Christophe Buet
email: christophebuet@cea.fr

Kim-Claire Le Thanh
email: kim-claire.le-thanh@cea.fr@cea.fr

Positive, conservative, equilibrium state preserving and implicit difference schemes

1 Introduction

The Fokker-Planck-Landau equation (FPLE) is commonly used in plasma physics when studying kinetic effects between charged particles under Coulomb interaction.

In particular, the isotropic Fokker-Planck-Landau operator is generally used in the modeling of inertial controlled fusion. More precisely, it is used to describe the electronic energy transport phenomena in laser produced plasma. In some conditions, it is well known that the fluid theory, for which the hydrodynamic equations are closed using the law for the thermal fluxes proposed by Spitzer-Harm [START_REF] Spitzer | Transport phenomena in a completely ionized gaz[END_REF] is not valid [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF]. A more accurate solution is to use a model based on the expansion of the FPL operator in spherical harmonics [START_REF] Shkarofsky | The Particle Kinetics of Plasmas[END_REF] and to retain the two first terms (P1 model), and the isotropic FPL operator is the leading order term [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF][START_REF] Kingham | An implicit Vlasov-Fokker-Planck code to model nonlocal electron transport in 2-D with magnetic fields[END_REF]. Within this inertial controlled fusion context P1 model is sufficient to describe physical phenomena and 3-D FPL equation unnecessarily complicates the issue. The most popular difference scheme for Fokker-Planck type equations is the Chang and Cooper method [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF]. This method was originally devoted to linear Fokker-Planck equations and it was shown in [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF] that in this case this method is positive and preserves the equilibrium states. On the contrary a paper of Larsen et al. [START_REF] Larsen | Discretization methods for one-dimensional Fokker-Planck operators[END_REF] shows that this method applied for non-linear Fokker-Planck equations could produce non-positive solutions. It is also used for the isotropic Landau equation by Langdon [START_REF] Langdon | Conservative differencing of the electron Fokker-Planck transport equation[END_REF], in the SPARK code by Epperlein [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF] and by Kingham and Bell [START_REF] Kingham | An implicit Vlasov-Fokker-Planck code to model nonlocal electron transport in 2-D with magnetic fields[END_REF] in the code IMPACT. At our knowledge, in this non-linear case, there is no rigorous proof of the positivity and the energy conservation. There exists also conservative or entropic schemes for this equation [START_REF] Berezin | Conservative finite difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy[END_REF][START_REF] Bobylev | On the numerical solution of Landau's kinetic equation[END_REF][START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Buet | Numerical analysis of the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Degond | An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory[END_REF][START_REF] Potapenko | A completely conservative difference scheme for the two-dimensional Landau equation[END_REF][START_REF] Potapenko | The completely conservative difference scheme for the nonlinear Landau-Fokker-Planck equation[END_REF] based on the symmetrized weak form or on the "Log" weak symmetric form of the Landau equation. Unfortunately on non-uniform grid in energy these methods suffer from some limitations. Some of them could not be positive [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Degond | An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory[END_REF], some others [START_REF] Berezin | Conservative finite difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy[END_REF][START_REF] Bobylev | On the numerical solution of Landau's kinetic equation[END_REF][START_REF] Potapenko | A completely conservative difference scheme for the two-dimensional Landau equation[END_REF][START_REF] Potapenko | The completely conservative difference scheme for the nonlinear Landau-Fokker-Planck equation[END_REF] are not equilibrium state preserving. And the one's using Logarithm of the distribution function [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Degond | An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory[END_REF] do not handle correctly highly peaked distribution functions. Spectral methods were also developed by Pareschi et al. [START_REF] Pareschi | Fast spectral methods for the Fokker-Planck-Landau collision operator[END_REF] and Filbet et al. [START_REF] Filbet | A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the nonhomogeneous case[END_REF] for the 3-D FPL equation and could be applied to the isotropic FPL equation. But these schemes approximate energy only with spectral accuracy, that means they are not conservative. Moreover discrete Maxwellians are not preserved and positivity is not ensured. Another disadvantage is that such method is not suitable to describe peaked functions.

It would be interesting to enlighten the situation about positive, conservative and equilibrium states preserving velocity difference schemes for the isotropic Landau equation. Conservation property is important. Especially conservation of the energy to avoid inopportune cooling or heating of the plasma. Preservation of equilibrium states, Maxwellian in this case, is important since it is the long time behavior for coupling with fluid limit or to have the right heat flux at equilibrium. But positivity is also important since it is true at the continuous level and for obvious physical reasons. Thus it would be in time discretization to ensure stability. It is quite surprising that this point is not analyzed in papers devoted to algorithms for the FPL equation [START_REF] Berezin | Conservative finite difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy[END_REF][START_REF] Bobylev | On the numerical solution of Landau's kinetic equation[END_REF][START_REF] Degond | An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory[END_REF][START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF][START_REF] Lemou | Implicit schemes for the Fokker-Planck-Landau equation[END_REF][START_REF] Potapenko | A completely conservative difference scheme for the two-dimensional Landau equation[END_REF][START_REF] Potapenko | The completely conservative difference scheme for the nonlinear Landau-Fokker-Planck equation[END_REF].

In the first part of this paper we focus our attention on the Chang and Cooper method. Contrary to the existing works, we start from the weak symmetrized form of the isotropic Landau operator, so that the conservation of energy is obvious. We construct two variants of the Chang and Cooper method, a new one, called S 1 , and an other one, called S 2 , that is described by Langdon et al. [8,[START_REF] Langdon | Conservative differencing of the electron Fokker-Planck transport equation[END_REF] in the Coulombian case but constructed directly from the Rosenbluth form of the operator. These two schemes only differ in the boundary condition. These two variants conserve the energy but we show that the second one, S 2 , is not positive. As we will also see, due to bad boundary conditions, the Chang and Cooper version used by Epperlein [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF] or by Kingham and Bell [17], although positive, is not conservative in energy.

In other hand we show that the Chang and Cooper method is not the only way to obtain positive, conservative and equilibrium preserving states even on non-uniform meshes. We propose a new scheme that shares the same properties with the Chang and Cooper method. This "equilibrium scheme", called S 3 , is based on the work of Larsen et al. [START_REF] Larsen | Discretization methods for one-dimensional Fokker-Planck operators[END_REF] for Fokker-Planck equations. In the case of uniform meshes this scheme is nothing but the conservative positive and entropy scheme developed by Berezin et al. [START_REF] Berezin | Conservative finite difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy[END_REF] and studied in detail by one of the authors [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Buet | Numerical analysis of the isotropic Fokker-Planck-Landau equation[END_REF]. Time discretization is also a crucial point in the numerical simulation of the Fokker-Planck-Landau equation. Implicit schemes are attractive: they make the code robust and able to use large time steps well exceeding the characteristic collision time. The explicit method cannot be eliminate, since for practical computations, the number of discretization points will lie between 20 and 150 approximately. Despite its parabolic time step restriction, it could be cheaper for coarse grids. We study several implicit schemes. We begin with the well known fixed point method, described in particular in [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF], that is conservative in mass and positive. However, this scheme is not conservative in energy. Then, we recall the contracted implicit method developed by Lemou and Mieussens [START_REF] Lemou | Implicit schemes for the Fokker-Planck-Landau equation[END_REF]. Their scheme is conservative in mass and energy, the positivity is lacking but it is cheap. The last method studied is the Newton method. The mass and energy conservation is ensured. For the "contracted implicit" method and the Newton method, we show that by rewriting the problem, in the particular case of coulombian potentials, the linear systems involved can be solved by direct method in only O(N ) operations (N is the number of unknowns). This strategy can be applied to Maxwellian potential for the isotropic FPL equation but also for the 3-D equation. The key point, for the Coulombian potentials, is to remember that drift and diffusion coefficients satisfy Poisson equation, which can be solved efficiently at a linear cost by direct method (see also [7]). In fact, such a strategy works as soon as the discrete operator can be evaluated using recurrences relations, and then at a linear cost . We show, on numerical examples, the gain in these implicit methods compared to the explicit time discretization in terms of CPU times. This paper is organized as follow: in the first part, we recall the continuous FPL equation in the homogeneous and isotropic case and its properties. In a second part we present the Chang and Cooper type schemes and an equilibrium scheme. We show all the properties of these schemes: energy and equilibrium states conservation, positivity. We also indicate the simplifications occurring in the case of a uniform mesh, or in the Coulombian case. The third part is devoted to the time discretization: explicit and implicit schemes. In the fourth part we present some numerical tests in the Coulombian case.

The isotropic Fokker-Planck-Landau equation

Let us recall the homogeneous non-linear Fokker-Planck-Landau equation (FPL equation) in the isotropic case where the distribution function f ( x, v, t) depends only on the modulus of the velocity v = v and on the time t, in other words f ( x, v, t) = f (v, t). We consider f as a function of ε = v 2 where ε is the energy variable. In this case, Fokker-Planck-Landau equation can be written:

∂f (ε, t) ∂t = Q(f )(ε) = 1 √ ε ∂ ∂ε ∞ 0 g(ε, ε ) f (ε ) ∂f (ε) ∂ε -f (ε) ∂f (ε ) ∂ε dε , (2.1)
where g(ε, ε ) is positive, symmetric and increasing (g(ε, ε ) = min(ε

3 2 , ε 3
2 ) for Coulombian interactions and g(ε, ε

) = ε 3 2 ε 3 2
for Maxwellian interactions) and Q(f ) is the so-called FPL collision operator. This operator can be written in the following weak form (let φ(ε) be any test function)

∞ 0 ∂f (ε, t) ∂t φ(ε) √ εdε = - 1 2 ∞ 0 ∞ 0 ( ∂φ(ε) ∂ε - ∂φ(ε ) ∂ε )g(ε, ε )(f (ε ) ∂f (ε) ∂ε -f (ε) ∂f (ε ) ∂ε )dε dε.
(2.2) Note that the right hand side of (2.2) can be equivalently written in the so-called "Log" weak form

- 1 2 ∞ 0 ∞ 0 ( ∂φ(ε) ∂ε - ∂φ(ε ) ∂ε )g(ε, ε )f (ε )f (ε)( ∂ log f (ε) ∂ε - ∂ log f (ε ) ∂ε )dε dε, (2.3)
or in (2.1), the collision operator can be written in the following diffusive form

Q(f )(ε) = 1 √ ε ∂ ∂ε (E(f )f (ε) + D(f ) ∂f ∂ε ), (2.4)
where the drift coefficient E and the diffusion coefficient D are given by

D(f ) = ∞ 0 g(ε, ε )f (ε )dε and E(f ) = - ∞ 0 g(ε, ε ) ∂f (ε ) ∂ε dε .
(2.5)

General properties

Let us recall the most important properties of the problem (2.1). This operator satisfies the conservation of mass (respectively energy) by choosing φ

(ε) = 1 (respectively φ(ε) = ε) in (2.2) ρ = ∞ 0 f (ε, t) √ εdε and ρE = ∞ 0 f (ε, t)ε 3 2 dε. (2.6)
Note that the conservation properties are a consequence of the symmetry property (between ε and ε ) of the collision operator. Let us also mention that the temperature T of the plasma is defined as 3 2 ρT = ρE.

Any function of the type ψ(ε) = α exp(-βε) where α and β are arbitrary constants (β > 0) is a stationary solution of equation (2.1). And we have the famous H-Theorem: for any solution f the entropy defined by

H(f ) = ∞ 0 f (ε) log(f (ε)) √ εdε, (2.7)
satisfies dH(f ) dt ≤ 0 with dH(f ) dt = 0 ⇐⇒ f (ε) = α exp(-βε), (2.8)
with α and β such that f and M = α exp(-βε) have the same mass and density.

