δt δx ≤ 1 Lip 1,u0 (F ) + Lip 2,u0 (F ) . (2.4)
Remark 2.2 We write the numerical method and make its theoretical study using basic 2-points fluxes, but nothing prevents us from using higher order fluxes (computing f (u) at t = nδt and x = iδx by means of u n i-p , . . . , u n i+q instead of only u n i-1 and u n i ), provided that the scheme they define for ∂ t u + ∂ x (f (u)) = 0 is stable with respect to the L ∞ and BV norms (see Section 3). In particular, in Section 4, we present numerical results involving 4-points MUSCL fluxes.

For our theoretical study, and as for the numerical fluxes above, the discretization g δx of g does not need to have a specific expression but is only required to satisfy a series of assumptions (the curious reader can refer to Section 4.1 for an example of g δx ). The first ones are not surprising since the operator g itself satisfies continuous equivalent formulations of these assumptions (this can easily be seen from (1.2), see [START_REF] Droniou | Fractal first order partial differential equations[END_REF]):

g δx : l ∞ (Z) → l ∞ (Z) is linear, (2.5) ∀v ∈ l ∞ (Z) , if (i k ) k∈N is a sequence in Z such that lim k→∞ v i k = sup j∈Z v j , then lim inf k→∞ g δx [v] i k ≥ 0 , (2.6) if τ : l ∞ (Z) → l ∞ (Z) is the left translation τ (v) i = v i+1 , then τ g δx = g δx τ . (2.7)
The next assumption is quite natural in the framework of numerical analysis where, eventually, everything has to be finite in order to be implemented.

∃A δx > 0 such that, for all v ∈ l ∞ (Z), g δx [v] 0 only depends on (v j ) |j|≤A δx . (2.8)
The last assumptions impose the consistency of g δx as δx → 0 and necessitate to introduce a few conventions and notations. If δx is a given space step and v ∈ l ∞ (Z), we identify v with the function

v δx ∈ L ∞ (R) which is piecewise constant equal to v i on [iδx, (i + 1)δx[ for all i ∈ Z; likewise, g δx [v] is either considered as an element of l ∞ (Z) or of L ∞ (R), depending on the context. If K is a com- pact subset of R, C 2 K (R) is the space of C 2 functions on R with support in K (it is endowed with the norm ||φ|| C 2 K = ||φ|| L ∞ (R) + ||φ || L ∞ (R) + ||φ || L ∞ (R) )
and, for such a function φ, we define Φ ∈ l ∞ (Z) by Φ i = 1 δx (i+1)δx iδx φ(x) dx. (g δx ) * is the formal adjoint operator of g δx defined by: for all v ∈ l 1 (Z), (g δx ) * [v] i = j∈Z g δx [e i ] j v j where e i ∈ l ∞ (Z) is the sequence which has 1 at the i-th position and 0 elsewhere (note that, since g is self-adjoint, it is probable, but not required, that g δx also is self-adjoint). The assumptions regarding the behavior of g δx as δx → 0 are:

∀K compact in R , ∃θ K :]0, 1] → R + non-decreasing such that lim s→0 θ K (s) = 0 and, for all φ ∈ C 2 K (R) and all δx ∈]0, 1[, ||(g δx ) * [Φ] -g[φ]|| L 1 (R) ≤ ||φ|| C 2 K θ K (δx) , (2.9)
∀r > 0 , g δx = g δx λ,r + g δx 0,r where: 1) g δx λ,r satisfies (2.5)-(2.8) and (2.9) with g replaced by

g λ,r , 2) ∀Q compact in R, ∃γ r,Q :]0, 1[→ R + such that lim s→0 γ r,Q (s) = 0 and, for all δx ∈]0, 1[ and all v ∈ l ∞ (Z), ||g δx 0,r [v] -g 0,r [v δx ]|| L 1 (Q) ≤ ||v|| l ∞ (Z) γ r,Q (δx).
(2.10)

This last assumption is in fact useful only in the case λ ≤ 1, where we have to consider entropy solutions to (1.1) (the entropy formulation of this equation requires to cut g into g λ,r and g 0,r ). Remark 2.4 Since the construction and theoretical study of the scheme does not rely on the precise expression of the non-local term in (1.1), but only on general properties enjoyed by this term and its discretization (2 ), Theorem 2.3 can easily be generalized to equations involving, for example, other kinds of Lévy operators (not only the stable operator g).

3 Theoretical study of the scheme 3.1 Properties of the approximation g δx The assumptions made above on g δx allow us to precise the structure of this discretization of g and to deduce additional properties.

v i ) i∈Z → (v i-1 ) i∈Z . 2) If v ∈ l ∞ (Z) and (i k ) k≥1 are such that lim k→∞ v i k = inf j∈Z v j , then lim sup k→∞ g δx [v] i k ≤ 0. 3) If v ∈ l ∞ (Z) is a constant sequence then g δx [v] = 0.
4) There exists non-negative real numbers (µ δx j ) j=-A δx ,...,A δx such that

∀v ∈ l ∞ (Z) , ∀i ∈ Z : g δx [v] i = - |j|≤A δx µ δx j (v i+j -v i ). (3.1)
5) For all v ∈ l ∞ (Z), all η : R → R convex function and all i ∈ Z, we have

g δx [η(v)] i ≤ η (v i )g δx [v] i (3.2) (if η is not regular, we let η (v i ) denote any sub-differential of η at v i ). 6) If v ∈ l ∞ (Z) and (v m ) m≥1 is a bounded sequence in l ∞ (Z) such that, for all i ∈ Z, lim m→∞ v m i = v i , then (g δx [v m ]) m≥1 is bounded in l ∞ (Z) and lim m→∞ g δx [v m ] i = g δx [v] i for all i ∈ Z. 7) There exists C δx ≥ 0 such that, for all v ∈ l ∞ (Z) and all N ≥ 1, N i=-N g δx [v] i ≤ C δx sup N -A δx ≤|i|≤N +A δx |v i |. 8) If w ∈ l 1 (Z) and v ∈ l ∞ (Z), then (g δx ) * [w] ∈ l 1 (Z) and i∈Z g δx [v] i w i = i∈Z v i (g δx ) * [w] i . (3.3) Remark 3.
2 It is shown in [START_REF] Bony | Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-diffrentiels du second ordre donnant lieu au principe du maximum[END_REF] that operators acting on spaces of functions on R and satisfying a reverse maximum principle similar to (2.6) have integral representations, generalizations of (1.2). Formula (3.1) can be seen as a discrete version of this result (see also Section 4.1 to understand the absence, with respect to the continuous case, of a discrete derivative in (3.1)) and, as in the continuous case, the reverse maximum principle (2.6) truly is the key point to the study of the discretized equation.

Proof of Lemma 3.1

We first notice that Item 1 is evidently true, as a consequence of (2.7) and of the general fact that if an operator commutes with an isomorphism, then it also commutes with its inverse mapping. It is also easy to see that Item 2 is a consequence of (2.6) applied to -v instead of v. If v is a constant sequence, then any i satisfies v i = sup j∈Z v j = inf j∈Z v j and, by (2.6) and Item 2, we must have g δx [v] i ≥ 0 and g δx [v] i ≤ 0, which proves Item 3. By assumptions (2.5) and (2.8), there exists (β δx j ) j=-A δx ,...,A δx such that, for all v ∈ l ∞ (Z),

g δx [v] 0 = |j|≤A δx β δx j v j .
Let |j| ≤ A δx , j = 0 and v ∈ l ∞ (Z) be defined by v j = -1 and v i = 0 if i = j; applying (2.6) with i k ≡ 0 (we have v 0 = 0 = sup i∈Z v i ), we obtain 0 ≤ g δx [v] 0 = -β δx j which proves that, for all j = 0, β δx j ≤ 0. From the invariance by translation ((2.7) and Item 1), we also have

g δx [v] i = (τ i g δx [v]) 0 = g δx [τ i v] 0 = |j|≤A δx β δx j (τ i v) j = |j|≤A δx β δx j v i+j .
But Item 3 implies |j|≤A δx β δx j = 0 and thus β δx 0 =j =0 β δx j , which gives

g δx [v] i = j =0 β δx j v i+j + β δx 0 v i = j =0 β δx j v i+j -   j =0 β δx j   v i = j =0 β δx j (v i+j -v i ) = |j|≤A δx β δx j (v i+j -v i ).
Item 4 follows if we define µ δx 0 = 0 and, for j

∈ [-A δx , A δx ]\{0}, µ δx j = -β δx j . If η is convex then η(v i+j ) -η(v i ) ≥ η (v i )(v i+j -v i )
and Item 5 is thus a corollary of Item 4. Item 6 is also an immediate consequence of Formula (3.1) and, to prove Item 7, we simply write

N i=-N g δx [v] i = |j|≤A δx β δx j N i=-N v i+j - N i=-N v i = 0≤j≤A δx β δx j N +j i=N +1 v i - -N +j-1 i=-N v i + -A δx ≤j<0 β δx j   -N -1 i=-N +j v i - N i=N +j+1 v i   and thus N i=-N g δx [v] i ≤ 4 |j|≤A δx jβ δx j × sup N -A δx ≤|i|≤N +A δx |v i |.
It remains to prove Item 8. First, by (3.1), it is easy to see that (g δx ) * satisfies the same formula with µ δx -j instead of µ δx j ; hence, if w ∈ l 1 (Z) then (g δx ) * [w] is also in l 1 (Z). By definition of (g δx ) * and linearity of g δx , (3.3) is true if v has only a finite number of non-zero terms; since we can approximate, term by term, any v ∈ l ∞ (Z) by such sequences which stay bounded in l ∞ (Z), (3.3) for a general v follows from Item 6.

