Z Belemaalem

P Redou
email: redou@enib.fr

M Parenthoën

Solving Partial Differential Equations with Chaotic Asynchronous Schemes in Multi-Interaction Systems

Keywords: Transport, Diffusion, Advection, Schrödinger equation, Chaotic asynchronous scheduling, Multi-interaction system (MIS)

published or not. The documents may come

Introduction

The multi-interaction system framework (MIS) is a computing paradigm for complex system modeling [START_REF] Desmeulles | In virtuo experiments based on the multi-interaction system framework: the RéIS-COP meta-model[END_REF]. Mostly used in virtual reality applications, its role is to enable multi-model/multi-scale, interactive and (as close as possible to) real-time simulations. It emanates from the multi-agent approach and proposes a shift of focus from individual-centered to interactioncentered for studying the living and virtual reality. The MIS structure depends on the reification of interactions, on chaotic asynchronous iterations and on an organisation in autonomous systems. More precisely, a MIS is a set of active autonomous objects modeling interactions betwen the passive structural elements named constituents. Interaction is therefore the autonomous elementary active unit of the MIS, and constituents are the medium by which the interactions are linked. The couples interaction/set of associated constituents form systems. These elementary systems are thus structurally coupled to one another whilst the interactions act on the same constituents. In a MIS, interactions are active processes scheduled by chaotic asynchronous iterations. The notion of asynchronous iteration is different from the one commonly used and explained in [START_REF] Frommer | On asynchronous iterations[END_REF]. In our context, it expresses the fact that activities 1 , each modeling a phenomenon, have their own lifetime and are processed one after the other: the constituents are changed after each interaction activity for the following one. The notion of chaotic iteration deals with activities which are supposed to intervene at the same time: they occur one after the other (as for asynchronous iterations), but in a random order to avoid bias in computations [START_REF] Redou | Formal validation of asynchronous interactionagents algorithms for reaction-diffusion problems[END_REF]. This paradigm involving the reification of interaction and asynchronous scheduling leads to imagine original algorithms for 1 An activity matches a date on the execution of a process. 1 solving partial differential equations (PDEs), as they can appear to model natural phenomena.

Our MIS approach for solving PDEs distinguishes itself from classical methods as finite differences methods, finite element method, finite volume method (see [START_REF] Kumar | Finite volume element method for second order hyperbolic equations[END_REF] for hyperbolic problems), spectral method, meshfree methods, or domain decomposition methods and multigrid methods. It also stands out more recent methods, as for example, the generalized finite difference method (GFDM) [START_REF] Benito | Application of the generalized finite difference method to improve the approximated solution of pdes[END_REF], the indirect RBFN method [START_REF] Nam | Indirect rbfn method with scattered points for numerical solution of pdes[END_REF] and the numerical meshfree technique [START_REF] Sirajul | A numerical meshfree technique for the solution of the mew equation[END_REF][START_REF] Bourantas | Adaptive support domain implementation on the moving least squares approximation for mfree methods applied on elliptic and parabolic pde problems using strong-form description[END_REF], or (for parabolic problems) monotone Jacobi and Gauss-Seidel methods [START_REF] Boglaev | Numerical solutions of nonlinear parabolic problems by monotone jacobi and gauss-seidel methods[END_REF], NIPG/SIPG discontinuous Galerkin methods for time-dependent diffusion equation [START_REF] Proft | Discontinuous galerkin methods for convection-diffusion equations for varying and vanishing diffusivity[END_REF], and for instance conservative local discontinuous Galerkin methods for time dependent Schrödinger equation [START_REF] Lu | Conservative local discontinuous galerkin methods for time dependent schrödinger equation[END_REF]. Indeed, all these methods are based on synchronous iterations, so that they are not well adapted to the kind of simulations described above [START_REF] Atlan | The living cell as a paradigm for complex natural systems[END_REF].

Constituents and interactions in a MIS for a PDE resolution scheme. In (A), the single constituent is the whole mesh or set of points and the unique interaction instanciated by the PDEbased phenomenon acts according to the classical scheme. In (B), the constituents correpond to mesh elements or points and the PDE-based phenomenon instanciates as much interactions as interfaces between constituents. Each interaction acts locally as a sort of Maxwell deamon associated to that phenomenon. The (B) formulation is much more compatible with MIS assumptions than the (A). Let us illustrate how phenomena modeled by PDEs can be included into a MIS: by using directly a classical synchronous scheme or by rewriting the schemes into asynchronous ones (figure 1).

(1) A first idea could be to take a numerical scheme associated to the resolution of those PDEs as the interaction of that phenomenon acting from t to t + δt on a single constituent : the whole mesh or the whole set of points. Since in that case there is a unique constituent and a unique phenomenon interacting with that constituent, only one interaction will be instanciated in the simulation. Its activity consists in computing each δt one step of the associated scheme to transform the single constituent from t to t + δt. If another phenomenon acts on the same constituent, the corresponding interaction can only act before or after the previous one. If this last phenomenon is modeled by PDEs with its own numerical scheme, the resulting numerical resolution of both phenomena within asynchronous iterations can be viewed as a sort of stochastic splitting method applied to both schemes. Another way to solve this problem could be to write a new single scheme valid for the PDE system obtained by both phenomena; this is not very accurate in terms of structural coupling, thus moves away from the MIS principles for complex system modeling. Anyway, if one of the phenomena involved in the complex system and acting on such a constituent is not modeled by PDEs, its intrication with PDE-based models won't be possible for time intervals smaller than the time step δt used by PDE schemes.