For more details concerning the Landau equation we refer to [START_REF] Degond | The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case[END_REF][START_REF] Desvillettes | On Asymptotics of the Boltzmann Equation when the Collisions Become Grazing[END_REF][START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. Part I: Existence, uniqueness and smoothness[END_REF][START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. Part II: H-Theorem and applications[END_REF].

Coulombian interactions

In the Coulombian case, g(ε, ε ) = min(ε

3 2 , ε 3
2 ) thus the drift and diffusion coefficients E(f ) and D(f ) are given by

E(f ) = 3 2 ε 0 √ ε f (ε )dε and D(f ) = 3 2 ε 0 √ ε ∞ ε f (ε )dε dε . (2.9)
Deriving once E(f ) and twice D(f ) we have

f (ε) = 2 3 1 √ ε ∂E(f ) ∂ε = - 2 3 ∂ ∂ε 1 √ ε ∂ ∂ε D(f ).
(2.10)

These relations are nothing else than the Poisson equations satisfy by the Rosenbluth potentials.

Semi-discretized problems

In this section we focus on the discretization in the energy variable. For numerical simulations, we reduce the integration domain in FPL equation to a bounded domain in the variable ε where ε ∈ [0, E]. Thus we consider the approximate problem of (2.2) defined by

E 0 ∂f (ε, t) ∂t φ(ε) √ εdε = - 1 2 E 0 E 0 ( ∂φ(ε) ∂ε - ∂φ(ε ) ∂ε )g(ε, ε )(f (ε ) ∂f (ε) ∂ε -f (ε) ∂f (ε ) ∂ε )dε dε.
(3.1) Let us introduce {ε i } 1≤i≤N an increasing sequence such that ε 1 = 0, ε N = E and ∆ε i+ 1 2 = ε i+1 -ε i . We suppose that {∆ε i+ 1 2 } 1≤i≤N -1 is constant or increasing sequence. By convention we set ε 1 2 = 0. Any function f (ε, t) is approximated on the grid by values {f i } 1≤i≤N supposed be approximations of {f (ε i )} 1≤i≤N . We also introduce the notations (∆φ)

i+ 1 2 = φ i+1 -φ i and (Dφ) i+ 1 2 = (∆φ) i+ 1 2 (∆ε) i+ 1 2
as an approximation of the partial derivative ∂ ε φ(ε i+ 1

2

).

First, we consider the discretization of the expression E 0 ∂f ∂t φ(ε) √ εdε for any function φ. By using the standard quadrature formula with respect to the measure √ εdε, we approximate it by

E 0 ∂f ∂t φ √ εdε N i=1 (φ i df i dt ε i+ 1 2 ε i-1 2 √ εdε) def = N i=1 c i φ i df i dt . (3.2)
Thus the weights c i are defined by c 1 = 2 3 ε

3 2 3 2 , c i = 2 3 (ε 3 2 i+ 1 2 -ε 3 2 i- 1
2

) for i = 2, • • • , N .
Below, we present various strategies to construct schemes that have properties of conservation and if possible positivity and entropy decaying. These schemes differ intrinsically in the way we discretize the right-hand side of (3.1):

(r.h.s.) = -1 2

N -1 i=1 N -1 j=1 ε i+1 ε i ε j+1 ε j ( ∂φ(ε) ∂ε - ∂φ(ε ) ∂ε )g(ε, ε )(f (ε ) ∂f (ε) ∂ε -f (ε) ∂f (ε ) ∂ε )dεdε .
(3.3) We must also introduce the Maxwellian associated to a distribution function f . For a distribution function f we define ρ and ρE the discretized analogous of the density and the energy It's easy to check that T =

N i=1 ε i c i M i N i=1 c i M i = N i=1 ε i c i exp(-βε i ) N i=1 c i exp(-βε i )
, is a strictly monotone (decreasing) function of β with lim

= L(f, t) such that L ij ≥ 0 for i = j, L ii ≤ 0.
If there exists a constant C such that ∀ i, j |L ij (f, t)| ≤ C, and if the initial data is nonnegative i.e. f 0 ≥ 0, then as long as the solution exists it is non-negative.

Chang and Cooper type schemes

One of the most popular method used for Fokker-Planck equations is due to Chang and Cooper [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF]. Originally this method was proposed for the linear Fokker-Planck equation and the construction is entirely devoted to the preservation of equilibrium states. The authors also show that in the linear case (and only in this case) the method provides non-negative solutions. At contrary others authors, see [START_REF] Larsen | Discretization methods for one-dimensional Fokker-Planck operators[END_REF], have shown that for the non-linear Fokker-Planck equation this method can produce non-positive solutions. Let us recall the spirit of the Chang and Cooper method on the simple Fokker-Planck equation,

∂f ∂t = ∂F ∂v = ∂ ∂v (vf + σ ∂f ∂v ) = σ ∂ ∂v (M ∂ ∂v ( f M )),
where v ∈ R and the equilibrim state M is a Maxwellian:

M = M (v) = exp(-|v| 2 /2σ
). On a uniform grid of velocity space step ∆v and v i = i∆v, the Chang and Cooper method consists in discretizing the diffusion as usual and the drift in such a way that for f = M the fluxes F are all equal to zero. One takes at cell interface v i+ 1 2

= (i + 1 2 )∆v F i+ 1 2 = σ ∆v (f i+1 -f i ) + v i+ 1 2 (1 -δ i+ 1 2 )f i+1 + δ i+ 1 2 f i ,
and forcing the fluxes to be zero for equilibrium leads to

δ i+ 1 2 = σ v i+ 1 2 ∆v -exp( v i+ 1 2 ∆v σ ) -1 -1
.

The scheme writes as

df i dt = 1 ∆v (F i+ 1 2 -F i-1 2
). This method is also one of the most used for FPL equation. And this is a non-linear problem. None of the main work in this area [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF][START_REF] Kingham | An implicit Vlasov-Fokker-Planck code to model nonlocal electron transport in 2-D with magnetic fields[END_REF][START_REF] Langdon | Conservative differencing of the electron Fokker-Planck transport equation[END_REF] contains the proof of the positivity of the Chang and Cooper method or the proof of the energy conservation. Moreover it is not clear that equilibrium states are preserved by this method for the FPL equation since, in this case, the coefficients of Chang and Cooper [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF] are defined in an implicit manner for the FPL equation. In this section we propose a variant of this method which intrinsically contains the conservation of the energy, preserves discrete Maxwellian and is positive. We recall also the variants developed by Langdon et al. [8,[START_REF] Langdon | Conservative differencing of the electron Fokker-Planck transport equation[END_REF] which is non positive and the one's developped by Epperlein in [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF] and used by other people (see [START_REF] Kingham | An implicit Vlasov-Fokker-Planck code to model nonlocal electron transport in 2-D with magnetic fields[END_REF] for example) which is in fact non conservative in energy. For these two variants the problem comes from the boundary condition used at one end of the domain of computation ε = E.

A new variant: scheme S 1

Using for each integrals of (3.3) a midpoint quadrature formula, we approximate it by (r.h.s.) = -1 2

N -1 i=1 N -1 j=1 ((Dφ) i+ 1 2 -(Dφ) j+ 1 2 )g i+ 1 2 ,j+ 1 2 (f j+ 1 2 (Df ) i+ 1 2 -f i+ 1 2 (Df ) j+ 1 2 )∆ε i+ 1 2 ∆ε j+ 1 2 , (3.4) with g i+ 1 2 ,j+ 1 2 = g(ε i+ 1 2 , ε j+ 1 2
). Hence, the weak formulation of the semi-discretized model reads

N i=1 c i ∂f i ∂t φ i = - N -1 i=1 (Dφ) i+ 1 2 ∆ε i+ 1 2 N -1 j=1 g i+ 1 2 ,j+ 1 2 (f j+ 1 2 (Df ) i+ 1 2 -f i+ 1 2 (Df ) j+ 1 2 )∆ε j+ 1 2 . (3.5)
To simplify we note

K i+ 1 2 = N -1 j=1 g i+ 1 2 ,j+ 1 2 (f j+ 1 2 (Df ) i+ 1 2 -f i+ 1 2 (Df ) j+ 1 2 )∆ε j+ 1 2
the numerical flux. By identifying the terms involving φ i in (3.5), we obtain the system of ordinary differential equation

df i dt = Q S 1 i 1 ≤ i ≤ N, (3.6)
with

Q S 1 1 = K 3 2 /c 1 , Q S 1 i = (K i+ 1 2 -K i-1 2 )/c i for 2 ≤ i ≤ N -1 and Q S 1 N = -K N -1 2
/c N . We can rewrite the numerical flux

K i+ 1 2 = E i+ 1 2 f i+ 1 2 + D i+ 1 2 f i+1 -f i ∆ε i+ 1 2 1 ≤ i ≤ N -1, (3.7)
and the drift and diffusion coefficients are given by

E i+ 1 2 = - N -1 j=1 g i+ 1 2 ,j+ 1 2 (Df ) j+ 1 2 ∆ε j+ 1 2 and D i+ 1 2 = N -1 j=1 g i+ 1 2 ,j+ 1 2 f j+ 1 2 ∆ε j+ 1 2 . (3.8)
By integrating by parts the drift term reads

E i+ 1 2 = N -1 j=1 (g i+ 1 2 ,j+ 1 2 -g i+ 1 2 ,j-1 2 )f j -g i+ 1 2 ,N -1 2 f N . (3.9) Remark 1. We assume that {g i+ 1 2 ,j+ 1 2 } 1≤i≤N,1≤j≤N is an increasing sequence. Then, even if the f i 's are positive, E i+ 1 2 can be negative if f N = 0. In other hand, since the f i+ 1 2 's are positive, D i+ 1 2 remains positive.
We turn now to the discretization of

f i+ 1 2 . We suppose that f i+ 1 2 is an approximation of f (ε i+ 1 2
) given by the following definition [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF] Definition 1. The Chang and Cooper average f i+ 1 2 of quantities f i and f i+1 is defined by

f i+ 1 2 = δ i+ 1 2 f i + (1 -δ i+ 1 2 )f i+1 1 ≤ i ≤ N -1, (3.10) with δ i+ 1 2 = 1 α i+ 1 2 - 1 exp(α i+ 1 2 ) -1 1 ≤ i ≤ N -1, (3.11)
and

α i+ 1 2 = E i+ 1 2 D i+ 1 2 ∆ε i+ 1 2 .
To be comfortable we denote

h(α) = 1 α - 1 (exp(α) -1) (thus δ i+ 1 2 = h(α i+ 1 2
)). This smooth function is decreasing and bounded such that h(-∞) = 1, h(0) = 1/2 and h(+∞) = 0. Moreover, his derivative is negative and bounded too (h (±∞) = 0). Note that whatever the value of α i+ . We recall that f is the N-dimensional column vector with components {f i } 1≤i≤N . Thus, assuming that δ is the (N-1)-dimensional column vector with components {δ i+ 1 2 } 1≤i≤N -1 , we can write δ i+ 1 2 as a function of f and introducing

H = {H i+ 1 2 = h(α i+ 1 2 )} 1≤i≤N -1 , the system to be solved for δ is δ(f ) = H(δ(f ), f ), (3.12)
where f is a distribution function that has density ρ and energy ρE and H is a (N-1)dimensional vector valued function. Using a Newton's method requires O(N 3 ) operations. But we will see that for Coulombian potentials this cost can be reduced to O(N ) operations (see also [7]). Note that this is also true for Maxwellian potentials [START_REF] Buet | About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation[END_REF].