Existence and uniqueness of an approximate solution

We prove in this section that there exists a unique solution to (2.1)-(2.2), and we establish a first series of properties of this solution.

Lemma 3.3 Under assumptions (2.5)-(2.8), for all α ≥ 0 and all h ∈ l ∞ (Z) there exists a unique solution v ∈ l ∞ (Z) to ∀i ∈ Z :

v i + αg δx [v] i = h i . (3.4)
Moreover, we have inf

i∈Z h i ≤ inf i∈Z v i ≤ sup i∈Z v i ≤ sup i∈Z h i (3.5)
and

i∈Z |v i | ≤ i∈Z |h i |. (3.6)
Proof of Lemma 3.3 Let us first prove (3.5) and the uniqueness of the solution. Let (i k ) k≥1 be a sequence in Z such that lim k→∞ v i k = sup i∈Z v i ; then applying (3.4) to i = i k and passing to the inferior limit as k → ∞ thanks to (2.6) we find sup i∈Z v i ≤ sup i∈Z h i . Doing the same along a subsequence which converges to inf i∈Z v i (see item 2 in Lemma 3.1), we obtain inf i∈Z v i ≥ inf i∈Z h i and (3.5) is proved. These inequalities show that if h = 0 then v = 0. System (3.4) being linear, this proves the uniqueness of its solution.

To prove the existence of a solution we consider, for m ≥ 1, the approximate problem

∀i ∈ Z : v m i + αg δx [v m ] i 1 [-m,m] (i) = h i (3.7)
where

1 [-m,m] (i) = 1 if |i| ≤ m and 1 [-m,m] (i) = 0 otherwise.
Using the same argument as before, we notice that any solution to (3.7) also satisfies (3.5) and, in particular, that this problem has at most one solution. Since (3.7) clearly defines v m i (equal to h i ) if |i| > m, solving this system comes down to solving a finite-dimensional square linear system (of size 2m + 1); the uniqueness of the solution therefore ensures its existence.

Since (v m ) m≥1 is bounded in l ∞ (Z) (it satisfies (3.5)), we can assume up to a subsequence that, for all i ∈ Z, (v m i ) m≥1 converges to some v i as m → ∞. We can then pass to the limit m → ∞ in (3.7) thanks to Item 6 in Lemma 3.1 to see that (v i ) i∈Z ∈ l ∞ (Z) thus defined satisfies (3.4).

We conclude by proving (3.6), assuming that h ∈ l 1 (Z) (otherwise nothing needs to be proved). Multiplying (3.4) by sgn

(v i ) = η (v i ) for η = | • | and using (3.2), we have |v i | + αg δx [|v|] i ≤ |h i |. Summing on i = -N, . . . , N , we deduce from Item 7 in Lemma 3.1 that N i=-N |v i | ≤ N i=-N |h i | + αC δx sup N -A δx ≤|i|≤N +A δx |v i | ≤ i∈Z |h i | + αC δx ||v|| l ∞ (Z) < +∞. (3.8)
Hence, v ∈ l 1 (Z) and lim |i|→∞ v i = 0. We infer that lim N →∞ sup N -A δx ≤|i|≤N +A δx |v i | = 0 and, letting N → ∞ in the first inequality of (3.8), this concludes the proof of (3.6).

We can now prove the existence and uniqueness of the solution to the scheme.

Corollary 3.4 (Existence and uniqueness of an approximate solution) Let δt > 0 and δx > 0. Under assumptions (2.3)-(2.8), there exists a unique bounded solution

(u n i ) n≥0 , i∈Z to (2.1)-(2.2). Moreover, it satisfies, for all n ≥ 1, inf R u 0 ≤ inf i∈Z u n i ≤ sup i∈Z u n i ≤ sup R u 0 . (3.9)

Proof of Corollary 3.4

As it is usual for schemes involving monotone fluxes, we re-write (2.2) in the following way:

u n+1 i + δtg δx [u n+1 ] i = u n i - δt δx (F (u n i , u n i+1 ) -F (u n i , u n i )) + δt δx (F (u n i-1 , u n i ) -F (u n i , u n i )) = u n i - δt δx F (u n i , u n i+1 ) -F (u n i , u n i ) u n i+1 -u n i (u n i+1 -u n i ) + δt δx F (u n i-1 , u n i ) -F (u n i , u n i ) u n i-1 -u n i (u n i-1 -u n i ).
Let us define

a n i = - δt δx F (u n i , u n i+1 ) -F (u n i , u n i ) u n i+1 -u n i and b n i = δt δx F (u n i-1 , u n i ) -F (u n i , u n i ) u n i-1 -u n i (if u n i+1 = u n i or u n i-1 = u n i ,
we let the corresponding coefficient be equal to zero). The scheme is thus equivalent to

u n+1 i + δtg δx [u n+1 ] i = u n i + a n i (u n i+1 -u n i ) + b n i (u n i-1 -u n i ) , (3.10)
which comes down to asking that u n+1 is the solution to (3.4) with α = δt and

h i = (1 -a n i -b n i )u n i + a n i u n i+1 + b n i u n i-1 . But, under (2.3) and (2.4), if u n satisfies (3.9) then a n i ≥ 0, b n i ≥ 0 and a n i + b n i ≤ 1; this means that h i is a convex combination of (u n j ) j∈Z and thus that inf i∈Z u n i ≤ inf i∈Z h i ≤ sup i∈Z h i ≤ sup i∈Z u n i .
Hence, reasoning by induction on n from (2.1), Lemma 3.3 ensures the existence and uniqueness of a bounded solution to (2.1)-(2.2), which satisfies moreover (3.9).

Compactness estimates

(2.3)-(2.8). If (u n i ) n≥0 , i∈Z is the solution to (2.1)-(2.2) then, for all n ≥ 1, i∈Z |u n i+1 -u n i | ≤ |u 0 | BV (R) . (3.11)
Proof of Proposition 3.5 Subtracting (3.10) for i + 1 and for i and since g δx commutes with the translation τ , we obtain, with

v n i = u n i+1 -u n i , v n+1 i + δtg δx [v n+1 ] i = (1 -a n i -b n i+1 )v n i + a n i+1 v n i+1 + b n i v n i-1 . Hence v n+1 is the solution to (3.4) with α = δt and h i = (1 -a n i -b n i+1 )v n i + a n i+1 v n i+1 + b n i v n i-1 and we deduce from (3.6) that i∈Z |v n+1 i | ≤ i∈Z |1 -a n i -b n i+1 | |v n i | + i∈Z |a n i+1 | |v n i+1 | + i∈Z |b n i | |v n i-1 |.
But the CFL (2.4) and Estimate (3.9) ensure that 1-a n i -b n i+1 ≥ 0, a n i+1 ≥ 0 and b n i ≥ 0 and we therefore find, by re-indexing the last two sums,

i∈Z |u n+1 i+1 -u n+1 i | ≤ i∈Z |u n i+1 -u n i |.
This estimate allows to conclude the proof by induction on n (because (3.11) is true for n = 0 from the definition of u 0 in (2.1), see [START_REF] Eymard | Finite Volume Methods, Handbook of Numerical Analysis[END_REF]).

Proposition 3.6 (Time estimates) Let δt > 0 and δx ∈]0, 1[. Assume that (2.3)-(2.9) hold and let u δt,δx = (u n i ) n≥0 , i∈Z be the solution to (2.1)-(2.2). Define u δt,δx as the affine by parts time interpolate of (u n i ) n≥0 , i∈Z :

∀t ∈ [nδt, (n + 1)δt] , ∀x ∈ R : u δt,δx (t, x) = t -nδt δt u n+1 (x) + (n + 1)δt -t δt u n (x).
Then for all K compact subset of R, there exists M K ≥ 0 not depending on δt or δx such that

||∂ t u δt,δx || L ∞ (]0,∞[;(C 2 K (R)) ) ≤ M K . (3.12) Remark 3.7 If λ < 1 then (1.2) shows that g[φ] ∈ L 1 loc (R) as soon as φ ∈ L ∞ (R) ∩ BV (R).
Hence, since Proposition 3.5 gives space BV estimates on u δt,δx , the choice of a proper approximation g δx and the scheme (2.2) could allow to deduce, in the case λ < 1, time-BV local estimates on u δt,δx (stronger estimates than (3.12)).

Proof of Proposition 3.6

We have

∂ t u = u n+1 -u n δt on [nδt, (n + 1)δt] × R. Let K be a compact subset in R and φ ∈ C 2 K (R); define Φ ∈ l ∞ (Z) by Φ i = 1 δx (i+1)δx iδx φ(x) dx. From (2.2) we deduce, for t ∈ [nδt, (n + 1)δt], R ∂ t u δt,δx (t, x)φ(x) dx = i∈Z δx δt (u n+1 i -u n i )Φ i = i∈Z (F (u n i-1 , u n i ) -F (u n i , u n i+1 ))Φ i - i∈Z δx g δx [u n+1 ] i Φ i . (3.13) Using (2.3), we have |F (u n i-1 , u n i ) -F (u n i , u n i+1 )| ≤ C 1 (|u n i+1 -u n i | + |u n i -u n i-1 |) with C 1 not
(F (u n i-1 , u n i ) -F (u n i , u n i+1 ))Φ i ≤ 2C 1 |u 0 | BV (R) ||φ|| L ∞ (R) . (3.14) Formula (1.2) clearly shows that g is continuous W 2,1 (R) → L 1 (R) and thus, since C 2 K (R) is continuously embedded in W 2,1 (R), there exists E K not depending on φ such that ||g[φ]|| L 1 (R) ≤ E K ||φ|| C 2
K ; using Item 8 in Lemma 3.1, (2.9) and (3.9), we deduce

i∈Z δx g δx [u n+1 ] i Φ i = i∈Z δx u n+1 i (g δx ) * [Φ] i ≤ ||u 0 || L ∞ (R) ||(g δx ) * [Φ]|| L 1 (R) ≤ ||u 0 || L ∞ (R) (θ K (1) + E K )||φ|| C 2 K . (3.15)
The proof is concluded by plugging (3.14) and (3.15) into (3.13).