(2) The second idea, that we defend, consists in building asynchronous schemes for interactions at each interface between two constituents and acting as Maxwell-daemons to transfer flow between several separate environments, namely constituents, according to PDEs. The constituents are not the whole mesh or sets of points but the elements of the mesh or the points themselves. Each phenomenon instanciates as much interactions as interfaces between constituents. When several PDE-based phenomena are involved, asynchronous scheduling of interaction activities can be viewed as a sort of generalized stochastic splitting method at the level of interfaces. The resulting simulation respects MIS principles and intrication with no PDE-based models should be facilitated as a consequence of the structural coupling provided by MIS.

This paper aims at proposing asynchronous numerical schemes for solving transport and diffusion problems, according to the assumptions of MIS. In this context, our work applies to the simulation of transport phenomena such as heat transfer, mass transfer, fluid dynamics (momentum transfer), thermal or chemical diffusion phenomena and the averaging of a wave packet. Our interactions complete previous work based on autonomous interface-agents [START_REF] Redou | Formal validation of asynchronous interactionagents algorithms for reaction-diffusion problems[END_REF][START_REF] Redou | Absolute stability of chaotic asynchronous multi-interactions schemes for solving ode[END_REF] for reaction-diffusion problems. Furthermore, as we show in this paper, our approach provides numerical results which are as relevant as classical methods.

In the next section, we present the chaotic and chaotic-asynchronous multi-interaction model for the numerical computation of transport phenomena. In section 3, we describe a chaotic multi-interaction model for thermal diffusion phenomena, and we show how simple it is to intricate such autonomous models with an application to the phenomenon of transportdiffusion. Finally (section 4), we introduce a chaotic multi-interaction model for the averaging of a wave packet. NB: we have programmed our methods by using the RÉISCOP meta-model which offers an implementation for MIS [START_REF] Desmeulles | In virtuo experiments based on the multi-interaction system framework: the RéIS-COP meta-model[END_REF].

Transport models for MIS

Principle

In physics, chemistry, biology and engineering, a transport phenomenon is any of various mechanisms by which particles or quantities move from one place to another. The laws which govern transport connect a flux with a "motive force". Three common examples of transport phenomena are diffusion, convection, and radiation. We consider these three examples to build our model, called transport-agent model.

Each transport-agent represents a flow of transport of particles or quantities from one place to another.

-Perception : measuring particles or quantities, and transport velocity inside each cell of its neighborhood given by a spatial resolution grid; -Decision : computing the quantities or the particles passing from one cell to the other; -Action : modifying the quantities or the particles in each cell of its neighborhood.

Illustration

Let us illustrate our views, and consider three given functions

c : (x, t) ∈ [a, b] × [0, T] → c(x, t) ∈ R, f : (x, t) ∈ [a, b] × [0, T] → f (x, t) ∈ R, w : x ∈ [a, b] → w(x) ∈ R.
The transport problem that we consider here consists in seeking a function

u : (x, t) ∈ [a, b] × [0, T] → u(x, t) ∈ R which satisfies: ∂u ∂t (x, t) + c(x, t) ∂u ∂x (x, t) = f (x, t), x ∈ [a, b], t ∈ [0, T], u(x, 0) = w(x), x ∈ [a, b] u(a) = u(b).
(1) In the following, we present our transport-agent model with various methods to approximate the solution u for the problem (1) above. We establish formal proofs of convergence for our methods in the simple case where c(x, t) = c is a positive constant, and f = 0. We begin with a space discretization (x i ) 0≤i≤N of the system, and consider that in the middle of each interval

[x i , x i+1 ] is located a transport-agent, noted A i in point x i+1/2 = x i +x i+1 2
which has a "lifetime" δt i , whose role will be, when it takes the hand, to update the values of the solution in cells C i and C i+1 with the same length

δx where C i = [x i-1/2 , x i+1/2 ].
The values in other cells are unchanged. Suppose that the transport-agent intervenes at moment n: it considers the overall state of the system at this time, updates the values of the solution of two adjacent cells, and leaves the other values unchanged. As in the finite volume method, and integrating into the control cell C i × [t n , t n+1 ] we obtain:

1 δx x i+1/2 x i-1/2 u(x, t n+1 )dx - x i+1/2 x i-1/2 u(x, t n )dx = - δt δx 1 δt t n+1 t n cu(x i+1/2 , s)ds - 1 δt t n+1 t n cu(x i-1/2 , s)ds .
(

) 2
This writing suggests that:

i) An interesting description is given by:

u n i ≃ 1 δx x i+1/2 x i-1/2
u(x, t n )dx, average values of u in each cell of control. Another benefit of this description is that u can be undefined in one point, for example when the solution is discontinuous. However, the solutions are integrable, so that their average value is always defined.

ii) the stock of increase of u n i during a time step is given by

1 δt t n+1 t n cu(x i+1/2 , s)ds - 1 δt t n+1 t n cu(x i-1/2 , s)ds.
The term 1 δt t n+1 t n cu(x i+1/2 , s)ds is interpreted as the average flow on time of outgoing or incoming by interface of C i located in x i+1/2 , and the manner of approaching this average flow determines the activity of transport-agent A i .