Proposition 1. The flux K i+ 1 2
satisfies the following relation

K i+ 1 2 = A i+ 1 2 f i+1 -B i+ 1 2 f i 1 ≤ i ≤ N -1, (3.13)
where

A i+ 1 2 = v(α i+ 1 2 ) D i+ 1 2 ∆ε i+ 1 2
, and

B i+ 1 2 = u(α i+ 1 2 ) D i+ 1 2 ∆ε i+ 1 2 , (3.14
)

and with u(x) = x exp(x) -1 and v(x) = x exp(x) exp(x) -1 .
Proof. The proof is the same as in Chang and Cooper's paper [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF] but in the non-linear case. By substituting the Chang and Cooper average f i+ 1 2 in the right-hand side of the relation (3.7) we get

K i+ 1 2 = E i+ 1 2 ((1 -δ i+ 1 2 )f i+1 + δ i+ 1 2 f i ) + D i+ 1 2 f i+1 -f i ∆ε i+ 1 2 1 ≤ i ≤ N -1. As α i+ 1 2 = E i+ 1 2 D i+ 1 2 ∆ε i+ 1 2 , we can write E i+ 1 2 = α i+ 1 2 D i+ 1 2 ∆ε i+ 1 2
. Thus, if we develop δ i+ 1 2

we have

K i+ 1 2 = α i+ 1 2 exp(α i+ 1 2 ) exp(α i+ 1 2 ) -1 D i+ 1 2 ∆ε i+ 1 2 f i+1 - α i+ 1 2 exp(α i+ 1 2 ) -1 D i+ 1 2 ∆ε i+ 1 2 f i .
And now, we can write

K i+ 1 2 = A i+ 1 2 f i+1 -B i+ 1 2 f i .

Properties

We summarize the conservation properties of the scheme S 1 in the following proposition: Proposition 2. The scheme S 1 conserves the mass and the energy and has the Maxwellians M = exp(-βε) as equilibrium states.

Proof. Mass and energy conservation properties come from (3.4) by taking φ = 1, ε .

Let us now check the second point which is not so clear as for linear Fokker-Planck equations.

First we have

M i+1 -M i = -M i+1 (exp(-β∆ε i+ 1 2 ) -1). Assuming that M i+ 1 2
is computed by Chang and Cooper average formula (3.10) with

δ i+ 1 2 = 1 β∆ε i+ 1 2 - 1 exp(β∆ε i+ 1 2 ) -1 1 ≤ i ≤ N -1, (3.15)
we get

M i+ 1 2 = 1 β∆ε i+ 1 2 - 1 exp(-β∆ε i+ 1 2 ) -1 (M i -M i+1 ) + M i+1 . Therefore M i+ 1 2 = ( 1 β∆ε i+ 1 2 + M i+1 M i+1 -M i )(M i -M i+1 ) + M i+1 = - M i+1 -M i β∆ε i+ 1 2 . (3.16)
Thanks to (3.16) the drift term reads for a Maxwellian

E i+ 1 2 = - N -1 j=1 g i+ 1 2 ,j+ 1 2 (M j+1 -M j ) = N -1 j=1 g i+ 1 2 ,j+ 1 2 M j+ 1 2 β∆ε j+ 1 2 = βD i+ 1 2 .
Thus the coefficients δ i+ 1 2 defined by (3.15) are solution of the equation (3.12) when f = M . Furthermore, in that case the fluxes K i+ 1 2 vanish. So Maxwellians are equilibrium states of the scheme S 1 . Now we analyse the positivity of the scheme. For linear Fokker-Planck equations, Chang and Cooper [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF] have shown that their scheme is non-negative. Afterwards, Larsen et al. [START_REF] Larsen | Discretization methods for one-dimensional Fokker-Planck operators[END_REF] have pointed out that for non-linear Fokker-Planck equations, but with local drift and diffusion coefficients, the Chang and Cooper method can produce negative solution. Our case is a nonlinear Fokker-Planck equation with non-local drift and diffusion coefficients. For this case we can show that the Chang and Cooper scheme S 1 is positive. Proposition 3. The solutions of (3.6) with non-negative initial conditions f i (t = 0) are non-negative.

We don't have existence result for (3.6). The problem is that we have very few information about the solutions of (3.12): existence can be obtain by Brouwer's fixed point theorem but we have nothing about uniqueness and regularity of the solutions as functions of f . Proof. From the Proposition (1) we get

K i+ 1 2 = A i+ 1 2 f i+1 -B i+ 1 2 f i where A i+ 1 2 and B i+
f j ≤ ρ max i c i , ∀j; 1 ≤ j ≤ N -1.
So that we can bound the drift term as following

| E i+ 1 2 |=| N -1 j=1 g i+ 1 2 ,j+ 1 2 (f j -f j+1 ) |≤ ρ max i c i 2 N -1 j=1 g i+ 1 2 ,j+ 1 2 ≤ ρ max i c i 2(N -1)g N -1 2 ,N -1 2 .
Consequently, there exists

C > 0 such that | E i+ 1 2 |< C.
In the same way we get an upperbound for the diffusion coefficients

D i+ 1 2 ≤ ρ max i c i 2 N -1 j=1 g i+ 1 2 ,j+ 1 2 ∆ε j+ 1 2 .
To continue we distinguish two cases. First we suppose that | α i+ 1 2

|≥ 1. Then we get is also bounded. Thus, thanks to Lemma 1 this ends the proof.

A i+ 1 2 = exp(α i+ 1 2 ) exp(α i+ 1 2 ) -1 E i+ 1 2 and B i+ 1 2 = 1 exp(α i+ 1 2 ) -1 E i+ 1 2 . And even if | α i+ 1 2 |→ ∞, A i+
It is easy to check that the above results about positivity, energy conservation and equilibrium states for S 1 don't depend on the definition of the coefficients c i , see (3.2), or on our choice for the value of ε 1 . If one takes the Chang and Cooper coefficients δ i+ 1 2 at equilibrium, that is given by (3.15), or freeze the value given by the initial data, it is easy to see that if it is done in the expression of the fluxes given by (3.7) the scheme is still positive but no more conservative in energy and if it is done in (3.13) this is the converse. The algebra that permits to pass from the form (3.7) to the form (3.13) is valid if and only if the coefficients verify at all times the relation (3.12). This remark is still valid if one does not calculate exactly the Chang and Cooper coefficients by an iterative method. We don't have any H-theorem but as we will see on numerical examples, the entropy is quite decaying with this scheme.

The case of physical interest: Coulombian potential

We write now the Rosenbluth coefficients in the case of Coulombian potentials

(k i+ 1 2 ,j+ 1 2 = min(ε 3 2 i+ 1 2 , ε 3 2 j+ 1 2
)). To simplify we denote ∆ε

3 2 i = ε 3 2 i+ 1 2 -ε 3 2 i-1 2
. From (3.8) the drift term reads

E i+ 1 2 = i j=1 ∆ε 3 2 j f j -f N ε 3 2 i+ 1 2 1 ≤ i ≤ N -1, (3.17)
and for the diffusive term we have

D i+ 1 2 = i j=1 ∆ε j+ 1 2 f j+ 1 2 ε 3 2 j+ 1 2 + ε 3 2 i+ 1 2 N -1 j=i+1 ∆ε j+ 1 2 f j+ 1 2 1 ≤ i ≤ N -1. (3.18) As D i+ 1 2
depends on the f midpoint values it is clear that (3.10) is non-linear in the f i+ 1 2 's. We therefore have to solve a full matrix that arises from the implicit differencing of the f i+ 1 2 's. A way to compute the Chang and Cooper average (3.10) is to consider the second equation of (2.9). Thus we have to solve an elliptic equation, namely -

∂ ∂ε

1 √ ε ∂ ∂ε (D(f (ε))) = f (ε).
The non-linear system which solution is

f δ = {f i+ 1 2
} 1≤i≤N -1 is then replaced by a non-linear system in D(f ) and the matrix involved is tri-diagonal (see [START_REF] Buet | About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation[END_REF]7]). We can use a new expression for f i+ 1 2

f i+ 1 2 = - 1 ∆ε i+ 1 2   D i+ 3 2 -D i+ 1 2 ∆ε 3 2 i+1 - D i+ 1 2 -D i-1 2 ∆ε 3 2 i   , (3.19)
which can be solved, for a given distribution function by a Newton's method, with the cost of a tridiagonal system that is O(N ) operations at each iteration.

About other existing variants of the Chang and Cooper method for the isotropic FPL equation

We want to emphasize that the two known other Chang and Cooper method for the isotropic FPL equation are not conservative and positive at the same time. As we will see the first one conserves the energy but is not positive. The second one does not conserve the energy but is positive.

The first one, we call it the scheme S 2 , reduces to the Langdon's scheme in the case of a Coulombian potential. Another option consists in integrating the right-hand side of (3.5) up to ε N +1 , assuming that f N +1 = 0 and constraining the flux at the last point ε N + 1 2 to be identically equal to zero. This strategy was carried by Langdon [START_REF] Langdon | Conservative differencing of the electron Fokker-Planck transport equation[END_REF] and Decoster and Langdon [8] in the case of Coulombian interactions. We extend their method to the general case. The derivation begins as for the S 1 scheme using the weak formulation (3.4) but with N + 1 points. We denote by

E i+ 1 2 , D i+ 1 2 , K i+ 1 2
, i = 1, ...N the drift coefficients, the diffusion coefficients and the fluxes using respectively (3.8) and (3.7) with N + 1 points.

Imposing

df

N +1 dt = K N + 1 2 = 0 with f N +1 = 0 leads to E N + 1 2 f N + 1 2 -D N + 1 2 f N ∆ε N + 1 2 = 0. ( 3.20)
Thanks to (3.20) the diffusion becomes

D i+ 1 2 = N -1 j=1 g i+ 1 2 ,j+ 1 2 f j+ 1 2 ∆ε j+ 1 2 + g i+ 1 2 ,N + 1 2 D N + 1 2 E N + 1 2 f N . (3.21)
We integrate by parts the drift term and get

E i+ 1 2 = N j=1 (g i+ 1 2 ,j+ 1 2 -g i+ 1 2 ,j-1 2 )f j ∆ε j+ 1 2 . ( 3.22)
We assume that {g i+ 1 2 ,j+

with α i+ 1 2 = E i+ 1 2 D i+ 1 2 ∆ε i+ 1 2
. As in the previous scheme, the (N-1)-dimensional vector δ with components { δ i+ 1 2 } 1≤i≤N is the root of a non-linear equation like (3.12). By construction this scheme conserves the energy and discrete Maxwellians (see [START_REF] Buet | About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation[END_REF] for more details). Comparing formally the drift and diffusion functional discretized respectively by S 1 and S 2 we get for 1 ≤ i ≤ N -1

E i+ 1 2 = E i+ 1 2 + f N g i+ 1 2 ,N + 1 2 and D i+ 1 2 = D i+ 1 2 + g i+ 1 2 ,N + 1 2 D N + 1 2 E N + 1 2 f N . (3.23)
In the case of the Coulombian potential one can check (see [START_REF] Buet | About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation[END_REF]) that leads to the relation

D N + 1 2 E N + 1 2 = D N -1 2 E N -1 2
and thus could become negative. To see that this scheme is not positive it suffices to considere the following initial data: let f a distribution function with fixed density ρ and energy ρE and of the form f = aδ ε 1 + bδ ε N . Thus b = ρE/(c N ε N ) and a = (ρ -ρE/ε N )/c 1 and a and b are positive. With such an initial condition and for sufficiently fine grids it's easy to verify that df N -1 dt < 0 and consequently f N -1 gets negative too. The second well known implementation of the Chang and Cooper method is the one proposed by Epperlein for Coulombian collisions [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF] and used in the code SPARK or by Kingham and Bell in the recent code IMPACT [START_REF] Kingham | An implicit Vlasov-Fokker-Planck code to model nonlocal electron transport in 2-D with magnetic fields[END_REF]. These authors assume that the diffusion term and the distribution function vanish at ε N + 1 2 and that the distribution function vanishes at ε N +1 . Thus the numerical flux is zero and consequently the density is conserved. But the Epperlein velocity discretization is nothing else than a miscellany of what we called scheme S 1 and scheme S 2 : the drift term is the one discretized by scheme S 2 , formula (3.22), and the diffusion term the one computed by S 1 , formula (3.18). So the discrete operator has no weak symmetrized form. Thus one can have a doubt concerning the energy conservation since the energy conservation's proof is based on the existence of a weak symmetrized form of the collision operator. Numerical tests confirm this fact. The proof of positivity of this scheme is like those of the S 1 scheme since the diffusion term is non-negative. For more details the reader can refer to [START_REF] Buet | About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation[END_REF]. We can remark that these three Chang and Cooper type schemes are equivalent when the integrating domain is not bounded. The only difference between these schemes lies in the manner to treat boundary conditions when the integrating domain is reduced to [0, E].