Corollary 3.8 (Compactness of the approximate solution) Assume that (2.3) and (2.5)-(2.9) hold.

Then as δt > 0 and δx > 0 tend to 0 while satisfying (2.4), up to a subsequence the solution u δt,δx to

(2.1)-(2.2) converges in L 1 loc ([0, ∞[×R).
Proof of Corollary 3.8 Let δt > 0 and δx ∈]0, 1[ satisfy (2.4) and define u δt,δx as the affine interpolate of (u n i ) n≥0,, i∈Z as in Proposition 3.6. Estimate (3.11) show that, for all n ≥ 0,

|u n | BV (R) ≤ |u 0 | BV (R) ; since, for all t > 0, u δt,δx (t, •) is a convex combination of u n and u n+1 (for some n ≥ 0), we deduce that | u δt,δx (t, •)| BV (R) ≤ |u 0 | BV (R) and, by (3.9), that || u δt,δx || L ∞ (]0,∞[×R) ≤ ||u 0 || L ∞ (R) . For all compact K ⊂ R, the set S = { u δt,δx ; δt > 0 and δx > 0 satisfy (2.4)} is therefore bounded in L ∞ ([0, ∞[; L ∞ (K)
) and, by Proposition 3.6, the time derivatives of the functions in this set are bounded in

L ∞ ([0, ∞[; (C 2 K (R)) ). Since L ∞ (K) is compactly embedded in (C 2 K (R)) (because C 2 K (R) is compactly and densely embedded in L 1 (K)), we deduce that S is bounded in W 1,∞ ([0, ∞[; (C 2 K (R))
) and, by Aubin-Simon's compactness theorem (see [START_REF] Aubin | Un théorème de compacité[END_REF][START_REF] Simon | Compact sets in L p (0, T ; B)[END_REF]), that S is relatively compact in

L 1 loc ([0, ∞[; (C 2 K (R))
). For all t ≥ 0, denoting by n the integer such that t ∈ [nδt, (n + 1)δt[, we have u δt,δx (t, •) = u δt,δx (nδt, •) and the bound in

W 1,∞ ([0, ∞[; (C 2 K (R)) ) thus shows that ||u δt,δx (t, •) -u δt,δx (t, •)|| (C 2 K (R)) ≤ C 2 δt with C 2 not depending on δt or δx. The compactness of S in L 1 loc ([0, ∞[; (C 2 K (R))
) therefore shows that, as δt → 0, u δt,δx is also relatively compact in this space. By (3.11), for all t ≥ 0 we have |u δt,δx (t,

•)| BV (R) ≤ |u 0 | BV (R) which implies, by a classical result on BV functions, for all ξ ∈ R, ||u δt,δx (t, • + ξ) -u δt,δx (t, •)|| L 1 (R) ≤ |u 0 | BV (R) |ξ|.
Associated with the relative compactness, as δt → 0, of

u δt,δx in L 1 loc ([0, ∞[; (C 2 K (R))
) for all K ⊂ R compact, this estimate makes it possible to apply Lemma 7.5 in [START_REF] Chainais-Hillairet | Convergence analysis of a mixed finite volume scheme for an elliptic-parabolic system modelling miscible fluid flows in porous media[END_REF] (or more precisely the technique of proof of this lemma) to conclude that this relative compactness also holds in

L 1 loc ([0, ∞[; L 1 loc (R)).
Remark 3.9 If u 0 does not belong to BV (R), then it is not possible in general to directly prove strong space BV estimates, and thus strong compactness, for u δt,δx . In this situation, one has to invoke the convergence of u δt,δx in the non-linear L ∞ weak- * sense (i.e. in the sense of Young measures), to prove that the limit of u δt,δx is an entropy process solution to (1.1) (this is done thanks to some space weak BV estimates on u δt,δx ) and to check, following [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF], that this entropy process solution is unique (see the general method for pure scalar conservation laws in [START_REF] Eymard | Finite Volume Methods, Handbook of Numerical Analysis[END_REF]).

Convergence

We can now prove the convergence of the solution of (2.1)-(2.2) toward the solution of (1.1), as δt and δx tend to 0 while satisfying (2.4). By Corollary 3.8 and since u δt,δx is bounded in L ∞ ([0, ∞[×R), up to a subsequence we can assume that it converges toward some u weakly- * in L ∞ ([0, ∞[×R) and strongly in L p loc ([0, ∞[×R) for all p < ∞. We now show that any such limit u of u δt,δx is the unique (entropy) solution to (1.1), which implies that the whole family u δt,δx converges to this solution and concludes the proof of Theorem 2.3.

Let φ ∈ C 2 c ([0, ∞[×R) and define Φ n i = 1 δx (i+1)δx iδx φ(nδt, x) dx. Multiplying (2.
2) by δtΦ n i and summing on n and i (all these sums are finite since Φ n i is equal to zero for n or |i| large), we obtain

T 1 + T 2 + T 3 = 0 where T 1 = n≥0 i∈Z δx(u n+1 i -u n i )Φ n i , T 2 = n≥0 δt i∈Z (F (u n i , u n i+1 ) -F (u n i-1 , u n i ))Φ n i and T 3 = n≥0 δt i∈Z δxg δx [u n+1 ] i Φ n i
Let us study the limit of each of these terms. We have

T 1 = n≥1 δt i∈Z δxu n i Φ n-1 i -Φ n i δt - i∈Z δxu 0 i Φ 0 i = ∞ 0 R u δt,δx (t, x)Ψ δt,δx (t, x) dtdx - R u 0 (x)Φ 0 (x) dx
where Ψ δt,δx is equal to 0 on [0, δt[×R and to

Φ n-1 i -Φ n i δt on [nδt, (n + 1)δt[×[iδx, (i + 1
)δx[ for all n ≥ 1 and all i ∈ Z. By regularity of φ, as δt and δx tend to 0, Ψ δt,δx and Φ 0 converge respectively to -∂ t φ in L 1 ([0, ∞[×R) and to φ(0, •) in L 1 (R). The weak- * convergence of u δt,δx then shows that

T 1 → - ∞ 0 R u(t, x)∂ t φ(t, x) dtdx - R u 0 (x)φ(0, x) dx. (3.16)
To handle T 2 we write, thanks to (2.3),

T 2 = n≥0 δt i∈Z δxF (u n i , u n i+1 ) Φ n i -Φ n i+1 δx = n≥0 δt i∈Z δxf (u n i ) Φ n i -Φ n i+1 δx + n≥0 δt i∈Z δx(F (u n i , u n i+1 ) -f (u n i )) Φ n i -Φ n i+1 δx = ∞ 0 R f (u δt,δx (t, x))Θ δt,δx (t, x) dtdx + n≥0 δt i∈Z δx(F (u n i , u n i+1 ) -F (u n i , u n i )) Φ n i -Φ n i+1 δx (3.17)
where Θ δt,δx =

Φ n i -Φ n i+1 δx on [nδt, (n + 1)δt[×[iδx, (i + 1
)δx[ for all n ≥ 0 and all i ∈ Z; as δt and δx tend to 0, this function converges to -∂ x φ in L 1 ([0, ∞[×R) by regularity of φ. Moreover, by local Lipschitzcontinuity of f , uniform bound on u δt,δx and convergence of this function toward

u in L 1 loc ([0, ∞[×R), f (u δt,δx ) → f (u) in L 1 loc ([0, ∞[×R) while staying bounded in L ∞ ([0, ∞[×R); the convergence of f (u δt,δx ) thus also holds in L ∞ ([0, ∞[×R)
weak- * and we therefore see that the first term in the right-hand side of (3.17) tends to -

∞ 0 R f (u(t, x))∂ x φ(t, x) dt dx.
Regarding the second term, we invoke (3.11) and the regularity of φ to write

n≥0 δt i∈Z δx(F (u n i , u n i+1 ) -F (u n i , u n i )) Φ n i -Φ n i+1 δx ≤ Lip 2,u0 (F )C 3 0≤n<T /δt δt i∈Z δx|u n i+1 -u n i | ≤ Lip 2,u0 (F )C 3 T |u 0 | BV (R) δx
where C 3 only depends on φ and T is such that supp(φ) ⊂ [0, T ] × R (so that Φ n i = 0 if n ≥ T /δt). This last right-hand side tends to 0 with δx and we conclude that

T 2 → - ∞ 0 R f (u(t, x))∂ x φ(t, x) dt dx. (3.18)
The convergence of T 3 is pretty straightforward from Item 8 in Lemma 3.1 and assumption (2.9): we have

T 3 = n≥0 δt i∈Z δxu n+1 i (g δx ) * [Φ n ] i = T 0 R u δt,
(t, •) -g[φ δt (t, •)]|| L 1 (R) ≤ ||φ|| L ∞ ([0,∞[;C 2 K (R)) θ K (δx) where K is a compact set such that supp(φ) ⊂ [0, ∞[×K. As δt → 0, the regularity of φ ensures that φ δt → φ in L ∞ ([0, T ]; C 2 K (R)), and thus in L ∞ ([0, T ]; W 2,1 (R)); since g : W 2,1 (R) → L 1 (R) is linear continuous (see Formula (1.2)), this shows that g[φ δt ] → g[φ] in L ∞ ([0, T ]; L 1 (R))
. We deduce that, as δt and δx tend to 0,