Multi-Interaction chaotic model 2.3.1 Chaotic scheduling of activities

Definition: We name cycle of actions a succession of actions of transport-agents involving each of them only once.

In our model, the cycles of actions follow, but the sequence of actions within a cycle is not imposed, and changes from one cycle to another, randomly. Figure 2 illustrates this scheduling strategy. Note: Let σ ∈ S N (S N permutation group). We define the following application, which describes a cycle of action.

φ σ = f σ(N) • f σ(N-1) • • • f σ(1)
where N is the number of transport-agents and f i describes the action of A i .

Let us see exactly how a transport-agent A i is involved: at time t n the order of its intervention is σ -1 (i), and when A i starts to operate, it perceives nearby cells

Approximation inspired by Upwind scheme: upwind chaotic transport

In the following,

u n+σ -1 (i)/N i
is the new value of the solution in the cell C i updating by A i . The action of A i , i ∈ {0, N}, is described by:

u n+ σ -1 (i) N i = u n+ σ -1 (i)-1 N i -c δt δx u n+ σ -1 (i)-1 N i u n+ σ -1 (i) N i+1 = u n+ σ -1 (i)-1 N i+1 + c δt δx u n+ σ -1 (i)-1 N i u n+ σ -1 (i) N j = u n+ σ -1 (i)-1 N j ∀j ∈ {i, i + 1} (3)
and the actions of A 0 and A N respectively by:

u n+ σ -1 (0) N 0 = u n+ σ -1 (0)-1 N 0 u n+ σ -1 (0) N 1 = u n+ σ -1 (0)-1 N 1 + c δt δx u n+ σ -1 (0)-1 N 0 u n+ σ -1 (0) N j = u n+ σ -1 (0)-1 N j ∀j ∈ {0, 1} u n+ σ -1 (N) N N-1 = u n+ σ -1 (N)-1 N N-1 -c δt δx u n+ σ -1 (N)-1 N N-1 u n+ σ -1 (N) N N = u n+ σ -1 (N)-1 N N + c δt δx u n+ σ -1 (N)-1 N N-1 -c δt δx u n+ σ -1 (N)-1 N N u n+ σ -1 (N) N j = u n+ σ -1 (N)-1 N j ∀j ∈ {N -1, N}.
Note: In the following of this section, we use the notation ω = c δt δx . We can describe the actions of A i by a matrix:

f i = (1 -ω)E ii + ωE i+1,i + ∑ j =i E jj
where E ij are elementary matrices.

Let us illustrate our point of view by two examples, and consider at first the following case: at moment n, a cycle begins, which we assume to be the cycle involving the A i s, in this order: A N-1 ,A N-2 , ...,A 0 . Using the notations of the previous paragraph, we get the fact that at time n + 1, the state of the system is given by

U n+1 = f 0 • f 1 • • • • • f N-1 (U n ) where U n = (u n 0 , u n 1 , • • • , u n N )
. Thus, in the choice of the precise sequence of actions in a cycle, we have for example with N = 4

U n+1 =     1 0 0 0 ω 1 -ω 0 0 0 ω 1 -ω 0 0 0 ω 1 -ω     U n .
Now, let us consider the case where at time n, a cycle begins, which we assume to be the cycle involving, in this order, A 0 , A 1 ,A 2 , and A 3 . Using the notations of the preceeding paragraph, we get the fact that at time n + 1, the state of the system is given by

U n+1 = f 3 • f 2 • f 1 • f 0 (U n ) =     1 0 0 0 ω(1 -ω) 1 -ω 0 0 ω 2 (1 -ω) ω(1 -ω) 1 -ω 0 ω 3 (1 -ω) ω 2 (1 -ω) ω(1 -ω) 1 -ω     U n .
Note: If we develop the coefficients of the matrix of the second case above into sums of powers of ω , and if we keep only the terms of order 1, we obtain a similar Upwind scheme.

In the general case the idea is to determine the matrix that allows us to move from one moment to another by a column by column computation.

Proof of convergence

From Lax theorem2 , since our scheme is linear, the following lemma proves the convergence: Lemma: The upwind chaotic transport scheme is stable and consistent with order 1 in space and 1 in time if |c|δt ≤ δx 3 Proof. After a simple calculation we find that

u n+1 i = (1 -ω)u n i + ωu n i-1 + R 1 + R 2 where R 1 = -ξ 1 ω 2 u n i-1 + ξ 2 ω 2 u n i-2
and

R 2 = i-2 ∑ k=0 p k .u n k .
Here ξ i are equal to 1 or 0, and for all k, p k is a polynomial, and we notice that in this scheme all the coefficients are positive or zero.