E i+ 1 2 = E i+ 1 2 -f N ε 3 2 i+ 1 2 D i+ 1 2 = D i+ 1 2 - D N -1 2 E N -1 2 ε 3 2 i+ 1 2 . ( 3

Equilibrium scheme (scheme S 3 )

The Chang and Cooper method is not the only way to provide positive, conservative and equilibrium states preserving schemes for the FPL equation on non-uniform grid. The scheme we'll propose in this section share also these properties and is simpler and thus cheaper. This scheme is based on the work of Larsen et al. [START_REF] Larsen | Discretization methods for one-dimensional Fokker-Planck operators[END_REF]. Their work is for linear and non-linear Fokker-Planck equations. But they do not consider non-linearity as in the Landau equation, that is drift and diffusion coefficients are functionals of the distribution functions. One of the two main ideas exposed in their paper to preserve Maxwellian states is to remark that ε-derivative that appears in the flux can be rewritten as ∂ ∂ε = -βy ∂ ∂y where y = exp(-βε).

Thus it becomes easy to preserve Maxwellians in the weak form of the Fokker-Planck equation.

Following these authors, in the case of the FPL equation, we set

∆ε i+ 1 2 = - (∆M ) i+ 1 2 βM i+1 = exp(β∆ε i+ 1 2 ) -1 β ∆ε i+ 1 2 . (3.25)
Thus we consider the following approximation of the weak symmetrized form of the problem

N i=1 c i ∂f i ∂t φ i = - 1 2 N -1 i=1 N -1 j=1 ∆φ i+ 1 2 ∆ε i+ 1 2 - ∆φ j+ 1 2 ∆ε j+ 1 2 g i+ 1 2 ,j+ 1 2 f j+ 1 2 ∆f i+ 1 2 ∆ε i+ 1 2 -f i+ 1 2 ∆f j+ 1 2 ∆ε j+ 1 2 ∆ε j+ 1 2 ∆ε i+ 1 2 , ( 3
= f i+1 .
As in the previous section we obtain the system of ordinary equation for the approximation of the FPL equation

df i dt = Q S 3 i 1 ≤ i ≤ N, (3.27)
where

Q S 3 1 = K 3 2 /c 1 , Q S 3 i = (K i+ 1 2 -K i-1 2 )/c i for 2 ≤ i ≤ N -1 and Q S 3 N = -K N -1 2 /c N with the numerical flux K i+ 1 2 = ∆ε i+ 1 2 ∆ε i+ 1 2 N -1 j=1 g i+ 1 2 ,j+ 1 2 (f j+1 ∆f i+ 1 2 ∆ε i+ 1 2 -f i+1 ∆f j+ 1 2 ∆ε j+ 1 2 )∆ε j+ 1 2 1 ≤ i ≤ N -1.
By factorizing the terms f i and f i+1 in the last sum, the numerical flux reads

K i+ 1 2 = A i+ 1 2 f i+1 -B i+ 1 2 f i , (3.28)
with

               A i+ 1 2 = ∆ε i+ 1 2 ∆ε i+ 1 2   1 ∆ε i+ 1 2 N -1 j=1 g i+ 1 2 ,j+ 1 2 f j+1 ∆ε j+ 1 2 + N -1 j=1 (g i+ 1 2 ,j+ 1 2 -g i+ 1 2 ,j-1 2 )f j -f N g i+ 1 2 ,N -1 2   , B i+ 1 2 = 1 ∆ε i+ 1 2 N -1 j=1 g i+ 1 2 ,j+ 1 2 f j+1 ∆ε j+ 1 2 .
(3.29) The fluxes can be also written using drift and diffusion coefficients 's it's easy to see that for f = M we have is obviously positive. We can write

K i+ 1 2 = E i+ 1 2 f i+1 + D i+ 1 2 f i+1 -f i ∆ε i+ 1 2 , ( 3
f j+ 1 2 ∆f i+ 1 2 ∆ε i+ 1 2 -f i+ 1 2 ∆f j+ 1 2 ∆ε j+ 1 2 = -β(M j+1 M i+1 -M i+1 M j+1 ) =
A i+ 1 2 = ∆ε i+ 1 2 ∆ε i+ 1 2 ( 1 ∆ε i+ 1 2 N -2 j=1 g i+ 1 2 ,j+ 1 2 f j+1 ∆ε j+ 1 2 + N -1 j=1 (g i+ 1 2 ,j+ 1 2 -g i+ 1 2 ,j-1 2 )f j + f N g i+ 1 2 ,N -1 2 ( ∆ε N -1 2 ∆ε i+ 1 2 -1)).
Assuming that g i+ 1 2 ,j+ 1 2 is an increasing sequence and that ∆ε i+

1 2 ≤ ∆ε N -1 2 leads to the positivity of A i+ 1 2 . Now, ∆ε i+ 1 2 /∆ε i+ 1 2 ∈ [0, 1] therefore A i+ 1 2 , B i+ 1 2
are bounded since, due to mass conservation, we have f i ≤ ρ min j c j . According to Lemma (1), f i cannot vanish in finite time. That completes the proof.

One could replace the exact equilibrium M i in the definition of the scheme by an approximate equilibrium state M i with same mass and energy and the resulting scheme is still conservative, positive but the equilibrium state is now M i . This could be useful since this scheme requires the knowledge of the equilibrium state and normally this can be done only by solving a non-linear equation with an iterative method.

If we stand are equal and up to a multiplicative constant the flux reduces to

f i+ 1 2 = f i instead of f i+ 1 2 = f i+1 in
K i+ 1 2 = N -1 j=1 g i+ 1 2 ,j+ 1 2 (f i+1 f j -f i f j+1 ).
The scheme is then nothing else than the scheme provided by Berezin and Pekker [START_REF] Berezin | Conservative finite difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy[END_REF] which is also an entropic scheme as shown in [START_REF] Buet | Numerical analysis of the isotropic Fokker-Planck-Landau equation[END_REF]. For Maxwellian (g(ε, ε

) = ε 3 2 ε 3 2 ) or Coulombian (g(ε, ε ) = min(ε 3 2 , ε 3
2 )) potentials the evaluation of all the coefficients A i+ 1 2 and B i+ 1 2 can be achieved in only O(N ) operations as explained in [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Buet | Numerical analysis of the isotropic Fokker-Planck-Landau equation[END_REF]. When applied to Coulombian potentials it can be easily checked that the D i+ 1 2 satisfies a relation of the type (3.19).

The second idea of Larsen et al. in [START_REF] Larsen | Discretization methods for one-dimensional Fokker-Planck operators[END_REF] to preserve equilibrium states in the discretization of Fokker-Planck equations and which could be applied to the isotropic FPL equation is to write

∂f ∂ε = M ∂ ∂ε ( f M
) -βf in the weak formulation (2.2). Proceeding as above leads to a scheme which is indeed conservative in mass and energy and preserves the Maxwellians. But at this time, we cannot show the positivity. Nevertheless it could be possible to derive a positive scheme since there are sufficiently degrees of freedom in the discretization, namely the factor f i+ 1 2 and M i+ 1 2 .

Time discretizations

Time discretization is also a key point in the numerical simulation of the FPL equation. As we have seen above, S 1 and S 3 can be evaluate in O(N ) operations in the Coulombian case (this is true for all known schemes dealing with the FPL equation for Coulombian potential). Thus the only problem could arise with the time discretization. For ICF applications the scheme must be able to take into account time steps in the range from .01τ to 10τ , where

τ = π 3 1 ρ ( ρE ρ ) 3 2
is the collision time. Moreover the number of discretization points N will be between 20 and 150. Despite the parabolic stability condition inherent to such an equation it is not clear that implicit time discretization could be cheaper than explicit time discretization. Thus we explore all the ways. Contrary to "Log" schemes for the isotropic FPL equation, [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Dellacherie | Sur un schéma numérique semi-discret appliqué à un opérateur de Fokker-Planck isotrope[END_REF], we can exhibit a good stability condition (positivity) for explicit schemes. It will be possible to measure the gain obtained using implicit discretization for example. We recall the more natural way to obtain an implicit solution, the fixed point method using an Mmatrix. One must converge to ensure energy conservation but at each iteration one has to invert a tridiagonal band matrix which is very cheaper. We examine also two other methods, the classical and widely used Newton's method and the "contracted implicit" method (noniterative method) developed by Lemou and Mieussens [START_REF] Lemou | Implicit schemes for the Fokker-Planck-Landau equation[END_REF]. For these two methods we show that by recasting them properly gives band diagonal system which can be solved by direct method like LU in O(N ) operations. The key point to achieve that is to remember that drift and diffusion coefficients satisfy (2.10) in continuous and it is anything else only to remember that for 3-D FPL equation Rosenbluth potentials satisfy Poisson equation. After that the game is to minimize the size of the band to have the minimum lower cost as possible.

Explicit time discretization

Let us introduce explicit scheme in time such as the well known Euler forward scheme

f n+1 -f n = ∆tQ(f n ), (4.1)
where ∆t is the time step, f n is the approximation of f (ε, t n ) and t n+1 = t n + ∆t and Q is the discrete operator defined by (3.6) for S 1 or by (3.27) for S 3 . We exhibit a condition such that the scheme remains positive. For the schemes S 1 and S 3 we get from relation (3.13) and (3.28)

f n+1 i = f n i (1 -∆tP n i ) + ∆tG n i ,
where

P n i = 1 c i (A n i-1 2 + B n i+ 1 2
) and

G n i = 1 c i (A n i+ 1 2 f n i+1 + B n i-1 2 f n i-1
). Above we saw, for S 1 and S 3 , that if f i ≥ 0 for all i ∈ [1, N ] then A i+ 1 2 and B i+ 1 2 are non-negative and bounded for all i. Consequently G n i and P n i are positive and bounded too. Now, if ∆t ≤ max i (P n i ) -1 then f n+1 i is positive. For a given time step we define the CF L number as CF L = ∆t max i P n i . Such time stability condition can't be obtained using "Log" schemes, that is, schemes based on the "Log" form of the equation (see [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF]). As proved in [START_REF] Buet | Numerical analysis of the isotropic Fokker-Planck-Landau equation[END_REF], our time restriction could be bounded below by using mass, energy and the L ∞ norm of the distribution function, but from a practical point of view it is unnecessary.

Implicit time discretization

The implicit scheme consists in solving

f n+1 -f n = ∆t Q(f n+1 ), (4.2)
where f n is supposed being the approximation of f (ε, t n ).