Ω δt,δx → g[φ] in L ∞ ([0, T ]; L 1 (R)) → L 1 ([0, T ] × R). Passing to the limit in (3.19) by weak- * convergence in L ∞ ([0, ∞[×R) of u δt,δx , we find T 3 → T 0 R u(t, x)g[φ(t, •)](x) dtdx. (3.20)
Gathering (3.16), (3.18) and (3.20) in

T 1 + T 2 + T 3 = 0 leads to ∞ 0 R u(t, x)∂ t φ(t, x) dtdx + ∞ 0 R f (u(t, x))∂ x φ(t, x) dtdx - ∞ 0 R u(t, x)g[φ(t, •)](x) dtdx = R u 0 (x)φ(0, x) dx.
This proves that u is a weak solution to (1.1). If λ > 1, this weak solution is in fact the unique solution in the sense of Duhamel's formula, and thus also the unique smooth strong solution (see [START_REF] Droniou | Global solution and smoothing effect for a non-local regularization of an hyperbolic equation[END_REF]), and the proof is complete. If λ ≤ 1, we must modify the preceding reasoning to show, using (2.10), that u is an entropy solution to (1.1).

Under assumptions (2.3) and (2.4), (2.2) can be written

u n+1 i = u n i - δt δx F (u n i , u n i+1 ) + δt δx F (u n i-1 , u n i ) -δtg δx [u n+1 ] i = H(u n i-1 , u n i , u n i+1 ) -δtg δx [u n+1 ] i
where H is non-decreasing with respect to each of its variable on [inf R u 0 , sup R u 0 ] 3 and, for all κ ∈

[inf R u 0 , sup R u 0 ], H(κ, κ, κ) = κ. Denoting a b = max(a, b), we have in particular H(u n i-1 , u n i , u n i+1 ) ≤ H(u n i-1 κ, u n i κ, u n i+1 κ) and κ ≤ H(u n i-1 κ, u n i κ, u n i+1 κ
) and we deduce, examining separately the cases

u n+1 i ≤ κ and u n+1 i > κ, u n+1 i κ ≤ H(u n i-1 κ, u n i κ, u n i+1 κ) -1 ]κ,∞[ (u n+1 i )δtg δx [u n+1 ] i .
Similarly, if a⊥b = min(a, b),

u n+1 i ⊥κ ≥ H(u n i-1 ⊥κ, u n i ⊥κ, u n i+1 ⊥κ) -1 ]-∞,κ[ (u n+1 i )δtg δx [u n+1 ] i
and therefore

u n+1 i κ -u n+1 i ⊥κ ≤ H(u n i-1 κ, u n i κ, u n i+1 κ) -H(u n i-1 ⊥κ, u n i ⊥κ, u n i+1 ⊥κ) -1 ]κ,∞[ (u n+1 i ) -1 ]-∞,κ[ (u n+1 i ) δtg δx [u n+1 ] i . Defining η κ (s) = |s -κ| = s κ -s⊥κ, we have η κ (s) = 1 ]κ,∞[ (s) -1 ]-∞,κ[ (
s) (this selects the subdifferential of η κ equal to 0 at s = 0) and the definition of H thus leads to

δx δt (η κ (u n+1 i ) -η κ (u n i ))+ F (u n i κ, u n i+1 κ) -F (u n i ⊥κ, u n i+1 ⊥κ) -F (u n i-1 κ, u n i κ) -F (u n i-1 ⊥κ, u n i ⊥κ) + δxη κ (u n+1 i )g δx [u n+1 ] i ≤ 0.
Taking r > 0, applying (2.10) and using (3.2) for g δx λ,r (which satisfies the assumptions of Lemma 3.1), we find

δx δt (η κ (u n+1 i ) -η κ (u n i )) + F (u n i κ, u n i+1 κ) -F (u n i ⊥κ, u n i+1 ⊥κ) -F (u n i-1 κ, u n i κ) -F (u n i-1 ⊥κ, u n i ⊥κ) +δxg δx λ,r [η κ (u n+1 )] i + δxη κ (u n+1 i )g δx 0,r [u n+1 ] i ≤ 0. (3.21)
These inequalities (for all r > 0) are discrete versions of the entropy inequalities for (1.1) and it is quite straightforward to deduce from them that the limit u of u δt,δx satisfies the entropy inequalities for (1.1). Indeed, taking a non-negative φ ∈ C 2 c ([0, ∞[×R), defining Φ n i from φ as before, multiplying (3.21) by δtΦ n i and summing on n and i, we obtain T 4 + T 5 + T 6 + T 7 ≤ 0 where

• T 4 is T 1 with u n i replaced by η k (u n i ), • T 5 is T 2 with F (u n i , u n i+1 ) replaced by F (u n i κ, u n i+1 κ) -F (u n i ⊥κ, u n i+1 ⊥κ) and F (u n i-1 , u n i ) re- placed by F (u n i-1 κ, u n i κ) -F (u n i-1 ⊥κ, u n i ⊥κ),
• T 6 is T 3 with g δx replaced by g δx λ,r and u n+1 replaced by η k (u n+1 ) and

T 7 = n≥0 δt i∈Z δxη κ (u n+1 i )g δx 0,r [u n+1 ] i Φ n i .
Using the same techniques as in the study of convergence of T 1 , T 2 and T 3 , the strong convergence of u δt,δx to u allows us to see that, as δt and δx tend to 0,

T 4 → - ∞ 0 R η κ (u(t, x))∂ t φ(t, x) dtdx - R η κ (u 0 (x))φ(0, x) dx, (3.22)
T 5 → - ∞ 0 R (f (u(t, x) κ) -f (u(t, x)⊥κ)) ∂ x φ(t, x) dtdx (3.23)
and

T 6 → ∞ 0 R η κ (u(t, x))g λ,r [φ(t, •)](x) dtdx. (3.24)
Regarding T 7 , we have

T 7 = ∞ 0 R η κ (u δt,δx (t + δt, x))V δt,δx (t, x)Φ δt,δx (t, x) dtdx (3.25)
where V δt,δx = g δx 0,r [u n+1 ] i and Φ δt,δx = Φ n i on [nδt, (n + 1)δt[×[iδx, (i + 1)δx[ for all n ≥ 0 and all i ∈ Z. By (2.10), for all compact Q and all t ≥ 0, taking n ≥ 0 such that t ∈ [nδt, (n + 1)δt[, we have

||V δt,δx (t, •) -g 0,r [u n+1 ]|| L 1 (Q) ≤ ||u n+1 || L ∞ (R) γ r,Q (δx) ≤ ||u 0 || L ∞ (R) γ r,Q (δx).
From the definition of g 0,r and the convergence of u δt,δx to u we see that the function defined by g 0,r [u n+1 ] on [nδt, (n + 1)δt[×R converges to g 0,r [u] in L 1 loc ([0, ∞[×R), and we therefore deduce that, as δt and δx go to 0, V δt,δx also converges to g 0

,r [u] in L 1 loc ([0, ∞[×R). The convergence of u δt,δx to u in L 1 loc ([0, ∞[×R)
shows that u δt,δx (• + δt, •) also converges in this space to u and thus, up to a subsequence, a.e. on ]0, ∞[×R; but, for a.e. κ ∈ R, the measure of {(t, x) ∈]0, ∞[×R , u(t, x) = κ} vanishes and, since η κ is continuous on R\{κ}, we have, for such κ, η κ (u δt,δx (• + δt, •)) → η κ (u) a.e. on ]0, ∞[×R. Combined with the fact that |η κ | ≤ 1, the convergence of V δt,δx to g 0,r [u] in L 1 loc , the uniform convergence of Φ δt,δx to φ and the fact that the support of Φ δt,δx stays in a compact subset of [0, ∞[×R, this allows to pass to the limit in (3.25) to find

T 7 → ∞ 0 R η κ (u(t, x))g 0,r [u(t, •)](x)φ(t, x) dtdx. ( 3
∞ 0 R η κ (u(t, x))∂ t φ(t, x) dtdx + ∞ 0 R (f (u(t, x) κ) -f (u(t, x)⊥κ))∂ x φ(t, x) dtdx - ∞ 0 R η κ (u(t, x))g λ,r [φ(t, •)](x) dtdx - ∞ 0 R η κ (u(t, x))g 0,r [u(t, •)](x)φ(t, x) dtdx + R η κ (u 0 (x))φ(0, x) dx ≥ 0 (3.27)
where we recall that η κ (s) = |s -κ| and η κ (s ) → η κ on R as m → ∞, all these functions staying bounded on bounded subsets of R; we can therefore take the mean value of (3.27) applied to κ m and κ m and let m → ∞ to see that (3.27) is also satisfied with κ. This shows that u is the unique entropy solution to (1.1) (see [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF]) and concludes the proof. Remark 3.10 We could as well consider the multi-dimensional form of (1.1) (i.e. with N space dimensions instead of one); on cartesian grids, the adaptation of the preceding reasoning is straightforward; on unstructured grids, however, the schemes for scalar conservation laws are not necessarily TVD (total variation decreasing) and it is therefore not possible to directly prove Corollary 3.8: even if u 0 ∈ BV (R N ), we have then to rely on the techniques sketched in Remark 3.9. 4 Implementation of the numerical method 4.1 A few words on the resolution procedure