Stability:

We have

|u n+1 i | ≤ (1 -ω)|u n i | + ω|u n i-1 | + ξ 1 ω 2 |u n i-1 | +ξ 2 ω 2 |u n i-2 | + A with A = i-2 ∑ k=0 p k .|u n k | so that u n+1 ∞ ≤ (1 + ξ 1 ω 2 + ξ 2 ω 2 + i-2 ∑ k=0 p k ) u n ∞
and in the conditions of this lemma, there are constants C 2 and C 3 such that

ξ 1 ω 2 + ξ 2 ω 2 ≤ C 1 δtδx and i-2 ∑ k=0 p k ≤ C 2 δtδx 2 .
In fact, for all k, p k is a polynomial of degree equal or higher than 3 whose coefficients of the terms of degree lower than 3 are null or p k is null. Thus, p k = O(δtδx 3 ) and more specifically ∀k, |p k | ≤ 2ω 3 ≤ 2δx 3 δt.

Therefore, we get i-2

∑ k=0 p k ≤ 2| i-2 ∑ k=0 δx 3 δt ≤ 2δx 3 δt.
However Nδx ≤ ba thus ∑ i-2 k=0 p k = O(δtδx 2 ), so that there exists a constant (small enough) such that:

u n+1 ∞ ≤ (1 + ρδt) n u 0 ∞ so that u n+1 ∞ ≤ e nρδt u 0 ∞ or nδt ≤ T thus u n+1
∞ ≤ e ρT u 0 ∞ . Thus the scheme is stable for the norm . ∞ . Consistency and order of the scheme:

Let v be a regular solution of the continuous problem (1), we will estimate the quantity

ξ = v(t + δt, x) -(1 -ω)v(t, x) -ωv(t, x -δx) +ξ 1 ω 2 v(t, x -δx) + ξ 2 ω i-2 ωv(t, x -2δx) -∑ i-2 k=0 p k .v(t, x -kδx)
where ξ 1 , ξ 2 . . . are null or equal to 1. To reduce the writing we consider the following notations:

v(t, x) = v, v tt (t, x) = v tt and v xx (t, x) = v xx
After an order 2 limited development one obtains:

ξ = δt 2 i-1 v tt -ωδx 2 v xx + ξ 1 ω 2 v(t, x -δx) + ξ 2 ω 2 v(t, x -2h) -∑ i-2 k=0 p k .v(t,
xkδx) Thus we get the relations:

1. δt 2 v tt = O(δt 2 ) 2. ξ 1 ω 2 v(t, x -δx) + ξ 2 ω 2 v(t, x -2δx) = O(δtδx) 3. ωδx 2 v xx = O(δtδx) 4. ∑ i-2 k=0 p k .v(t, x -kδx) = O(δtδx 2 ). Finally, ξ δt = O(δx) + O(δt).

Numerical illustration

In figure 3, we compare our Transport-Agent approach (T-A) to upwind scheme (UW) and the exact solution(EX), with w(x) = e -x 2 . We note that our ap- proach is more accurate than the upwind scheme for the "small" iteration count (200 interventions of each transport-agent), and still similar to upwind scheme(cfl=0.1) for "great" iteration count.

Approximation inspired by Lax-Wendroff scheme: Lax-Wendroff chaotic transport

Now we approximate the flow by the flow of the Lax-Wendroff scheme [START_REF] Anderson | Computational fluid mechanics and heat transfer[END_REF]. The orders of intervention of A i , A 0 , A N are respectively σ -1 (i), σ -1 (0) and σ -1 (N) and their actions are defined respectively by:

u n+σ -1 (i) i = u n+ σ -1 (i)-1 N i -c δt δx u n+ σ -1 (i)-1 N i + 1 2 1 -δt δx u n+ σ -1 (i)-1 N i+1 -u n+ σ -1 (i)-1 N i u n+σ -1 (i) i+1 = u n+ σ -1 (i)-1 N i+1 + c δt δx u n+ σ -1 (i)-1 N i + 1 2 1 -δt δx u n+ σ -1 (i)-1 N i+1 -u n+ σ -1 (i)-1 N i u n+σ -1 (i) j = u n+ σ -1 (i)-1 N j ∀j ∈ {i, i + 1} u n+ σ -1 (0) N 0 = u n+ σ -1 (0)-1 N 0 u n+σ -1 (0)/N 1 = u n+ σ -1 (0)-1 N 1 + c δt δx u n+ σ -1 (0)-1 N i + 1 2 (1 -δt δx )(u n+ σ -1 (0)-1 N i+1 -u n+ σ -1 (0)-1 N i ) u n+σ -1 (0)/N j = u n+ σ -1 (0)-1 N j ∀j ∈ {0, 1} u n+σ -1 (N)/N N-1 = u n+ σ -1 (N)-1 N N-1 -c δt δx u n+ σ -1 (N)-1 N N-1 + 1 2 (1 -δt δx )(u n+ σ -1 (N)-1 N N -u n+ σ -1 (N)-1 N N-1 ) u n+σ -1 (N)/N N = u n+ σ -1 (N)-1 N N + c δt δx u n N-1 +u n+ σ -1 (N)-1 N N -1 2 (1 -δt δx )u n+ σ -1 (N)-1 N N-1 u n+σ -1 (N)/N j = u n+ σ -1 (N)-1 N j ∀j ∈ {N -1, N} (4)
In figures 4 and 5, we compare our Transport-Agent (T-A) approach to upwind scheme (UW) and the exact solution(Ex), with w(x) = e -x 2 . We note that our approach is more accurate than the upwind scheme for the "small" values of cfl, but for values of cfl ≥ 0.3 accuracy decreases (cf figure 5).