A positive fixed point method

The first discretization uses a Picard fixed point method based on a semi-implicit algorithm, namely

f k i -f n i ∆t = 1 c i A k-1 i+ 1 2 f k i+1 -(A k-1 i-1 2 + B k-1 i+ 1 2 )f k i + B k-1 i-1 2 f k i-1 ,
which can be written in the compact form

f k -f n = ∆tL(f k-1 )f k , (4.3)
where f k is the k th iterate and the coefficients A i+ 1 2 and B i+ 1 2 are defined by (3.14) for S 3 and (3.29) for S 1 . At convergence we naturally obtain f n+1 = f k . As we have seen above L(f k ) is an L-matrix for S 1 and S 3 , his transpose is diagonally dominant, and the kernel is the space of vectors whose all components are equal. Thus Id -∆tL(f k ) is an M-matrix and the iterative procedure (4.3) preserves mass and positivity. Due to the tridiagonal form of Id -∆tL(f k ) , it only requires O(N ) operations to compute f k (in fact the cost is the same as the explicit method). We show on numerical examples that this fixed point method converges rather well.

The "contracted implicit" method

This scheme was developed by Lemou and Mieussens [START_REF] Lemou | Implicit schemes for the Fokker-Planck-Landau equation[END_REF]. By noticing that conservation and entropy properties are a consequence of the symmetry property of the collision operator, the authors make implicit in time the diffusive term (f (ε )∂ ε f (ε)) in (2.1) without breaking the symmetry between ε and ε . They get the following contracted implicit scheme

f n+1 -f n = ∆tq ci (f n , f n+1 ), (4.4)
with

q ci (f, g) = 1 √ ε ∂ ∂ε ∞ 0 k(ε, ε ) f (ε ) ∂g(ε) ∂ε -f (ε) ∂g(ε ) ∂ε dε .
This scheme conserves mass, energy, see [START_REF] Lemou | Implicit schemes for the Fokker-Planck-Landau equation[END_REF], and discrete Maxwellians but it can produce negative solution. In our case this time discretization writes as

f n+1 i -f n i = ∆t c i (E n+1 i+ 1 2 f n i+ 1 2 + D n i+ 1 2 f n+1 i+1 -f n+1 i ∆ε i+ 1 2 -E n+1 i-1 2 f n i-1 2 -D n i-1 2 f n+1 i -f n+1 i-1 ∆ε i- 1 2
), (

for S 1 and for S 3 we just have to replace

f n i+ 1 2 by f n i+1 ∆ε i+ 1 2 ∆ε i+ 1 2 (see (3.25)) and D n i+ 1 2 by D n i+ 1 2 .
For the sake of simplicity we write them in the compact form (Id -

∆tL ci (f n ))f n+1 = f n .
Using the definition of the drift coefficients E i+ 1 2 it is obvious that Id -∆tL ci (f n ) is a nonsparse matrix, the drift term fills the matrix with his coefficients and we get a N × N matrix. The procedure to solve (4.4) by a direct method requires O(N 3 ) operations (note that L ci is nearly a lower triangular matrix, thus the cost can be reduced to O(N 2 ) operations). To bring the matrix sparse (and in fact band diagonal with the last column if the unknowns are stored correctly) the idea is to introduce the drift coefficient E as unknown associated with the equation (2.10). Namely we rewrite the "contracted implicit" as

f n+1 -f n -∆tq ci ((f n , E n ), (f n+1 , E n+1 )) = f n+1 -f n -∆tq ci (f n , f n+1 ) = 0, f n+1 -R(E n+1 ) = 0. ( 4.6)
where

R(E n+1 ) = 2 3 1 √ ε ∂E n+1 ∂ε .
If one discretizes in energy by the Chang and Cooper type scheme we get from (3.17)

             f n+1 i -f n i - ∆t c i (E n+1 i+ 1 2 f n i+ 1 2 + D n i+ 1 2 f n+1 i+1 -f n+1 i ∆ε i+ 1 2 -E n+1 i-1 2 f n i-1 2 -D n i-1 2 f n+1 i -f n+1 i-1 ∆ε i-1 2 ) = 0, f n+1 i -f n+1 N - E n+1 i+ 1 2 -E n+1 i-1 2 ∆ε 3 2 i = 0. (4.7)
To solve (4.7) the number of arithmetic operations required is N -1 2 operations (subtractions)

to eliminate the last column and m 2 × (2N -1) operations to solve one m-diagonal system by LU factorization (here m = 5). We can also substitute the function distribution f n+1 i in the first equation of (4.7) by his expression in the drift term given by the second equation. Thus the operator is a N × N sparse matrix and the number of operations required to solve the scheme reduces to O(N ) and the number of diagonal is now m = 5 (as done by Decoster for the Newton's method [7]). Therefore we gain a factor 2 in terms of CPU time. For the equilibrium scheme S 3 we just have to replace f n It is obvious that the scheme (4.7) is still conservative in mass and energy. This type of procedure can be easily extended to all variants of the "Log" scheme. It is also straightforward to generalize it to Maxwellian potential since in this case we have recurrences relation to evaluate drift and diffusion coefficients. It is also interesting to notice that the "contracted implicit" method applied to S 1 , S 3 or the "Log" scheme depicted by Dellacherie in [START_REF] Dellacherie | Sur un schéma numérique semi-discret appliqué à un opérateur de Fokker-Planck isotrope[END_REF] gives exactly the same system to solve for uniform grid. The only differences is in the manner to compute the diffusion coefficients from the distribution function. Obviously, at equilibrium, the diffusion coefficients are the same.

The Newton's method

The last method uses a Newton algorithm to solve equation (4.2).

Introducing F (f ) = f -f n -∆tQ(f ) the resulting iteration procedure is

F (f k-1 ) • (f k -f k-1 ) = -F (f k-1 ), (4.8)
where F (f k-1 ) is the Jacobian and we choose the starting point f 0 with the same mass and energy as f n . Introducing the linearized operator Q l around f we have

Q l (f, g) = 1 √ ε ∂ ∂ε ∞ 0 k(ε, ε ) f (ε ) ∂g(ε) ∂ε -f (ε) ∂g(ε ) ∂ε + g(ε ) ∂f (ε) ∂ε -g(ε) ∂f (ε ) ∂ε dε , (4.9
) and thus we can write

F (f k-1 ) • (f k -f k-1 ) = f k -f k-1 -∆tQ l (f k-1 , f k -f k-1 ).
This is the method used by Epperlein in [START_REF] Epperlein | Implicit and conservative difference schemes for the Fokker-Planck equation[END_REF]. The Newton's sequence becomes

f k -f n = ∆t Q l (f k-1 , f k ) -Q(f k-1 , f k-1 ) .
Because Q l and Q have a symmetric weak form, the Newton's process conserves mass and energy at each iteration. Now we assume that the discretization in energy is provided by the Chang and Cooper type scheme. This scheme is naturally expensive because it generates a N × N non-sparse matrix when we discretize in energy. The cost is then in O(N 3 ) at each iteration. In the case where potentials are Coulombian an other strategy is to remember the equations satisfied by the drift and the diffusive terms, namely (2.9), and to introduce E and D as unknowns [7]. Thus the implicit scheme writes

         f n+1 -f n ∆t = Q(f n+1 ) = Q(f n+1 , E(f n+1 ), D(f n+1 )) = Q(f n+1 , E n+1 , D n+1 ), f n+1 = R(E(f n+1 )) = R(E n+1 ), f n+1 = -S(D(f n+1 )) = -S(D n+1 ),
where the unknowns are now f n+1 , E n+1 and D n+1 and S(D) = 2 3

∂ ∂ε

1 √ ε ∂D ∂ε
. We solve the previous system by the Newton's method. The Jacobian matrix involved in the method is now a sparse matrix. If one discretizes in energy by the Chang and Cooper type scheme this yields to

                         f i -f n i - ∆t c i (E i+ 1 2 f i+ 1 2 + D i+ 1 2 f i+1 -f i ∆ε i+ 1 2 -E i-1 2 f i-1 2 -D i-1 2 f i -f i-1 ∆ε i-1 2 ) = 0, f i -f N - E i+ 1 2 -E i-1 2 ∆ε 3 2 i = 0, f i+ 1 2 + 1 ∆ε i+ 1 2   D i+ 3 2 -D i+ 1 2 ∆ε 3 2 i+1 - D i+ 1 2 -D i-1 2 ∆ε 3 2 i   = 0, where f i+ 1 2
is given by the Chang and Cooper average (3.10) for S 1 . Note that f i+ 1 2 depends on E, D and f = {f i } 1≤i≤N . Thus we can write the above system in f, E, D as

     f -f n -∆t Q(f, E, D) = 0, f -R(E) = 0, h(f, E, D) + S(D) = 0. (4.10)
Starting from (f 0 , E 0 , D 0 ) verifying the second and third equation of (4.10) the iteration procedure is defined as:

Iteration process 1. Steps: 1. we solve                      f k -f n = ∆t ∂ f,E,D Q(f k-1 , E k-1 , D k-1 ).(f k -f k-1 , E k -E k-1 , D k -D k-1 ) -Q(f k-1 , E k-1 , D k-1 ) f k -R(E k ) = 0, ∂ f,E,D h(f k-1 , E k-1 , D k-1 ) • (f k -f k-1 , E k -E k-1 , D k -D k-1 ) + S(D k ) = -h(f k-1 , E k-1 , D k-1 ) (4.11) where ∂ f,E,D Ql (f, E, D) is the linearized of Q(f, E, D
), and we have taken into account all the simplifications coming from the fact that R and S are linear operators.

Set

D k as the solution of h(f k , E k , D k ) -S(D k ) = 0.
Proposition 5. The sequence f k defined by the iteration process (1) is conservative in mass and energy.

Proof. We proceed by induction. By construction f k-1 , E k-1 , D k-1 satisfy the second and third equations of (4.10). Thus the second and third equations of (4.11)

give E k -E k-1 = ∂ f E(f k-1 ).(f k -f k-1 ) and D k -D k-1 = ∂ f D(f k-1 ).(f k -f k-1
). Plugging these relations in the first equation of (4.11), we have nothing else than the sequence of the Newton's method (4.8) with only the unknown f , which is conservative in mass and energy since the linearized operator Q(f ) defined by (3.6) is too. Thus f k and f k-1 have the same mass and energy. Since (f 0 , E 0 , D 0 ) verify the second and third equation of (4.10) that ends the proof. The starting point f 0 is obviously chosen such that it has the same mass and energy as f n .

The linear system involved by (4.11) when the vector of the unknowns is stored as

((f i , E i+ 1 2 , D i+ 1 2 ) i=1,N -1 , f N )
gives a (3N -2) × (3N -2) matrix like a block-tridiagonal matrix where 3(N -1) blocks represent a square matrix of order 3 and the last row (respectively column) is a 3N -2 vector (respectively column-vector). The last row can be easily eliminated by subtracting lines and that gives a band diagonal system which can be solved by a direct method like LU in O((3N -2)m 2 ) operations, m = 11 is the number of diagonals. Applied to the equilibrium scheme S 3 this strategy is more simple since the function h becomes linear: the second step of the iteration process (1) is useless. Additional optimization can be done in the case of Coulombian potentials, by replacing the

f i ; i = 1, N -1 by E i+ 1 2
and by rearranging a little bit the linear system involved by the Newton's method, the matrix is then only (2N -1) × (2N -1), the number of diagonals is m = 9 and we gain approximately a factor 2 in terms of CPU time. This the Newton's implementation of Decoster [7] for S 2 for which we have a substantial gain since the last row is null. Such a strategy can be also be applied with "Log" schemes. It can also be applied for Maxwellian potentials even for the full 3-D Landau equation. The key point is still the fact that in these cases drift and diffusion coefficients can be computed trough recurrences relations. But for the full Landau equation with Coulombian potentials the question remains open.

Numerical results

We compare S 1 and S 3 and the various implicit schemes on two numerical test cases: the first one is the classical Rosenbluth test [START_REF] Rosenbluth | Fokker-Planck equation for an Inverse-Square Force[END_REF] and the second one is a δ function in energy.