) = 1 ]κ,∞[ (s) -1 ]-∞,

Example of g δx

A space step δx > 0 being chosen, Formula (1.2) makes it easy to write a discretization of g: we approximate each integral sign using a basic quadrature rule on the mesh ([jδx, (j +1)δx[) j∈Z (for example the right rectangles for z > 0 and the left rectangles for z < 0 -this avoids the singularity of 1/|z| 1+λ at z = 0 and preserves the symmetry between z > 0 and z < 0) and we use a finite difference approximation of the derivative (for example a centered one). However, such an approximation would use all the (v j ) j∈Z in order to compute g δx [v] i ; in practical application, the considered functions are usually constant near -∞ and +∞: it is therefore safe to assume this when discretizing g and to use the mesh ([jδx, (j+1)δx[) j∈Z only up to |z| = J δx δx (for some integer J δx such that J δx δx → +∞ as δx → 0), approximating the remaining parts with two unbounded space steps ] -∞, -J δx δx] and [J δx δx, +∞[. This leads to

g δx [v] i = -c(λ) 0<|j|≤r/δx δx v i+j -v i -vi+1-vi-1 2δx jδx |jδx| 1+λ -c(λ) r/δx<|j|≤J δx δx v i+j -v i |jδx| 1+λ -c(λ) v i-J δx -1 -v i λ(J δx δx) λ -c(λ) v i+J δx +1 -v i λ(J δx δx) λ . (4.1)
But 0<|j|≤r/δx j |jδx| 1+λ = 0 by symmetry and we can in fact drop the discretization of the derivative:

g δx [v] i = -c(λ) 0<|j|≤r/δx δx v i+j -v i |jδx| 1+λ -c(λ) r/δx<|j|≤J δx δx v i+j -v i |jδx| 1+λ -c(λ) v i-J δx -1 -v i λ(J δx δx) λ -c(λ) v i+J δx +1 -v i λ(J δx δx) λ (4.2) = -c(λ) 0<|j|≤J δx δx v i+j -v i |jδx| 1+λ -c(λ) v i-J δx -1 -v i λ(J δx δx) λ -c(λ) v i+J δx +1 -v i λ(J δx δx) λ . (4.3)
This dropping of the discretization of the derivative is in concordance with the reason behind the existence of φ (x) in (1.2); in fact, g is essentially the principal value of (| • | 1-λ ) and g[φ] is therefore the limit as

ε → 0 of -c(λ) |z|≥ε φ(x+z)-φ(x) |z| 1+λ
dz (see [START_REF] Droniou | Fractal first order partial differential equations[END_REF]); the term φ (x)z is introduced on ] -r, r[ because, by symmetry, it does not modify this integral sign but makes it possible to write the limit as ε → 0 as an integral sign on R. In the framework of numerical analysis, there is no such question of principal value and integrability at 0, and the disappearance of the discrete derivative is therefore not surprising.

It is easy to prove the first properties (2.5)-(2.8) for g δx defined by (4.3). Indeed, if lim k→∞ v i k = sup j v j then, for all ε > 0 and for k large enough, v i k +j -v i k ≤ ε for all j ∈ Z and thus

g δx [v] i k ≥ -c(λ)ε 0<|j|≤J δx δx 1 |jδx| 1+λ - 2c(λ)ε λ(J δx δx) λ = -C(δx)ε
and (2.6) is obtained by taking the lim inf k→∞ of this inequality and then letting ε → 0. The linearity (2.5), the invariance by translation (2.7) and the dependence on a finite number of values (2.8) are obviously satisfied. The proof of Properties (2.9) and (2.10) is way more technical and is therefore given in the appendix (Lemma 5.1).

Choice of the parameters

The practical implementation of the scheme (2.1)-(2.2) requires to make some choices of truncation parameters. First of all, we cannot obviously compute the approximate solution on the whole of [0, ∞[×R, we have to select a bounded domain on which we intend to obtain the solution: assume that this domain is [0, T ] × [-D, D]. To simplify the presentation, we also assume that δt = T /N δt and δx = D/N δx for some integers N δt and N δx . If we forget for a moment the operator g δx in (2.2), we notice that the calculation of (u n+1 i ) |i|≤N δx (in order to obtain the approximate solution at time step n + 1 on [-D, D]) necessitates to know (u n i ) |i|≤N δx +1 (or (u n i ) |i|≤N δx +2 in the case of 4-points numerical fluxes instead of 2-points fluxes). Hence, the hyperbolic part of the scheme imposes to begin at t = 0 with the indexes |i| ≤ N δx + N δt (or N δx + 2N δt in the case of the 4-points scheme) in order to obtain the approximate solution at time t = T on [-D, D]; this is the discrete counterpart of the well-known finite speed propagation of the scalar conservation laws. But we must also consider the operator g δx , which makes of the scheme a non-trivial infinite linear system. The proof of Lemma 3.3 however gives a way to approximate the solution to (2.2): h i being the right-hand side of (3.10) and α being equal to δt, an approximation of (u n+1 i ) i∈Z is given by the solution to (3.7) for m "large enough"... but which m? It is not obvious to give an analytical answer to this question: it is possible, from (4.3), to estimate the convergence as m → ∞ of the solution of (3.7) to the solution of (3.4); however, this general estimate is very slow (of order ξ m/J δx δx for some ξ δx < 1) and imposing m using this error bound leads to unreasonable values. The same holds for the choice of J δx in the definition of g δx : it is easy to see that the difference between g δx defined by (4.3) and the same expression with an infinite series (J δx = +∞) is of order ||v|| l ∞ (Z) (J δx δx) -λ and thus, if we take J δx = -1/λ δx , that the error, in the definition of g δx , due to the truncation of the sum at J δx is of order at most ... however, the value thus chosen for J δx is not reasonable, especially if λ is small. These general findings are in concordance with the estimate on the infinite speed propagation phenomenon of (1.1): it is proved in [START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF] that the influence of u 0 (x) on u(1, y) decreases as |x -y| -λ (i.e. very slowly). However, in practical situations, things behave much better than the preceding reasoning might let believe (partly because the above bounds are quite rough, partly because the considered initial conditions are not any kind of function). Consider for example T = 0.5, D = 1, λ = 0.5, a Burgers flux f (s) = s 2 /2 and a Riemann initial condition u 0 (x) = 1 if x < 0 and u 0 (x) = -1 if x > 0 (we also use a 4-points MUSCL method based on the Godunov numerical flux, see [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF], instead of a simple 2-points fluxes in (2.2)). Due to the hyperbolic part of the equation, we compute the solution for at least the indexes |i| ≤ N δx + 2N δt , and it seems wise to take this value as a lower bound for the choice of m in (3.7) (in order that the non-local operator influences all the terms coming from the hyperbolic part of the 1, for various values of N δt and N δx (all satisfying the CFL condition associated with the MUSCL scheme), the difference between these solutions computed with the values m = N δx + 2N δt and m = 3(N δx + 2N δt ) (and in either case for J δx large enough to have a minimal interference): the very small difference between the two solutions shows that the choice m = N δ + 2N δx is sufficient to obtain, in most cases, a good approximate solution to the scheme. As for J δx , a minimal value appears to be 2m in order that, when solving (3.7), the computation of g δx [v m ] i takes into account all the (v m j ) j∈Z which are influenced by g δx in this system of equations. Here again, there is in fact little gain to be found in using a much larger value for J δx than this estimated minimum, as shown by Table 2 (in which we present the L ∞ difference of the solutions computed with m = N δx + 2N δt and either J δx = 2m or J δx = 6m). Fixing J δx = 2m seems sufficient to obtain acceptable numerical approximations. Notice that, once m and J δx are chosen, we know exactly which indexes are to be considered in the implementation: the indexes |i| ≤ m + J δx + 1 (this can be seen from (3.7), since the computation of g δx [v m ] i for all |i| ≤ m uses only (v m j ) |j|≤m+J δx +1 ).

N δx N δt m L ∞ difference

Efficient numerical computation of the solution

Once the truncation parameters m and J δx are chosen, computing an approximate solution to the scheme requires to solve the following systems of the kind (3.7):

∀i ∈ Z : u n+1 i + δtg δx [u n+1 ] i 1 [-m,m] (i) = h n i , (4.4)
where h n is obtained by an iteration of the scheme for the pure scalar conservation law, i.e.

h n i = u n i + δt δx F (u n i-1 , u n i ) -δt δx F (u n i , u n i+1 ). This system imposes u n+1 i = h n i for |i| > m; defining then v = (h n i 1 Z\[-m,m] (i)) |i|∈Z and W = (u n+1 i
) |i|≤m , (4.4) reduces to a square system of size 2m + 1 on W :

W + δtG δx W = (h n i -δtg δx [v] i ) |i|≤m , (4.5)
in which the matrix

G δx comes from g δx (G δx W = (g δx [ W ] i ) |i|≤m with W i = W i if |i| ≤ m and W i = 0 if |i| > m).
It is easy to see from the definition of g δx that G δx is a symmetric semi-definite positive (it is diagonal-dominant) Toeplitz matrix, and thus that the matrix I + δtG δx of (4.5) is symmetric definite positive Toeplitz; solving this system can therefore be done in an extremely fast way by using a preconditioned Conjugate Gradient method and multiplication algorithms coming from the FFT framework (see [START_REF] Chan | Conjugate gradient methods for Toeplitz systems[END_REF][START_REF] Van Loan | Computational frameworks for the fast Fourier transform[END_REF] and also [START_REF] Salapaka | Analysis of a circulant based preconditioner for a class of lower rank extracted systems[END_REF] for a possible adaptation to "more local" operators). Moreover, because of the Toeplitz form of g δx , it is also possible to use FFT-based algorithms to achieve a very fast computation of the right-hand side of (4.5).