Multi-interactions chaotic asynchronous method 2.4.1 Formalization of actions of transport-agents

In this section,we keep all the previous notations and consider that the order of interventions of the transport-agents is carried out by temporal cycles, containing at least one intervention of each transportagent (and thus at least two updates for each value in a given cell).

Here again, the cycles of actions follow each other, but the sequence of actions within a cycle is not imposed, and changes from one cycle to another, randomly. We thus consider a time step δt as the unit time for the transport-agents, and that each A i operates p i times, and has one "lifetime" p i δt = δt i ; we also consider the relation pδt ≤ δx 3 , with p = max i (p i ) and thus the time step of our system is naturally △t = mδt where m = lcm(p 1 , . . . , p N-1 ), where lcm is the Least Common Multiple. Figure 6 illustrates this asynchronous scheduling strategy.

ρ : {1, 2, • • • , m} -→ {A N , A N-1 , • • • , A 0 } an application with: card(σ -1 (A i )) = m p i
, the number of times A i operates during one cycle.

We define the following application, which describes a cycle of actions.

φ ρ•σ = g σ(ρ -1 (A N )) • g σ(ρ -1 (A N-1 )) • • • g σ(ρ -1 (A 1 ))
where g is a function such that g σ(ρ -1

(A i )) = f σ(a) i • f σ(b) i • • • • • f σ(e) i when ρ -1 (A N ) = {a, b, • • • , e}
and f j i describes the action of transport-agent A i and order of intervention j.

Approximation inspired by upwind scheme: upwind chaotic asynchronous transport

By the same reasoning as in the previous section, we will define the action of the transport-agents following Upwind and Lax-Wendroff schemes. We start with upwind scheme, and consider at time t n the order of interventions of A i A 0 and A N are respectively j ∈ σ(ρ -1

(A i )), k ∈ σ(ρ -1 (A 0 )) and l ∈ σ(ρ -1 (A N )): u n+j/m i = u n+(j-1)/m i + -c δt i δx u n+(j-1)/m i u n+j/m i+1 = u n+(j-1)/m i+1 + c δt i δx u n+(j-1)/m i u n+j/m j = u n+(j-1)/m j ∀j ∈ {i, i + 1} u n+k/m 0 = u n+(k-1)/m 0 u n+k/m 1 = u n+(k-1)/m 1 + c δt 0 δx u n+(k-1)/m 0 u n+k/m j = u n+(k-1)/m j ∀j ∈ {0, 1} u n+l/m N-1 = u n+(l-1)/m N-1 -c δt N-1 δx u n+(l-1)/m N-1 u n+l/m N = u n+(l-1)/m N + c δt N-1 δx u n+(l-1)/m N-1 -δt N-1 δx u n+(l-1)/m N u n+l/m j = u n+(l-1)/m j ∀j ∈ {N -1, N}.
To reduce the writings we adopt the notation: ω i = a p i δt δx , ω = ap δt δx and each A i is represented by the linear application f i .

Consider for instance the following case: at time n, a cycle begins, we assume to be the cycle involving, A 0 , A 1 2 times and A 2 within the same cycle, in this order: A 0 , A 1 ,A 2 and A 1 . Using the notations of the previous paragraph, we get the fact that at time n + 1, the state of the system is given by

U n+1 =     1 0 0 0 ω 0 (1 -ω 1 ) 2 0 0 0 2ω 1 -(ω 1 ) 2 -ω 1 ω 2 1 -ω 2 0 0 ω 1 ω 2 ω 2 1 -ω 2     U n .

Proof of convergence

As in the previous case, the matrix making it possible to go from one time step to the following is lower triangular, so that to determine the form of the resulting numerical scheme it is enough to determine column by column the matrix by taking account of the fact that f i (e i ) = (1ω i )e i + ω i e i+1 where (e i ) 1≤i≤N is the canonical basis of R N .