Test 1: the classical Rosenbluth test

This numerical test is extracted from the work of Rosenbluth et al. [START_REF] Rosenbluth | Fokker-Planck equation for an Inverse-Square Force[END_REF]. The initial data is a gaussian velocity distribution given by f 0 (ε) = 0.01 exp -10(( √ ε -0.3)/0.3) 2 . We use a regular velocity discretization. Thus the discrete energy points are ε i = ( i N ) 2 for i = 1 to N . Let ∆t exp the largest step ensuring the stability of the explicit scheme. As the grid is irregular, the time step of the explicit scheme is constrained by the smallest step size of the mesh. We denote τ the characteristic collision time. In figure [START_REF] Berezin | Conservative finite difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy[END_REF] we plot the numerical kinetic entropy N i=1 f n i log f n i c i as a function of time, obtained by the "contracted implicit" method (4.4) and the full implicit method (4.11). Tolerance for the iterative schemes is 10 -12 . We consider three time steps namely: ∆t = 10 -1 τ , ∆t = τ and ∆t = 10τ . The final physical time corresponds to 20τ . We observe that the equilibrium state is reached in only four collision times. The results given by the full implicit method are the more accurate. This is clear also in figure [START_REF] Bobylev | On the numerical solution of Landau's kinetic equation[END_REF] where the fourth moment N i=1 f n i ε 2 i c i is plotted as a function of time or for the distribution function in figure [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF]. The time step ∆t = 10τ gives a CF L of about 23000. For our two schemes we show in table [START_REF] Berezin | Conservative finite difference schemes for the Fokker-Planck equation not violating the law of an increasing entropy[END_REF], for all the time discretization, the CPU time required to attain a final time of 30τ . The "contracted implicit" scheme is the cheaper method but gives poor results as the time step exceed τ and it could give negative solutions.

Explicit Contracted implicit Fixed point Newton S 1 , ∆t = 10

Test 2: a mono-energetic initial data

This is typically a test that can't be handle by "Log" schemes : either the scheme is not defined [START_REF] Degond | An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory[END_REF] or the initial data don't evolve through time [START_REF] Buet | Numerical analysis of the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Dellacherie | Sur un schéma numérique semi-discret appliqué à un opérateur de Fokker-Planck isotrope[END_REF]. One consider thus the initial data f 0 (ε) = δ ε=0.3 and we do the same computations as for the Rosenbluth test case, that is with the same parameters. Now the time step ∆t = 10τ gives a CF L of about 23000. As for test 1, the full implicit scheme via fixed point method or the Newton method gives the more accurate results even if the time step is much larger: see figures [START_REF] Chang | A practical difference scheme for Fokker-Planck equations[END_REF] and [START_REF] Buet | About positive, energy conservative and equilibrium state preserving schemes for the isotropic Fokker-Planck-Landau equation[END_REF] where the fourth moment (4). And it could produce negative solutions even with large time steps. At contrary the full implicit scheme using fixed point iteration or Newton method gives always a positive solution and a better relaxation despite their extra cost. At this point, we want to emphasize that, our schemes S 1 and S 3 are robust whatever the time discretization. This point out the interest of designing positive schemes for the FPL equation, not as the "Log" schemes rightly. This last class of scheme is very useless with the "contracted implicit" method or the Newton method since they used Logarithm of the distribution function. And the fixed point iterative method cannot be defined in general for them. This is in relationship with the fact that for them, excepted for the Berezin and Pekker one on uniform grid in energy, it is impossible to exhibit a condition on the time step to ensure the positivity (see [START_REF] Buet | Conservative and entropy schemes for the isotropic Fokker-Planck-Landau equation[END_REF][START_REF] Buet | Numerical analysis of the isotropic Fokker-Planck-Landau equation[END_REF]).

Conclusions

In this work, we have introduced a new variant of the Chang and Cooper method S 1 for the isotropic FPL equation and also a new scheme called equilibrium scheme S 3 . These two methods, on non-uniform grid, are both positive, conservative in mass and energy and preserve the discrete Maxwellians, contrary to other existing methods for the isotropic FPL equation. This last method is cheaper than the Chang and Cooper one. Nevertheless this one could be more precise if the temperature of the plasma becomes low compared to the mesh size. These two schemes should apply to the full 3-D Landau equation. We have also shown that time discretization like the "contracted implicit" method and the Newton's method can be formulate in such a manner so that they lead to the inversion of a band diagonal system which can be solved by direct method with a linear cost. Let us also emphasize that if such technics could be also applied to "Log" schemes, they will be more expensive due to the more complex structure of such algorithms. Despite it's lack of conservation of energy, the fixed point approach method works well on numerical tests and it has a cost comparable to the "contracted implicit" method or the Newton's method. Moreover since we have positive scheme, we are able to give a time step restriction for the explicit scheme which could be the cheaper if the number of points is of the order of 20 and if the time step is little compared to the collision time (approximately 10 -2 τ ). With these four time's discretization it could be possible, by switching between them, to minimize the cost of a time step according to the number of points and the ratio ∆t τ . In conclusion, through this study we have also emphasized the importance of positive schemes, especially when time discretization is approached. The extension of these tools to the full Landau equation will be the subject of a future work.

1 Introduction 2 2 5 3-discretized problems 5 3 . 1 13 4discretizations 16 4 . 1 18 5results 21 5 . 1

 1253113411851 The isotropic Fokker-Planck-Landau equation 4 2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Coulombian interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SemiChang and Cooper type schemes . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1 A new variant: scheme S 1 . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.3 The case of physical interest: Coulombian potential . . . . . . . . . . 11 3.1.4 About other existing variants of the Chang and Cooper method for the isotropic FPL equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Equilibrium scheme (scheme S 3 ) . . . . . . . . . . . . . . . . . . . . . . . . . Time Explicit time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Implicit time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.1 A positive fixed point method . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.2 The "contracted implicit" method . . . . . . . . . . . . . . . . . . . . 17 4.2.3 The Newton's method . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Test 1: the classical Rosenbluth test . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 Test 2: a mono-energetic initial data . . . . . . . . . . . . . . . . . . . . . . . 22 6 Conclusions 24

( 2 . 6 )

 26 as ρ = N i=1 c i f i and ρE = N i=1 ε i c i f i . The temperature is still defined by 3 2 ρT = ρE. For a distribution function f , we denote by M = α exp(-βε) the Maxwellian which has the same mass and energy as f .

TLemma 1 .

 1 = T min = 0.. Thus for any distribution function f such that 0 ≤ T ≤ T max there is a unique β ≥ 0 such that T = T and consequently M is unique. Note that β 1 T . For the rest of this work we consider only distribution functions such that 0 ≤ T ≤ T max , that is, we exclude distribution function for which M is an increasing function of ε (β ≤ 0) (that means we exclude distribution function such that T max ≤ T ≤ 2 3 E). To check the positivity of the schemes we need the following well-known result: Consider the Cauchy problem for the ordinary differential equation df dt = Lf and f (t = 0) = f 0 , with f = {f i } 1≤i≤N and with the square matrix L

1 2(positive or negative) if δ i+ 1 2 exists we get 0 ≤ δ i+ 1 2 ≤ 1 . 2 '

 12212 Therefore, since the f i 's are positive the f i+ 1 s remain positive. Now we clarify the expression of δ i+ 1 2

1

 1

2 2 's and B i+ 1 2 ' 2 ≥ 2 and B i+ 1 2 2 and B i+ 1 2

 2222121 are given by(3.14). The function u(x) =x exp(x) -1 is positive, decreasing from -∞ to 0, while v(x) = x exp(x) exp(x) -1 is positive too, increasing from 0 to +∞. To prove the positivity we use Lemma 1. Thus all we need to do is showing that the A i+ 1 s are positive and bounded. As a result (see Remark 1) we have D i+ 1 0. Moreover u and v are positive functions therefore A i+ 1 are positive too. Now we have to show that A i+ 1 are bounded. At first we assume that f is a distribution function that has density ρ and energy ρE thus we get

1 2and B i+ 1 2 2 |≤ 1 then u(α i+ 1 2 ) and v(α i+ 1 2 ) 2 and B i+ 1 2

 11212221 remain bounded. Now we assume that | α i+ 1 are bounded and A i+ 1 too since D i+ 1 2

.30) where the drift coefficients ∆ε i+ 1 2 ∆ε i+ 1 2 E i+ 1 2 2 are defined by ( 3 . 8 ) with f i+ 1 2 = f i+1 and ∆ε i+ 1 2 in place of ∆ε i+ 1 2 .2 ,j+ 1 2 2 ≤

 12123822212 are defined by (3.9) and the diffusion coefficients D i+ 1 We can summarize the properties of this scheme in the following proposition Proposition 4. If g i+ 1 is an increasing sequence and ∆ε i+ 1 ∆ε N -1 2 the scheme S 3 is conservative in mass and energy. Moreover it is positive and preserves the equilibrium state when is reached. Proof. Mass and energy conservation are easily checked by setting φ = 1 and φ = ε in the weak symmetrized form of the discrete FPL equation (3.26). Using the definition (3.25) of ∆ε i+ 1 2 and the definition of the f i+ 1 2

2 and B i+ 1 2 1 2 ,j+ 1 2

 2111 0 thus the scheme preserves the Maxwellians states. Let us now check the positivity. The fluxes are given by (3.28) and (3.29). Now, to use Lemma 1 we have just to check that A i+ 1 are positive and bounded. As g i+ is positive, B i+ 1 2

Figure 1 :Figure 2 :

 12 Figure 1: Test 1. Kinetic entropy for contracted implicit scheme (CI) and full implicit (FI), S 1 vs S 3 .

N i=1 f n i ε 2 i

 2 c i is plotted as a function of time. The contracted implicit scheme is still the cheaper but to give the right relaxation, the time step would not exceed 10 -1 τ , see

Figure 3 :

 3 Figure 3: Test 1. Distribution function, ∆t = 0.1τ , S 1 vs S 3 .

Figure 4 :

 4 Figure 4: Test 2. Distribution function, ∆t = τ , S 1 vs S 3 .

Figure 5 :

 5 Figure 5: Test 2. Kinetic entropy for contracted implicit scheme (CI) and full implicit (FI), S 1 vs S 3 .

Figure 6 :

 6 Figure 6: Test 2. Fourth order moment for contracted implicit scheme (CI) and full implicit (FI), S 1 vs S 3 .

  .24) This is the Langdon version of the Chang and Cooper method for the FPL equation derived originally only for Coulombian potential and using the Rosenbluth form of the equation. But using (3.24) one can see that D N -1 2

  the numerical flux and if we take ∆ε i+ 1

	2	=	(∆M ) i+ 1 2 βM i	,
	the Maxwellians are also preserved but we can show that S 3 is positive if and only if the energy
	grid is uniform.
	On a uniform grid all the terms ∆ε i+ 1 2

Table 1 :

 1 Test 1. Comparison of the total costs involved by S 1 and S 3 for three values of the time step, ∆t = 10 -1 τ, τ, 10τ , and a final time T = 30τ .

	-1 τ	6	7.9E-002	0.26	0.24
	S 1 , ∆t = τ	6	9.6E-003	7.32E-002 2.46E-002
	S 1 , ∆t = 10τ	5.7	1.6E-003	5.1E-002	3.9E-003
	S 3 , ∆t = 10 -1 τ	.6	3.0E-002	6.6E-002	0.16
	S 3 , ∆t = τ	0.75	4.2E-003	1.7E-002	1.7E-002
	S 3 , ∆t = 10τ	0.78	1.0E-003	1.0E-002	1.7E-003


Camille Allaria

Keywords:

published or not. The documents may come

Electronic monitoring of offenders on home detention sentences in France

Camille Allaria

Doctor of Sociology, research associate, Laboratoire Méditerranéen de Sociologie, MSH, Aix-Marseilles University Camille Allaria

INVISIBLE CELL WALLS

Electronic monitoring of offenders on home detention sentences in France

Electronic monitoring consists in confining convicted offenders to their residence during time frames set by the sentencing judge. Information relative to an offender's movements is electronically relayed to a computer in a monitoring station. The computer monitor displays alert messages in case of delay or absence. This technically provides supervising officers with an opportunity to track offenders' whereaboutswhich they do by phone at first (1). How will the criminal sanction evolve as digital incarceration (electronic monitoring of offenders' movements) progressively replaces physical imprisonment and makes cell walls invisible?