Hence, the scheme (2.1)-(2.2) is not only a proper theoretical approximation of (1.1), but also a very efficient one in terms of computational cost. We now give some numerical results to show that this scheme also provides nice practical approximations of solutions to fractal conservation laws.

Numerical results

In the following numerical tests, we consider a Burgers flux f (s) = s 2 /2 and, in order to avoid introducing too much numerical diffusion, we use a 4-points MUSCL method based on the Godunov flux [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF] to compute the numerical fluxes associated with f . Except in Section 4.2.3, we present snapshots of the approximate solutions ( 3) at time T = 0.5 on the domain [-1, 1], computed with a space step δx = 6.67 × 10 -3 and a time step δt = 1.67 × 10 -3 (with our choices of initial conditions, these values satisfy the CFL associated with the MUSCL method); we use g δx given by (4.3), the parameters m and J δx being chosen according to the discussion in the preceding section (δx and δt correspond to the choices N δx = 150 and N δt = 300, so m = N δx + 2N δt = 750 and J δx = 2m = 1500). Note that δx = 6.67 × 10 -3 and δt = 1.67 × 10 -3 are not very small steps; thanks to the algorithms mentioned in Section 4.1.3, each of the following numerical test only takes a few seconds on a personal computer and it would not be a strong computational issue to reduce the size of the time-space grid. We choose to present the results using these values of δx and δt in order to show that the numerical outputs of the scheme are quite good even without using a very fine grid.

Shock preservation and creation

If λ > 1, the solution to (1.1) is C ∞ -regular for any bounded initial data (see [START_REF] Droniou | Global solution and smoothing effect for a non-local regularization of an hyperbolic equation[END_REF]). If λ < 1, however, it is proved in [START_REF] Alibaud | Occurrence and non-appearance of shocks in fractal Burgers equations[END_REF] that the diffusion properties of g are not always strong enough, when in presence of a Burgers flux, to smoothen discontinuous initial data; moreover, in this situation, even C ∞ -regular initial data can give rise to discontinuous solutions. These two different behaviors (smoothing or shock preservation) with respect to a discontinuous initial condition are illustrated in Figure 1 (in which the initial condition is of Riemann type: u 0 (x) = 1 if x < 0 and u 0 (x) = -1 if x > 0). The figure clearly shows that the solution corresponding to λ = 0.3 presents a shock at x = 0, whereas the solution for λ = 1.5 is smooth. The phenomenon of shock creation if λ < 1 is shown in Figure 2; in this test, we take a kind of initial data which, although Lipschitz continuous, ensures that the solution develops a shock in finite time (see [START_REF] Alibaud | Occurrence and non-appearance of shocks in fractal Burgers equations[END_REF]): u 0 (x) = min(1, max(-3x, -1)) (u 0 is in fact piecewise linear, with a strong negative slope around 0 which provokes the creation of a shock; we could have smoothen u 0 around its slope discontinuities at x = -1/3 and x = 1/3 without changing much the behavior of the solution).

Speeds of diffusion

Let us consider for a moment the pure fractal equation, i.e. f = 0 in (1.1). It is known that, for any λ ∈]0, 2[, the solution to ∂ t u + g[u] = 0 is regular. The diffusive effects of the operator g, which explains this regularizing effect, however depend on the value of λ; indeed, taking the Fourier transform of ∂ t u+g[u] = 0 we see that ∂ t F(u)+|ξ| λ F(u) = 0: thus, during the evolution, the larger λ the more high frequencies are reduced and the less low frequencies are diffused. This property explains in particular the different behaviors in presence of a Burgers flux with respect to shocks (Section 4.2.1), but is also illustrated, for the pure fractal equation, in Figure 3: the initial data used in this test (-1 if x < 0, +1 if x > 0) has mainly low frequencies and is globally less diffused for a higher λ, except around the discontinuity (high frequency) where the smoothing is stronger (the slope of the solution is smaller).

-1.00

1.00 The presence of a flux can also interact with the different diffusive properties of g for various λ. If, keeping the same non-decreasing discontinuous initial data, we add a Burgers flux (i.e. we consider (1.1) with f (s) = s 2 /2), then the hyperbolic part of the equation generates a rarefaction wave: the initial shock is transformed into a piecewise-linear solution; the high frequencies are therefore killed by the flux and it can be seen in Figure 4 that the behaviors of the solutions for various λ no longer differ around the initial shock (in fact, from (1.2) we can see that g vanishes on affine functions for any λ). The stronger diffusive effect for low λ is however still perceptible in the zones of lower frequencies of the solution.

Asymptotic behavior

In [START_REF] Biler | Asymptotics for conservation laws involving Lévy diffusion generators[END_REF], the asymptotic behavior as t → ∞ of the solution to

∂ t v + ∂ x (f (v)) + g[v]
-∆v = 0 is studied; the addition, with respect to (1.1), of the Laplacian term provokes little disturbance in the long-time behavior of the solution and the results of this reference are also valid for (1.1). Let us try and illustrate them with the help of the numerical scheme.

We take λ = 0.5, and an initial data u 0 equal to 1 on [-0.2, 0.2] and to 0 elsewhere; the time-space domain of discretization is [0, 30] × [-1, 1] and, to avoid that the rarefaction wave and the shock generated by a Burgers flux for u 0 leave the domain of study, we reduce the strength of the flux by taking f (s) = s 2 /6. Denoting by K(t, x) the kernel of

∂ t + g = 0 (i.e. ∂ t K(t, x) + g[K(t, •)](x) = 0 and K(t, •) → δ 0 as t → 0), it is proved in [7] that t 1/λ [u(t) -K(t) * u 0 ] → 0 in L ∞ (R) as t → ∞.
Figures 5 and6 illustrate this property, by showing on one side the functions x → u(t, x) and x → K(t) * u 0 (x) for various times, and on the other side the plot in log-log scale of the L ∞ norm of t 1/λ [u(t) -K(t) * u 0 ] on [-1, 1] versus the time; the approximations of u and K * u 0 (solution to (1.1) with f = 0) used to draw these figures have been computed on [0, 30] × [-1, 1] using the numerical scheme with N δx = 100 and N δt = 4000. It is proved in [START_REF] Biler | Asymptotics for conservation laws involving Lévy diffusion generators[END_REF] that the next term in the asymptotic expansion of u(t) is of order t -2/λ , i.e. that ||t 1/λ [u(t) -K(t) * u 0 ]|| L ∞ (R) = O(t -1/λ ) as t → ∞, and the reference slope t → t -1/λ in Figure 6 confirms this (see below regarding the change of behavior after t = 10).

In fact, the second term in the asymptotics of u(t) is known: it is proportional to ∂ x K(t) (see [START_REF] Biler | Asymptotics for conservation laws involving Lévy diffusion generators[END_REF]); the numerical capture of this term is however quite challenging. Indeed, since K(t, x) = t -1/λ K(1, t -1/λ x), we have ∂ x K(t, x) = t -2/λ ∂ x K(1, t -1/λ x) and ∂ x K(t) is of order t -2/λ in L ∞ (R) but, as ∂ x K(1, 0) = 0, its maximal absolute values are attained at points x which go to ±∞ with t 1/λ ; restricted to [-1, 1], ∂ x K(t) is in fact of order t -3/λ and can thus interact with a possible third -and yet unknownterm in the expansion; this behavior is in concordance with the acceleration of convergence which clearly appears in Figure 6: restricted to [-1, 1], t 1/λ [u(t) -K(t) * u 0 ] seems to be asymptotically more of order t -2/λ than t -1/λ . Numerically illustrating the second term in the asymptotics of u(t) would therefore require to approximate this solution on a large time-space scale (including the extremal values of ∂ x K) and with a very high degree of precision (so that the numerical error is negligible with respect to t -2/λ ), which is beyond standard computational power.

Notice that this problem does not appear for the first term K(t) * u 0 in the asymptotic expansion: its maximum absolute value is of order t -1/λ and is attained in [-1, 1] for all t > 0; a reasonable approximation of u(t) on [-1, 1] thus suffices to capture this term. This is shown in Figure 6, and also confirmed if look at the relative L ∞ error on [-1, 1] between u(t) and K(t) * u 0 : for t = 1, this computed error

||u(t)-K(t) * u0|| L ∞ ([-1,1]) ||K(t) * u0|| L ∞ ([-1,1])
is around 0.63, whereas it is around 0.047 for t = 10 and around 0.0014 for t = 30. We are thus confident that the numerical scheme really has captured the proximity of u(t) and K(t) * u 0 for t large, not only a small quantity due to the difference of two small functions.