In this case, the scheme is written in the following form:

u n+1 i = αu n i + βu n i-1 + γu n i-2 + R 1 + R 2 After a simple calculation one finds α = 1 - m/p i ω i , β = m/p i-1 ω i-1 , |R 1 | ≤ C 1 ω 2 (|u n i | + |u n i-1 |) and |R 2 | ≤ C 2 ω 3 (∑ i k=0 u n k )
where C 1 and C 2 are constants. Knowing that m/p i ω i = m/p i-1 ω i-1 and with the same type of calculation that the previous paragraph one proves the stability and the consistency of order 1 in time and space, and thus the convergence.

Numerical illustration

The graphs on figure 7 represent the results of this method (transport-agent asynchronouschaotic method). We compare our Transport-Agent asynchronous-chaotic method (T-A) to upwind scheme (UW) and the exact solution (Ex), with w(x) = e -x 2 . NB: Each A i has its own " lifetime" δt i . As in the and δx = 0.05) , UW (cfl=0.1 and 0.9) and EX.

previous paragraph, we note that our approach is more accurate than the upwind scheme for the "small" iteration count (200 interventions of each transportagent), and still similar to upwind scheme(cfl=0.1) for "great" iteration count.

Approximation inspired by Lax-Wendroff scheme: Lax-Wendroff chaotic asynchronous transport

Now we will illustrate (figure 8) the results of this method (transport-agent chaotic asynchronous method), for which the approximation of action of a transport-agent is inspired by the scheme of Lax-Wendroff. We compare our transport-agent asynchronous-chaotic method (T-A) to upwind scheme (UW) and the exact solution(Ex). Therefore, as in the previous case we have a better accuracy than the upwind scheme for the values of c f l max below 0.3, but beyond this value it loses accuracy (cf figure 9).

Diffusion model for MIS

Principle

In this section, we present an alternative to the traditional approach, for the resolution of the phenomenon of thermal transfer within a homogeneous environment. This approach is an extension of the one exposed in [START_REF] Redou | Formal validation of asynchronous interactionagents algorithms for reaction-diffusion problems[END_REF], it is based on the multi-interactions approach, whose main role is the transfer of energy. Firstly each diffusion-agent perceives its environment, then decides on what it must do, taking into account the laws of behavior, which are allocated to him. Finally the diffusion-agent acts by modifying its environment. We describe in the continuation of this section, the action of the diffusion-agents.

Illustration

The parabolic problem considered here consists in seeking a function u : (x, t) ∈ [a, b] × R + → u(x, t) ∈ R, giving the temperature of a metal bar, such that:

∂u ∂t (x, t) -k ∂ 2 u ∂x 2 (x, t) = f (x, t), ∀x ∈]a, b[, t > 0, u(a, t) = u(b, t) = 0, ∀t > 0, u(x, 0) = w(x), ∀x ∈]a, b[, (5)
where w : x ∈ [0, 1] → w(x) ∈ R is a given initial condition, k and f are respectively thermal conductivity and the power per unit of length provided to the bar, both of them are divided by the voluminal density and the mass specific heat.In the following, we present diffusion-agents model to approach the solution u of the problem described above. We compare the numerical solutions with the Finite volume method within a simple framework where f are identically null (no heat source brought to the bar). It is necessary to note that the parabolic equations corresponding to the phenomena of diffusion admit well known analytical solutions [START_REF] Reinhard | Equations aux dérivées partielles[END_REF], and can in the same way be solved numerically by various methods (DF, EFF, Monte Carlo [START_REF] Lapeyre | Introduction to Monte-Carlo Methods for Transport and Diffusion Equations[END_REF]).

Main Results

Multi-interaction chaotic model

In the following of this section, we adopt the notation :

λ = k δt δx 2 .
Keeping the same ratings as before, we define the action of diffusion-agent A i by:

u n+j/n i = u n+(j-1)/N i + λ u n+j/n i+1 -u n+j/n i u n+j/n i+1 = u n+(j-1)/N i+1 -λ u n+j/n i+1 -u n+j/n i u n+j/n j = u n+(j-1)/N j ∀j ∈ {i, i + 1} (6)
where j denotes the order of intervention of A i inside the cycle starting at t n .

In figure 10 we compare our Diffusion-Agent approach (D-A) to the finite difference method (FDM).

Intricating transport and diffusion models

We show here how can be intricated previous choatic schemes for transport and diffusion into a new scheme for the mixed transport-diffusion problem:

∂u ∂t (x, t) + c ∂u ∂x (x, t) -k ∂ 2 u ∂x 2 (x, t) = 0.
We provide A i with two methods, transport and diffusion: 1. for the transport method, we use the scheme described by equations (4) ;

2. for the diffusion method, we take equations [START_REF] Besse | A relaxation scheme for nonlinear schrödinger equation[END_REF].

Each of these methods is associated to an activity with the same frequency. Activities are scheduled by chaotic iterations.

By this way, we relocalise at the level of each transport-dissusion-agent the method of splitting [START_REF] Ropp | Stability of operator splitting methods for systems with indefinite operators: Advection-diffusion-reaction systems[END_REF].