With the introduction of the electronic ankle bracelet (which at first was thought to provide relief to prison overcrowding), the home has become a restrictive environment. If house arrest is by no means a new phenomenon, electronic monitoring of offenders on home detention, however, is. It has become an inescapable issue raising new challenges and renewing the relations between detainees and the penal institution. It is also the source of a double process whereby the private sphere becomes "visible," and the prison institution becomes « invisible.» At work here is a double territorial shift: the place where the sentence is served is de-territorialized and materialized by the electronic tracking process, which consists in fixing the mark of the criminal sanction -the electronic ankle bracelet -to the offender's body. In the process, the private sphere becomes « visible » to the monitoring station thanks to the ubiquity of digital technology.

Restrictive use of the private space

Installing the monitoring system on the new territory of the sentence, the offender's home, means defining boundaries between allowed and forbidden spaces. This is given full significance by negotiations over the demarcation of space between offenders and supervising officers. The rules are for the most part set by the technology, insofar as the allowed range defined as normative space is set up according to the range of the system. « He measured my ankle, installed the sensor, put the bracelet, then he had me walk around the house in order to figure out its surface area and which places I could go. If I go off-limits, the alarm goes off. I get a phone call from the monitoring station right away. » Interviewed bracelet wearer « They put the bracelet round your ankle, you walk around the house, you go from wall to wall, from one room to another… Then, he asked me to go downstairs. When I reached the foot of the stairs, he said, 'Come back up! See, as soon as you step outside of this perimeter, the alarm will go off.' So you don't take the risk. »

Interviewed bracelet wearer (2)

In these conditions, the home acquires a double status: it is both the place where the offender and their family live and a place of incarceration. The family circle serves as a link in the monitoring chain and is to some extent submitted to the same restric-

(1) According to the figures of the French Ministry of Justice, on January 1, 2015, there were in France 66,270 persons detained in prison and 10,767 on electronically monitored home detention.

(2) We conducted over fifty interviews with bracelet wearers, reintegration and probation officers tions as the offender.

« Oftentimes, [my wife] says, 'It's time to go back home,' or 'There, we can't go out now!' Like on Sunday, she says, 'It isn't worth going out because we won't make it.»

Interviewed bracelet wearer « And when I give instructions, you see, I give them to everyone, to the whole family, because I think everyone's involved. So I explain the guy everything, when he can go out and how much time he's allowed to spend out of the house, what he must and must not do…»

Interviewed supervising officer

Besides, the information automatically relayed about offenders' whereabouts provides indications that institutional agents exploit when assessing offenders' « selfsufficiency,» ability to comply with their approved schedule, and even how functional their families are.

« In fact, monitoring provides us with indications on how self-sufficient a person is […]

. There are many things to consider: is the person self-sufficient? Is the person able to bear and comply with restrictions? Does the person's environment make it harder for them to stabilize their life? » Interviewed reintegration and probation officer However, this technological device cannot give a full representation of the monitored persons' whereabouts. The track data it provides, that is, information it is able to capture, is fragmented at best. Supervising officers thus feel greatly uncertain when it comes to evaluating the monitored persons' compliance. They make up for this lack of information by calling them up and gathering all the information needed for closely evaluating their degree of compliance. This raises a number of issues such as, for instance, the emergence of negotiation practices about how compliance is to be defined. These will not be dealt with here, however.

Insofar as the monitoring device is installed in the offender's home, it requires the intervention of a supervising officer from the monitoring station not only at the moment of installation but also throughout the duration of the sentence should technical failures occur.

The penal institution becomes invisible

Making the private sphere « visible » goes together with the dematerialization of the penal institution. This phenomenon manifests in mutations, albeit minor, on the body and in the way in which officers of the penal institution present themselves to the community. When the device is installed in the offender's home -which represents the first, and sometimes only, meeting between the offender and the supervising officer, the uniform, one of the tangible representations of the institution, is not visible.

« How did it feel to see a prison guard in uniform in your home?

No, no, he was wearing plain clothes! That's right, he came wearing plain clothes with his briefcase, tools, and everything.»

Interviewed bracelet wearer

Unlike prison guards, supervising officers do not wear a uniform, not because they remotely supervise detainees or just for comfort. Plain clothes are the most appropriate outfit for officers because they may have to visit offenders at home and find themselves on their own searching for a building in a neighborhood.

« Why aren't you wearing a uniform?

Because at some point, you may have to go to a bracelet wearer's home, just like G. did yesterday, and you just don't want to walk around in projects wearing a uniform… unless there is an important official visit, in which case you must wear it…»

Interviewed supervising officer

The interesting point in this statement is that the opening of the prisons administration has a specific impact on the body itself. If the primary function of the uniform is to make holders of legitimate violence visible and identifiable, to give precedence to function over identity, and contribute to constructing the collective identity of state civil servants, what are the implications of wearing plain clothes and being stripped of any symbol of might and sovereign power for officers when they are in face-to-face situations? During the home confinement procedure, monitored people and supervising officers are mutually anonymous and incorporeal. However, the moment of installation of the monitoring device gives them an opportunity, sometimes the only one, to give a concrete form, a face, a build, to the other person. « The electronic monitoring program only allows authorities to continue, as before, to exert their power to punish, with the difference that what they did through a visible system, they now do unseen.» (3) TEPSIS PAPER 13 / September 2016 Laboratoire d'Excellence TEPSIS, 190-198 av. de France, 75013 Paris, https://tepsis.fr 4

(3) Jean-Charles Froment, « La surveillance électronique à domicile: Une nouvelle économie du pouvoir de punir?», Les cahiers de la sécurité intérieure 34 (1998): 158.

The technological evolution of the means of separation between allowed and forbidden spaces inevitably raises the question of the body and its relationship to space. How is one to perceive an authorized boundary if it is invisible? Will there ever be a means of perceiving this boundary? What are the implications of the protagonists' mutual invisibility? The digital aspect of home detention goes along with a fundamental characteristic: the distance between the supervising officer's and the monitored person's bodies. The two protagonists are nonetheless connected, and this, by a bond that only the monitoring device placed in the home and on the body of the offender can make visible. This distance has a corollary: « deprived of all territory, the sanction is attached to and moves along with the detainee.» (4) If one considers the body as the core of an individual's identity, the fact that the punishment is implemented through attaching a bracelet to the offender's leg deserves analysis. The bracelet materializes a « virtual » punishment and, through various appropriation and stigmatization mechanisms, becomes the material « mark » of the offense itself. In these conditions, offenders wear the mark of their guilt everywhere and at all times. These various elements raise the more general question of the relationship between individuals and the state and on the nature of the latter. « Elites, ruling classes, bosses, adults, men, Caucasians -superordinate groups generally -maintain their power as much by controlling how people define the world, its components, and its possibilities as by the use of more primitive forms of control. They may use more primitive means to establish hegemony.

But control based on the manipulation of definitions and labels works more smoothly and costs less.» (5) The electronic bracelet defines and maintains a boundary between offenders and law-abiding citizens. This surveillance system is made possible by the body. « Concretely, the bracelet extends the prison system's management of space by means of virtual confinement. This spatial extension is correlative with a temporal extension of the sentence to an indefinite duration and, above all, calls for permanent surveillance of bracelet wearers' behavior -a surveillance overwhelmingly internalized by offenders themselves, who then become the main agents of their own punishment.» (6)

( 4 )

 4 Ibid., 155. (5) Howard Saul Becker, Outsiders: Studies in the Sociology of Deviance (New York: Free Press, 1973). (6) Olivier Razac, « Le placement sous surveillance électronique mobile: un nouveau modèle pénal?», CIRAP Report 6 (2010), https://detentions.wordpress.com/2011/05/02/le-placement-soussurveillance-electronique-un-nouveau-modele-penal/ TEPSIS PAPER 13 / September 2016 Laboratoire d'Excellence TEPSIS, 190-198 av. de France, 75013 Paris, https://tepsis.fr

Laboratoire d'Excellence TEPSIS, 190-198 av. de France, 75013 Paris, https://tepsis.fr


Romain Tiquet

Romain Tiquet Penal

LABOR AND PRODUCTIVE INCARCERATION MOBILE PRISON CAMPS IN SENEGAL DURING THE INTERWAR PERIOD 14

Keywords:

French research on colonial imprisonment in Africa is significantly lagging behind (1). For instance, it has particularly overlooked the economic dimension of imprisonment in the colonies -i.e., penal labor, considering it marginal or not characteristic of the situation in the colonies (2). Yet, the institution, functioning, and evolution of penal labor in the colonies provide a relevant analysis framework as regards the realities of a colonial system primarily based on coercion and compulsory labor. In Senegal, for instance, the creation of mobile prison camps has actively contributed to the construction and renovation of the Senegalese road network. The colonial prison system was installed in societies for the most part unfamiliar with incarceration. In French West Africa and more particularly in Senegal, the first incarceration venues (prison, solitary confinement and security cells) appeared when trading posts were established, in the late eighteenth century, and developed in direct connection with the slave trade. However, colonial prison was not just a mere importation of the mainland model to the colonies. Even though imprisonment in the colonies was a product of reflections, ideologies and practices held worldwide, it was constantly (re)formulated, reinvented and recomposed according to local situations, generating a great number of hybrid models with diverse functions.

Generally speaking, incarceration in the colonies was rather an instrument of conquest than an application of the Foucauldian view of the penitentiary as a setting of punishment and moral reform. Contrary to the prison in mainland France, which defines individuals as citizens and subjects of rights and duties, colonial prison took part in the construction of « indigenous » populations as objects of power. In a way, it mirrored within walls what colonial society did outside. It was used as a plural instrument of social control and involved more than just the penal dimension.

In this regard, the creation of mobile prison camps in Senegal is both original and emblematic. These camps, established during the interwar period, housed inmates condemned to long sentences in mobile prisons that moved camp depending on the needs of private and public roadwork sites, providing these sites with labor they could exploit at will. Inmates were not expected to participate in their own moral reform but rather to contribute to the public interest by taking part, among other things, in road works. The camps acted as a real labor pool, and in this, were characteristic of this economy of coercion. Our analysis purports to highlight the role played by the use of penal labor in French colonies and more particularly in Senegal through the study of the organization of penal camps and life conditions of inmates. Effectively, the use of inmates on Senegal's roadwork sites partook of the colonial obsession with putting the native populations to work in the name of the sacrosanct « development » of colonies -the purpose being to secure a labor pool. (2) As concerns Senegal, Ibra Sene's is the only article dealing with penal camps, although it tends to be rather descriptive and falls short of stressing their economic dimension. Ibra Sene, « Colonisation française et main-d'oeuvre carcérale au Sénégal: De l'emploi des détenus des camps pénaux sur les chantiers des travaux routiers (1927)(1928)(1929)(1930)(1931)(1932)(1933)(1934)(1935)(1936)(1937)(1938)(1939)(1940),» French Colonial History, 5/1 (2004): 153-171.

ROADWORK SITES AND COLONIALOBSESSION WITH WORK: CREATION OF MOBILE PRISON CAMPS

In 1936, Inspector Monguillot conducted a general inspection of prisons and proposed to reform the prison service in Senegal by making a more rational use of the penal workforce. His report suggested a reorganization of penal labor aiming to decongest civilian prisons by concentrating long-term offenders in three penal camps and having them work on roadwork sites. Monguillot noted that given the state of roads, « there will be work for a long time.»(3).