About the explicit scheme

The explicit form of the scheme consists in replacing (2.2) with

∀n ≥ 0 , ∀i ∈ Z : δx δt (u n+1 i -u n i ) + F (u n i , u n i+1 ) -F (u n i-1 , u n i ) + δxg δx [u n ] i = 0. (4.6)
The computation of the approximate solution to this scheme, on the contrary to the implicit scheme, does not require to solve a linear system at each time step (a truncation parameter m is however still needed for the practical implementation), but it is known that the CFL condition, ensuring the L ∞ stability of

6: Plot in log-log scale of t → ||t 1/λ [u(t) -K(t) * u 0 ]|| L ∞ ([-1,1]) (continuous line) and of t → t -1/λ (dashed line), for λ = 0.5, u 0 (x) = 1 [-0.2,0.2] (x) and u solution to ∂ t u + ∂ x (u 2 /6) + g[u] = 0.
the method, is usually more binding than in the implicit case. Using Formula (3.1), we can make this condition precise: with the same notations as in the proof of Corollary 3.4, and since the index j = 0 in (3.1) plays no role, the equivalent of (3.10) for the explicit scheme gives

u n+1 i =   1 -a n i -b n i -δt 0<|j|≤A δx µ δx j   u n i + a n i u n i+1 + b n i u n i-1 + δt 0<|j|≤A δx µ δx j u n i+j .
A sufficient condition for the L ∞ stability of the scheme is that u n+1 i is a convex combination of (u n j ) j∈Z , which is, from the definition of a n i and b n i , ensured by the preceding relation if

δt δx Lip 1,u0 (F ) + Lip 2,u0 (F ) + δt 0<|j|≤A δx µ δx j ≤ 1.
For the particular example of g δx given by (4.3), this comes down to

δt δx Lip 1,u0 (F ) + Lip 2,u0 (F ) + δt (δx) λ c(λ)   0<|j|≤J δx 1 |j| 1+λ + 2 λJ λ δx   ≤ 1. (4.7)
This condition on the time and space steps is more restrictive than (2.4), but in general not terribly more since c(λ) is small (c(0.5) ≈ 0.08, c(1) ≈ 0.05, c(1.5) ≈ 0.02); this is especially true if λ < 1: asymptotically as the space step tends to 0, the term coming from the hyperbolic part of the equation is then leading in (4.7). This is however the opposite if λ > 1, and this CFL condition is also very sensitive to the ratio diffusion/hyperbolic flux: if the hyperbolic flux is smaller than the diffusion term (e.g. if Lip u0 (f ) is small -which entails in general that Lip 1,u0 (F ) + Lip 2,u0 (F ) is also small -or if we multiply g in (1.1) by a coefficient), (4.7) can be much more demanding than (2.4); at the level of discretization used in the preceding tests and for λ > 1, a ratio of 5 between the coefficient of g and Lip u0 (f ) is enough to find a noticeable difference between these two CFL (recall also that the g we used is in fact g = (2π) -λ (-∆) λ/2 ).

From a practical point of view, if the parameters are chosen so that the explicit scheme is stable (in which case the implicit scheme is of course also stable), the solutions given by both forms (explicit and implicit) of the scheme are very similar: for example, in the preceding numerical tests, the relative L ∞ norm of their difference is lower than 3 × 10 -4 .

Since the explicit scheme demands to compute g δx [u n ] at each time step, its implementation can take advantage of FFT-based algorithms as in Section 4.1.3. In fact, as explained in this section, the implicit scheme also requires such a computation at each time step, and the difference of cost between the explicit and implicit methods therefore lies in the preconditioned CG iterations needed to solve the system in the implicit case. Although very fast, these CG iterations are not negligible in the overall cost and, in situations where (4.7) is not much more demanding than (2.4), the explicit scheme is clearly faster than the implicit scheme. However, for a diffusion-dominated problem (for example f (s) = s 2 /2, λ = 1.5 and (2π) λ g = (-∆) λ/2 instead of g), (4.7) can impose a much smaller time step than (2.4) and the implicit method then remains way more efficient than the explicit method.

5 Appendix: technical lemmas Lemma 5.1 If J δx is such that J δx δx → +∞ as δx → 0, then g δx defined by (4.3) satisfies (2.9) and (2.10).

Proof of Lemma 5.1

Step 1: proof of (2.9). We notice first, from (4.3), that (g δx ) * = g δx . Let K be a compact subset of R and define

A δx : C 2 K (R) → L 1 (R) by A δx φ = g δx [Φ] -g[φ]
, where Φ is defined from φ as before (2.9) ( 4 ). Proving (2.9) is equivalent to proving that A δx → 0 in L(C 2 K (R); L 1 (R)) as δx → 0, which we intend to do by applying Lemma 5.3 (stated after this proof). Let r > 0 and x ∈ R; by definition, choosing i ∈ Z such that x ∈ [iδx, (i + 1)δx[ we have

g δx [Φ](x) = -c(λ) 0<|j|≤r/δx δx Φ i+j -Φ i |jδx| 1+λ -c(λ) r/δx<|j|≤J δx δx Φ i+j -Φ i |jδx| 1+λ -c(λ) Φ i-J δx -1 -Φ i λ(J δx δx) λ -c(λ) Φ i+J δx +1 -Φ i λ(J δx δx) λ = -c(λ) 0<|j|≤r/δx δx |jδx| 1+λ 1 δx (i+1)δx iδx φ(ξ + jδx) -φ(ξ) dξ -c(λ) r/δx<|j|≤J δx δx |jδx| 1+λ 1 δx (i+1)δx iδx φ(ξ + jδx) -φ(ξ) dξ -c(λ) 1 λ(J δx δx) λ 1 δx (i-J δx )δx (i-J δx -1)δx φ(ξ) dξ - (i+1)δx iδx φ(ξ) dξ -c(λ) 1 λ(J δx δx) λ 1 δx (i+J δx +2)δx (i+J δx +1)δx φ(ξ) dξ - (i+1)δx iδx φ(ξ) dξ .
The same way we went from (4.1) to (4.2), we can add to each term in the first sum of the right-hand side anything of the form p i jδx |jδx| 1+λ without changing the value of the sum (these additional terms cancel out each other by symmetry). We choose to add - We then define the operators A δx 0,r and A δx 2,r by A δx 0,r φ = T δx,r The definition of g 0,r clearly shows that A δx 0,r φ is defined for any φ ∈ C 0 K (R). For all x ∈ R we have We handle T δx,r 8

[φ] in a similar way: integrating its definition with respect to x and using a comparison between discrete and integral sums, we find Together with (5.4), this proves that A δx 2,r satisfies Item 3 in Lemma 5.3 and concludes the proof of (2.9).

||T δx,r
Step 2: proof of (2.10) The cutting of g δx in g δx λ,r and g δx 0,r is of course the one given by (4.2), g δx λ,r being the first sum and g δx 0,r the rest of the right-hand side. The proof that g δx λ,r satisfies (2.5)-(2.8) and (2.9) with g λ,r instead of g is done exactly as for g δx (the proof of (2.9) is done by cutting the sum defining g δx λ,r at a level r /δx with r < r, by introducing the derivative of φ in the lower part of the sum and by replacing, in the reasoning of Step 1 above and in Lemma 5.3, r with r ). Let us study g δx 0,r . For all v ∈ l ∞ (Z) and all x ∈ R, choosing i ∈ Z such that x ∈ [iδx, (i + 1)δx[ we have (notice that if z ∈ E j (i, x) then |z| > r/2). We deduce

δx v i+j -v i |jδx| 1+λ - Ej (i,x) v(x + z) -v(x) |z| 1+λ dz ≤ 2||v|| l ∞ (Z) δx(1 + λ) Ej (i,x) dz (|z| -r/4) 2+λ
and, plugging this into (5.5) and defining E(i, x) = ∪ r/δx<|j|≤J δx E j (i, x) ⊂ {|z| > r/2}, g δx 0,r [v](x) + c(λ)

E(i,x) v(x + z) -v(x) |z| 1+λ dz ≤ 2||v|| l ∞ (Z) δx(1 + λ)

Lemma 3 . 1

 31 If g δx satisfies (2.5)-(2.8) then:1) g δx commutes with the right translation τ -1 : (

Proposition 3 . 5 (

 35 BV estimates) Let δt > 0 and δx > 0 and assume

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Smoothing effect for λ > 1 and preservation of shock for λ < 1 (the dotted line is the common initial condition of these tests).

Figure 4 :

 4 Figure 4: Solutions, for various λ, to (1.1) in the case of an initial data and a flux f (s) = s 2 /2 generating a rarefaction wave.

Figure 5 :

 5 Figure 5: Initial condition u 0 (dotted line), K(t) * u 0 (dashed line) and u(t) solution to ∂ t u + ∂ x (u 2 /6) + g[u] = 0 (continuous line), for λ = 0.5 and various times.

Figure

  Figure 6: Plot in log-log scale oft → ||t 1/λ [u(t) -K(t) * u 0 ]|| L ∞ ([-1,1]) (continuous line) and of t → t -1/λ (dashed line), for λ = 0.5, u 0 (x) = 1 [-0.2,0.2] (x) and u solution to ∂ t u + ∂ x (u 2 /6) + g[u] = 0.

( 8 [

 8 i+1)δx iδx φ (ξ) dξ jδx |jδx| 1+λ and we obtaing δx [Φ](x) = -c(λ) + jδx) -φ(ξ) -φ (ξ)jδx dξ -c(λ) + jδx) -φ(ξ) dξ -c(λ) 1 λ(J δx δx) λ 1 δx (i-J δx )δx (i-J δx -1)δx φ(ξ) dξφ](x) + T δx,r 9 [φ](x) + T δx 10 [φ](x) + T δx 11 [φ](x).

9 [ 8 [

 98 φ] + T δx 10 [φ] + T δx 11 [φ] -g 0,r [φ] and A δx 2,r φ = T δx,r φ] -g λ,r [φ].