For the splitting method, we consider the system as

du dt = O A (u) + O D (u) ( 7
)
where u is the solution vector and O A and O D are the advection and diffusion terms. We then split the terms, creating two systems of equations. Thus, a single step of a first-order splitting method advancing the solution from t n to t n+1 = t n + ∆t amounts to an application of time discretization applied to the system

du * dt = O A (u * ) on [t n , t n+1 ], u * (t n ) = u n , ( 8
)
du * * dt = O D (u * * ) on[t n , t n+1 ], u * * (t n ) = u * (t n+1 ), (9)
with u n+1 = u * * (t n+1 ).

Using operators notation we denote the solution of a step as u * = S X , where X = A or D to denote the different substep solution operators. Thus, the above method can be written as u n+1 = S A • S D u n .

Our model goes further than the splitting method because transport and diffusion activities are fully mixed. Each transport-diffusion-agent is able to apply its role of transport-agent and/or diffusion-agent in random order exactly once per cycle. Figure 11

The averaging of a wave packet 4.1 Principle

After having studied the problem of diffusion, we want now to extend the work to the spreading phenomenon of a wave packet. We are interested in Schrödinger equation, which describes the temporal evolution of the wave function ψ which characterizes a system whose squared module, assessed at position x and time t, |ψ(x, t)| 2 , is interpreted as the probability density of occurrence of the position x and time t. Schrödinger equation is a partial differential equation in the form ıh∂ t ψ = Hψ where H is the Hamiltonian operator of the system (the sum of operators of kinetic and potential energy)

H(t) = T + V(t).
The Hamiltonian operator depends on time if the potentials involved are explicitly dependent themselves on time, for example in such case of dealing with some problems in the semi-classical way [START_REF] Yamanaka | Timedependent coupled-channel calculations of positronium-formation cross sections in positronhydrogen collisions[END_REF][START_REF] Melezhik | Nonperturbative time-dependent approach to breakup of halo nuclei[END_REF] or when we study the effect on an atom or molecule of an external electric field variable [START_REF] Dundas | Efficient grid treatment of the ionization dynamics of laser-driven h2+[END_REF]. However when the operator H does not depend on time it is reduced to an Eigenvalue equation called stationary Schrödinger equation. From the properties of the Schrödinger equation we can show that there is an operator of evolution which applied to the state of a system at a given time, provides later system state [START_REF] Baye | Fourth-order factorization of the evolution operator for timedependent potentials[END_REF]. The resolution of the Schrödinger equation is then reduced to the search for this operator which is itself solution of an evolution equation. But except for rare cases, the Schrödinger equation has no analytical solutions. The numerical resolution of Schrödinger equation allows to circumvent this problem. Additionally, a numerical approach allows a concrete representation of the probability density which gives it an educational aspect. There are several ways for this numerical resolution: the scheme due to Delfour, Fortin and Payre [START_REF] Delfour | Finitedifference solutions of a non-linear schrödinger equation[END_REF], relaxation scheme [START_REF] Besse | A relaxation scheme for nonlinear schrödinger equation[END_REF], the variational method of Lagrange networks for solving the stationary Schrödinger equation [START_REF] Hesse | Lagrange-mesh calculations of three-body atoms and molecules[END_REF] and splitting schemes [START_REF] Besse | Order estimates in time of splitting methods for the nonlinear schrödinger equation[END_REF].

Illustration

The Schrödinger equation for a free particle (V = 0) in one dimension is:

ıh∂ t ψ(x, t) = -h 2 /2m∂ xx ψ(x, t).
The general solution of this equation can be expressed by a linear combination of solutions which are defined by plane wave

ψ(x, t) = A exp i(kx -ω(k)t) where ω(k) = hk 2 /2m.
Such a solution, called wave packet, takes the following form:

ψ(x, t) = ∞ -∞ g(k) exp i(kx -ω(k)t)
where g(k) is an arbitrary square sommable complex function. The wave packet is said Gaussian if g(k) is a Gaussian centered around k 0 :

g(k) = exp -α 2 (k -k 0 ) .
The squared module of the wave function is:

|ψ(x, t)| 2 = π α 4 + h2 t 2 4m 2 exp   - α 2 (x -vt) 2 α 4 + h2 t 2 4m 2  
where v = hk 0 m . Subsequently we consider that the spreading-agents A i all have the same "lifetime" δt, and that their order of intervention is carried out by temporal cycles containing one intervention and only one of each spreading-agent (and thus two updates for each value in a given cell of control).