Penal camps were mobile and moved according to the needs for maintenance and repair jobs on the road network. Inmates sentenced to more than a year's imprisonment were directed to one of the three camps depending on the duration of their sentence. Penal camp A, in Thiès region, housed inmates sentenced to less than five years' imprisonment. Penal camp B, Kaolack region -prisoners serving sentences of more than five years. Last, penal camp C, Louga region, comprised a "maximum-security section" and housed recidivists and « dangerous offenders.» The three camps stood at strategic spots between the two political capitals Dakar and Saint-Louis and Sine-Saloum in the heart of the Senegalese peanut economy. The governor of Senegal himself stated that the use of this free labor provided « major benefits for the repair of the road network.» (4).

One must keep in mind that the international economic context was decisive in the setting up of penal camps. Initially, road maintenance was the responsibility of workers subjected to prestation, a tax in kind to be paid in a number of free days of work on roadwork sites. During the 1929 Geneva debates on forced labor, this form of compelled employment was greatly criticized and condemned by Convention no. 29 of the International Labour Organization (ILO). One may then argue that the creation of penal camps in Senegal was a strategy adopted by colonial authorities both to appease international opinion and secure the workforce necessary for the maintenance of the road network. The gain was triple: first, colonial authorities protected themselves from international criticism of forced labor as the use of penal labor was tolerated. Second, the creation of penal camps allowed for the provision of extra workforce for roadwork sites in a colony where the number of prestation days was particularly low (four days per year). Last, the colony secured an inexhaustible labor pool in the event that the prestation system disappeared.

(3) National Archives of Senegal (NAS), Dakar, 3F100, p. 14. Colony Inspector Monguillot Inspection report on prison services in Senegal, February 1936.

(4) NAS, 3F101. Handwritten letter by the governor of Senegal. Note on penal camps, 5 January 1938.

TEPSIS PAPER 14 / SEPTEMBER 2016

And disappear it did, in 1937-1938, when the Popular Front created an additional tax.

The creation of penal camps thus sheds light on the colonial authorities' strategy: the various forms of forced labor were used as communicating vessels. In the context of a dwindling supply of prestation labor, roadwork was transferred to penal camps.

« Hope is dormant or destroyed »: Living conditions in penal camps The working day in penal camps was ten hours with a one-and-a-half-hour break. Inmates had to walk dozens of kilometers every day to reach roadwork sites in remote, deserted areas. For example, a report on penal camp C states that inmates left at 6:40 AM for roadwork site seven kilometers away (5). They spent most of their time breaking and carrying laterite blocks extracted in nearby quarries for the extension of the road network. Equipped with rudimentary tools, they also had to clear the undergrowth, dig ditches, and pack down new road sections on irregular or level ground.

In addition to these trying working conditions, living and hygiene conditions in the camps rendered everyday life even more difficult to bear. Architecturally, penal camps had an oversimplified design facilitating dismantling and moving. For instance, penal camp C had three rows of barbed wire, and guards released a dozen hounds in the camp at night to prevent escape (6). The camp comprised two twenty-by-five-meter steel-sheet barracks used as dormitories and housing up to a hundred inmates. One can easily imagine how tight and cramped space was. These conditions were far removed from those of the panopticon of the Foucauldian prison. Promiscuity was the rule and living conditions reduced to minimum needs.

ANNEX: REPORT ON KELLE PENAL CAMP, 1941 (NAS, 3F110)

As no lighting was allowed in the dormitories, inmates were immersed in total darkness upon their return from roadwork sites. No claims or collective complaints were permitted, the mail was subject to censorship by the camp management, and visits were allowed once a month -in theory at least -at the pleasure of guards and the manager (7).

In addition, penal camps had deleterious effects on inmates' physical and psychological health. Due to budget restrictions and mobility, sanitary conditions

(5) NAS, 3F107. Inspection report on penal camp C, 1938.

(6) Ibid.

TEPSIS PAPER 14 / SEPTEMBER 2016 were anything but a priority for colonial authorities. Several complaints by inmates, which got past censorship, mentioned insufficient food, clothing, sleeping accommodations and hygiene conditions.

A 1942 report following an unexpected inspection evokes the physical and mental state of the inmates of camp C. The police officer in charge of this inspection noted « that more than half of the inmates [have] festering sores or […] seemingly recent scars on their shoulders, arms and backs, and sometimes on their inner thighs. » According to the report, these injuries were caused by three separate factors.

First, the continuous friction due to carrying pails and rails on long distances irritated their shoulders and left deep wounds. Second, some injuries on the body or inner thighs came from intensive scratching caused by the presence of vermin in the dormitories. Last, inmates suffered much caning and physical abuse at the hands of camp guards. This daily suffering drove inmates to despair. Some of them developed incurable chronic diseases, leading the doctor of penal camp C to describe them as « human wrecks fated to certain death.» (8)

(7) Articles 25, 28, and 29. NAS, K237 (26). Acts adopted at a private meeting.

Order on the regulation of the service and system of penal camps, 7 January 1939.

(8) ANS, 3F136, Surprise Inspection of the Camp C by Lieutenant Boivin, 12 August 1942.

CONCLUSION

The economic logic of production and « development » of territories prevailed over surveillance and moral reform. This makes penal camps appear as venues of « open incarceration.» This interesting oxymoron evokes both a closed space under surveillance and an open space in which inmates moved between the spatial confinement of the prison and the open space of the roadwork sites where they worked. It that, compounded by degrading living and working conditions, incarceration gave rise to various forms of refusal of prison and penal work such as desertion, mutiny, complaints, or even, in some extreme cases, self-mutilation (9). In the 1930s, nearly 50 percent of inmates escaped from Senegalese prisons -and some did so several times. Complicity with guards was widespread and refusals to work occurred daily. The diversity and scale of these reactions show the inability of colonial authorities to address this resistance and more generally highlight the inertia of the colonial machine, which was far from omnipotent and often ineffectual.

(9) We have noted a dozen or so cases of self-mutilation in penal camps in archives covering the interwar years.

( 1 )

 1 See Frédéric Le Marcis and Marie Morelle, « Pour une pensée pluridisciplinaire de la prison en Afrique,» Afrique contemporaine, 253 (2015): 117-129; Florence Bernault, (ed.), Enfermement, prison et châtiments en Afrique du 19e siècle à nos jours (Paris: Karthala, 1999); Franck Dikötter, Ian Brown, (eds.), Cultures of confinement: A history of the prison in Africa, Asia and Latin America (Ithaca: Cornell University Press, 2007).

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Julie Guillot

Julie Guillot Lawyer

SUPPORT IN COMMUNITY LEGAL ACCESS POINTS: A LAWYER'S PERSPECTIVE

Keywords:

Imprisonment cuts off inmates from society. The lack of social, family, or professional bonds leaves inmates feeling empty, forgotten and left out. Time behind bars, regardless of its duration, causes disruption. Not only do inmates have their rights temporarily suspended, but they also experience their accelerated loss -in terms of housing, employment, family, or residence on French territory, etc. Life in prison has an inherent "unsocializing" effect on inmates in that it structures their lives for them and robs them of a large part of their daily responsibilities. However, sentences are finite, and men and women behind bars eventually return to the community… They must literally re-learn to live and function in society.

Recently, a man released after serving a ten-year sentence remained prostrate in front of the office door after an interview with SPIP (1) officers in Paris, waiting for the authorization to open the door. He was no longer used to doing things himself.

TEPSIS PAPER 06 / APRIL 2015 Laboratory of Excellence TEPSIS, 96 Bd Raspail 75006 Paris, http://tepsis. hypotheses.org 2

In addition to impacting an individual's ability to do things and make decisions, the return to freedom and the community constitutes a real shock and an additional challenge. For if time stops within prison walls, it goes on in the free world outside. Returning prisoners do not recognize the places and people they knew before incarceration.

WORK IN COMMUNITY LEGAL ACCESS POINTS

Droits d'Urgence runs legal access points at the Fresnes penitentiary and in the community in Paris that provide legal advice to inmates, returning prisoners and people with commuted sentences. Callers are employed or not, are consumers, have children, etc. Our task is to receive people with a social situation aggravated by affective, moral and economic distress in the privacy of an office, inform them on their basic rights and help them perform administrative tasks. The problems related to social exclusion that we handle daily at Droits d'Urgence have been exacerbated by criminal conviction.

Community legal access points are available to reentering prisoners and former inmates on probation or with commuted sentences. They provide them with information on their rights and obligations. They also help them continue dealing with the administrative tasks they undertook while still behind bars. Long-term support and follow-up of procedures engaged during detention are key in ensuring effective access to law. The higher the difficulties and vulnerabilities (isolation, illiteracy, language or psychological problems), the more urgent the need to help these men and women. The task is easier in the community because former inmates with commuted sentences can communicate and move freely to perform administrative tasks.

At the community legal access point of La Santé prison in Paris, access to law is a major issue for foreign nationals. Most of them encounter difficulties related to their administrative status, such as stalled procedures and denial of rights, which jeopardize the commutation of their sentences and, in the long run, their family or professional lives. The volunteers' active part in informing and assisting them in gaining recognition of their rights, which are often ignored or trampled on, favors the restoration of their dignity and of their standing as subjects of law. It helps provide answers to reentering prisoners and inmates in need, and plays a role in their reintegration. A PAST THAT WILL NOT PASS: A DOUBLE PENALTY

The criminal and prison records of foreign nationals constitute a genuine obstacle to their regularization even if they are entitled to residence permits corresponding to their personal situation.

For example, the number of years spent in France that is taken into account for the delivery of residence permits is impacted by a stay behind bars: if detention lasts more than a year, préfecture services only count the years starting from the release date even though the person has been in France for much longer. They also systematically put forward the threat to public order to hinder applications for residence or residence renewal.

A 21-year-old man was denied renewal of his "Private and Family Life" permit despite the fact that he had been living in France with his family since the age of five.

The commission in charge of residence permits gave an unfavorable recommendation, and the préfecture complied. In support of this denial, it was stated that the person posed a threat to public order (he had been given a two-month suspended prison sentence for violence during a fight). Préfecture services suggested that he find professional training in order to reapply for residency a year later. The question arises, however, as to how an illegal immigrant can find professional training. How can this young man be reintegrated? An appeal was filed against this decision on the grounds that it was a flagrant misjudgment and that it violated his right to respect for his private and family life, as set forth in Article 8 of the European Convention on Human Rights.

Permanent or temporary banishment from French territory imposed as an additional sentence to "protected" persons, i.e., persons protected against this very type of measure, is also regularly exposed by the lawyers volunteering at legal access points for prisoners and former prisoners. It constitutes a genuine "double penalty" leading to pauperization and restricting the possibility of their sentences being commuted. It also jeopardizes their integration into French society even when they cannot return to their countries of origin, where they face the risk of inhuman and degrading treatment (persecutions, deteriorating health for lack of medical treatment, etc.). Obtaining redress is a daily challenge. One such challenge is disputing orders of the prefect requiring immediate removal from the territory -that is, without waiting for the expiration of the 30 day period of voluntary departure. Appeals must be filed with territorially competent administrative courts within 48 hours following notification -which means that, in record time, the person concerned must understand the nature of the measure, find writing assistance, find arguments, gather documents and be able to send their file to the court… when they do not necessarily have access to a fax 24 hours a day! These persons' actual experience with legal access reflects the mindset of society.

It shows the place that we are willing, or not willing, to give them, and our respect, or lack thereof, for their dignity. Successful reintegration requires that everything be done so that they find their rightful place. For this, they must receive strong and regular legal support. This will help them become aware of their rights and duties, restore their self-confidence, and engage for recognition of their fundamental and inalienable rights. All association volunteers and institutional actors must be involved in this process.