1 |jδx||T δx,r 9 [ 9 [ 1 0 ( 1 -

 19911 |T δx 10 [φ](x) + T δx 11 [φ](x)| ≤ c(λ) λ(J δx δx) λ sup |s|≤δx |φ(x -J δx δx -δx + s)| + 2 sup |s|≤δx |φ(x + s)| + sup |s|≤δx |φ(x + J δx δx + δx + s)| ,which shows, integrating and using some changes of variables, that||T δx 10 [φ] + T δx 11 [φ]|| L 1 (R) ≤ 4c(λ) λ(J δx δx) λ R sup |s|≤δx |φ(z + s)| dz ≤ 4c(λ) λ(J δx δx) λ ||φ|| L ∞ (R) mes(K + [-δx, δx])and thus, since J δx δx → ∞ as δx → 0,T δx 10 + T δx 11 → 0 in L(C 0 K (R); L 1 (R)) as δx → 0. (5.1)We let I(j, δx) = [jδx, (j + 1)δx[ if j > 0 and I(j, δx) =](j -1)δx, jδx] if j < 0, and we define H δx : R → R by: for all r/δx < |j| ≤ J δx , H δx = 1+λ on I(j, δx), andH δx = 0 on R\ ∪ r/δx<|j|≤J δx I(j, δx). By regularity of z → |z| -1-λ on |z| ≥ r, we have, if δx < r/2, ∀z ∈ r/δx<|j|≤J δx I(j, δx) : H δx (z) -1 |z| 1+λ ≤ (1 + λ)δx sup |s|≤δx 1 |z + s| 2+λ ≤ δx 1 + λ (|z| -r/2) 2+λ . Since r/δx<|j|≤J δx I(j, δx) = {z ∈ R | r + α r,δx δx ≤ |z| < J δx δx + δx} for some α r,δx ∈ [0, 1], we deduce that H δx → 1 |z| 1+λ in L 1 (|z| > r) as δx → 0; for all φ ∈ C 0 K (R)and all x ∈ R, by uniform continuity of φ we infer thatT δx,r 9 [φ](x) → -c(λ) |z|>r φ(x + z) -φ(x) |z| 1+λ dz = g 0,r [φ](x) as δx → 0. (5.2) Moreover, still assuming that δx < r/2, φ](x)| ≤ c(λ) |z|>r H δx (z) sup |s|≤r |φ(x + z + s)| + sup |s|≤r/2 |φ(x + s)| dz (5.3)and, by convergence of H δx in L 1 (|z| > r), the right-hand side of (5.3) converges, as a function of x, in L 1 (R) as δx → 0. The dominated convergence theorem and (5.2) then show T δx,r9[φ] → g 0,r [φ] in L 1 (R) and, together with (5.1), this proves that A δx 0,r satisfies Item 1 in Lemma 5.3.Since (Hδx ) δx∈]0,r/2[ is bounded in L 1 (|z| > r), (5.3) gives ||T δx,r φ]|| L 1 (R) ≤ C 4 ||φ|| L ∞ (R) mes(K + [-r, r])with C 4 not depending on φ or δx ∈]0, r/2[; recalling (5.1) and since g 0,r ∈ L(C 0 K (R); L 1 (R)) (see the definition of g 0,r ), this shows that A δx 0,r satisfies Item 2 in Lemma 5.3. Let us now turn to A δx 2,r . Writing φ(x + z) -φ(x) -φ (x)z = s)φ (x + sz)z 2 ds, we have ||g λ,r [φ]|| L 1 (R) ≤ c(λ)||φ || L 1 (R) |z|≤r |z| 1-λ dz ≤ ||φ || L 1 (R)

8 [

 8 φ](x)|| L 1 (R) ≤ c(λ) 0<|j|≤r/δx δx |jδx| 1+λ R |φ(ξ + jδx) -φ(ξ) -φ (ξ)jδx| dξ ≤ c(λ) 0<|j|≤r/δx δx |jδx| 1+λ ||φ || L 1 (R) |jδx| 2 ≤ ||φ || L 1 (R) c(λ) |z|≤r+δx |z| 1-λ dz ≤ ||φ || L 1 (R) 2c(λ) 2 -λ (r + δx) 2-λ .

  g δx 0,r [v](x) = -c(λ) r/δx<|j|≤J δx δx v i+j -v i |jδx| 1+λ -c(λ) v i-J δx -1 -v i λ(J δx δx) λ -c(λ) v i+J δx +1 -v i λ(J δx δx) λ . (5.5) Let |j| > r/δx. We have x + z ∈ [(i + j)δx, (i + j + 1)δx[ (which implies v(x + z) = v i+j ) if and only if z ∈ [jδx, (j + 1)δx[+(iδx -x) =: E j (i, x), in which case |z -jδx| ≤ δx and, if δx ≤ r/4, 1 |z| 1+λ -1 |jδx| 1+λ ≤ (1 + λ)δx sup |s|≤δx 1 |z + s| 2+λ ≤ δx(1 + λ) 1 (|z| -r/4) 2+λ

|z|>r/ 2 dz

 2 (|z| -r/4) 2+λ + 4c(λ)||v|| l ∞ (Z) λ(J δx δx) λ . But E(i, x) = [-J δx δx + ρ 1 δx, -r + ρ 1 δx[∪[r + ρ 2 δx, J δx δx + ρ 2 δx[ with (ρ 1 , ρ 1 , ρ 2 , ρ 2 ) ∈ [-1,[START_REF] Alibaud | Entropy formulation for fractal conservation laws[END_REF] and the symmetric difference between E(i, x) and {|z| > r} is therefore contained in {r -δx ≤ |z| ≤ r +δx}∪{|z| ≥

  Moreover, if λ > 1 or if (2.10) holds, then, as δt and δx tend to 0 (while satisfying (2.4)), u δt,δx → u weakly- * in L ∞ ([0, ∞[×R) and strongly in L p loc ([0, ∞[×R) for all p < ∞, where u is the unique entropy solution to (1.1).

	Time and space steps δt and δx being given, in a similar way as above we identify a family (u n i ) n≥0 , i∈Z with the function u δt,δx : [0, ∞[×R → R equal to u n i on [nδt, (n + 1)δt[×[iδx, (i + 1)δx[, and u n : R → R is the function equal to u n i on [iδx, (i + 1)δx[. Our main result is the following.
	Theorem 2.3 (Existence, uniqueness and convergence of the approximate solution) Assume that (2.3)
	and (2.5)-(2.9) hold. Then, for all δt > 0 and all δx > 0 satisfying (2.4), there exists a unique bounded
	solution u δt,δx = (u n i ) n≥0 , i∈Z to (2.1)-(2.2).

  depending on δt, δx, n or i, and (3.11) therefore gives

	i∈Z

  δx (t + δt, x)Ω δt,δx (t, x) dtdx(3.19) where Ω δt,δx = (g δx ) * [Φ n ] i on [nδt, (n + 1)δt[×[iδx, (i + 1)δx[ for all n ≥ 0 and all i ∈ Z and T is as before. Let φ δt : [0, ∞[×R → R be the function equal to φ(nδt, •) on [nδt, (n + 1)δt[×R for all n ≥ 0. From (2.9) we have, for all t ≥ 0, ||Ω δt,δx

  .26) Gathering (3.22), (3.23), (3.24) and (3.26) in T 4 + T 5 + T 6 + T 7 ≤ 0, we conclude that

  κ[ (s). This inequality has been proved up to now only for almost every κ ∈ R; but for any κ ∈ R we can choose (κ m ) m≥1 and ( κ m ) m≥1 such that (3.27) is valid with κ = κ m and κ = κ m and such that κ m κ and κ m κ, and we have then

	1 2 (η κm + η κm ) → η κ and 1 e 2 (η κm + η

e κm

Table 1 :

 1 L ∞ difference between the approximate solutions computed with m = N δt + 2N δx and with m = 3(N δt + 2N δx ) (and J δx = 4m in either case).

	50 100 250 and 750	2.02E-4
	100 200 500 and 1500	2.05E-4
	150 300 750 and 2250	2.06E-4
	N δx N δt	J δx	L ∞ difference
	50 100 500 and 1500	1.76E-5
	100 200 1000 and 3000	8.82E-6
	150 300 1500 and 4500	5.88E-6

Table 2 :

 2 L ∞ difference between the approximate solutions computed with J δx = 2m and with J δx = 6m (and m = N δ + 2N δx in either case).

equation). To understand if a higher value of m can improve the precision of the approximate solution, we show in Table

Some of these properties (such as the invariance by translation (2.7)) being moreover stated and used only to simplify the presentation.

Or rather of affine interpolates of the constant-by-parts approximate solutions (these affine interpolates also converge to the solution of (1.1)).

It might not be straightforward that A δx takes its values in L 1 (R), because of the term g δx [Φ], but this will be made clear by the reasoning to come.

Acknowledgements: the author would like to thank the referee for his/her useful suggestions, which have led to a reduction of technicality in some reasonings (in particular in Step 1 of the proof of Lemma 5.1).

J δx δx -δx}; we conclude that

and Item 2 of (2.10) follows, the estimate being in fact valid in L ∞ (R) and not only in L 1 (Q).

Remark 5.2 It is also possible, by some direct estimates rather than using the abstract lemma 5.3, to give an explicit θ K such that (2.9) holds; such an expression could be useful, for example, to establish error estimates for the scheme (2.1)-(2.2). However, getting this θ K is much more technical than the arguments used in Step 1 of the preceding proof.

) the space of continuous (resp. twice continuously differentiable) functions with support included in K. Assume that (A δx ) δx>0 is a family of linear continuous operators C 2 K (R) → L 1 (R) such that, for all r > 0, we can write

Proof of Lemma 5.3

We take

Using the compactness of the embedding C 2 K (R) → C 0 K (R), we can assume that φ δx converges in C 0 K (R) to some φ as δx → 0. We then take r > 0 and write

+||A δx 2,r || L(C 2 K (R);L 1 (R)) + δx, and the conclusion follows by taking first the upper limit as δx → 0 and then the limit as r → 0.