Formalization and main results

Let us see precisely how a spreading-agent A k intervenes : this spreading-agent separates two cells C k and C k+1 . We suppose that the spreading-agent intervenes at moment n, the updates of ψ k and ψ k+1 are carried out as follows: Integrating on the cell of control C k × [t n , t n+1 ], and choosing an implicit estimate of the term on the right, one obtains:

1 h x i+1/2 x i-1/2 iψ(x, t n+1 )dx - 1 h x i+1/2 x i-1/2 iψ(x, t n )dx = - δt h h2 2m 1 δt t n+1 t n ∂ x ψ(x i+1/2 , s)ds -1 δt t n+1 t n ∂ x ψ(x i-1/2 , s)ds (10
) We now follow the same reasoning for the construction of the model as in the previous part, and make of it an implicit approximation of the gradient of function ψ. The action of the spreading-agents A k , A 0 and A N is defined by We can describe this action by the matrix

ψ n+o k /N k = ψ n+(o k -1)/N k + i δt δx 2 (ψ n+o k /N k+1 -ψ n+o k /N k ) ψ n+o k /N k+1 = ψ n+(o k -1)/N k+1 -i δt δx 2 (ψ n+o k /N k+1 -ψ n+o k /N k
f k = 1+κ 1+2κ (E kk + E k+1,k+1 ) + κ 1+2κ (E k,k+1 + E k+1,k ) + ∑ j =k E jj
The matrices f i , f 0 and f N are explicitly given by:

f i =             1 0 • • • 0 0 0 1 • • • 0 0 . . . . . . 0 1+κ 1+2κ κ 1+2κ κ 1+2κ 1+κ 1+2κ . . . 1             f 0 =             0 0 • • • 0 0 0 1 1+κ • • • 0 0 . . . . . . 0 1 1 . . . 1             f N =             1 0 • • • 0 0 1 • • • 0 . . . . . . 0 1 . . . 1 1+κ 0 0 0 0            
Figure 12 shows the evolution of Gaussian wave packet. This simulation was conducted in an interval [-5, 5] with a step space discretization δx = 0.1 and the duration of each spreading-agent is δt = 0.01

Conclusion and outlook

In the framework of the simutation of complex systems, we need numerical schemes for solving PDEs, resting on asynchronous iterations. We have proposed a model which is adapted to such simulations for transport phenomena, thermal diffusion phenomena, transport-diffusion phenomena and the spreading phenomenon of a wave packet.

We have demonstrated the convergence for the transport-agent method when its action is inspired by the upwind scheme, using both an asynchronous and a chaotic-asynchronous method. As we dispose of local conditions of stability, each interaction chooses activities in a way which is compatible with these conditions. We have illustrated numerically the relevance of our method for diffusion phenomena. We have shown through the transport-diffusion problem that such asynchronous models can be coupled easily. Finally we achieved a numerical explanation of a wave packet spreading, using an asynchronous model.

We plan to demonstrate convergence properties for diffusion-agent and transport-diffusion-agent methods. Furthermore, we will add an activity within each mesh which role is to take into account the potential in the Schrödinger equation, and aim at demonstrating the convergence of our scheme for Schrödinger equation.

Figure 1 :

 1 Figure 1: Two points of view for modeling PDE-based phenomena into MIS.

Figure 2 :

 2 Figure 2: Classical and transport-agent points of view for transports scheduling. Case of 4 transport-agents a i , 1 ≤ i ≤ 4.

Figure 3 :

 3 Figure 3: Comparison between T-A("lifetime" =9.10 -4 and δx = 0.1), UW(cfl=0.1 and 0.9) and EX.

Figure 4 :

 4 Figure 4: Comparison between T-A("lifetime"=0.005 and δx = 0.05), UW (cfl=0.1) and Ex.

Figure 5 :

 5 Figure 5: Comparison between T-A("lifetime"=0.015 and δx = 0.05) , UW (cfl=0.3) and EX.

Figure 6 :

 6 Figure 6: Classical and transport-agent points of view for transports asynchronous scheduling. Case of 3 transport-agents a i , 1 ≤ i ≤ 3, δt 1 = δt 2 = δt and δt 2 = 2δt thus ∆t = 2δt.

Figure 7 :

 7 Figure 7: Comparison between (T-A)( δt min = 2.10 -4 , δtmax = 9.10 -4

Figure 8 :

 8 Figure 8: Convective transport of the function e -x 2 (Comparison between (T-A), UW and EX).

Figure 9 :

 9 Figure 9: Convective transport of the function e -x 2 (Comparison between (T-A), UW and EX)

Figure 10 :

 10 Figure 10: Comparison between (D-A) and (FDM), according to the values of cfl and in several moments for w(x) = e -x 2 .

  compares our transport-diffusion-agent (T-D-A) method with finite volume (VF). Although we did not demonstrate it formally, for both c f l = 0.1 and 0.2, our T-D-A model seems to behave as well as the VF scheme does.

Figure 11 :

 11 Figure 11: Comparison between (T-D-A) and (VF), according to the values of cfl, at several moments.

  n+o N /N j = ψ n+(o N -1)/N j ∀j ∈ {N -1, N} where o i , o 0 and o N are respectively the order of intervention of A i , A 0 and A N inside the cycle starting at t n . Notation 4.3.1. Subsequently, we adopt the notation κ = i δt δx 2 .

Figure 12 :

 12 Figure 12: Evolution of Gaussian wave packet: |ψ(x, t)| 2 according to x for different values of t (h = m = 1, α = 2, k 0 = 1).

The theorem of Lax provides that in a well-posed mathematical problem, and with a consistent discretization scheme, stability is a necessary and sufficient condition for convergence.