L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

CONCENTRATIONS IN AGE-STRUCTURED **MODELS** ARISING IN
ADAPTIVE DYNAMICS ∗
LUIS ALMEIDA † , BENOIT PERTHAME † , AND XINRAN RUAN †
Abstract. We propose **an** **asymptotic** **preserving** (A-P) **scheme** **for** a population model struc- tured by age and a phenotypical trait with or without mutation. As proved in [ 24 ], Dirac concen- trations on particular phenotypical traits appear in the case without mutation, which makes the numerical resolution of the problem challenging. Inspired by its **asymptotic** behaviour, we apply a proper WKB representation of the solution to derive **an** A-P **scheme**, with which we can accurately capture the concentrations on a coarse, ε-independent mesh. The **scheme** is thoroughly analysed and important properties, including the A-P property, are rigorously proved. Furthermore, we observe nearly spectral accuracy in time in our numerical simulations. Next, we generalize the A-P **scheme** to the case with mutation, where a nonlinear Hamilton-Jacobi equation will be involved in the limiting model as ε → 0. It can be formally shown that the generalized **scheme** is A-P as well, and numerical experiments indicate that we can still accurately resolve the problem on a coarse, ε-independent mesh in the phenotype space.

En savoir plus
Let the dimensionless parameter ε denote the ratio of the mean free path and the typical length scale. Numerical solutions to the transport equation are challenging when ε is small, since it requires the numerical resolution of the small scale. To develop a multi-scale **scheme** whose stability and convergence are independent of ε refers to the **asymptotic** **preserving** (AP) property. When ε = O(1), the **models** behave like hyperbolic equations with source terms. When the source terms in the system become stiff, the usual numerical methods may give poor approximations to the steady state solutions [4, 25]. To maintain the steady states or to achieve them in the long time limit with **an** acceptable level of accuracy refers to the well balanced (WB) property.

En savoir plus
The purpose of this paper is to extend Godunov-type **asymptotic** **preserving** schemes **for** the Friedrichs systems on unstructured meshes. Firstly we introduce the Friedrichs systems and give a formal proof of the existence of the diffusion limit. In the second part we define a numerical strategy based on a decomposition between a "diffusive" part similar to the hyperbolic heat equa- tion and a "non diffusive" part which is negligible in the diffusion regime. This decomposition, close to the micro-macro decomposition [ LM07 ] allows to design a very simple method to discretize stiff hyperbolic systems. Indeed, using **an** **asymptotic** **preserving** **scheme** **for** the "diffusive" part (nodal **asymptotic** **preserving** **for** example [ BDF11 ]) and a classical hyperbolic **scheme** **for** the "non diffusive" part we obtain **an** **asymptotic** **preserving** discretization **for** the complete system. After this, we show how angular discretizations such as P N and S N **models** fall within this framework.

En savoir plus
equation, known to be a degenerate parabolic equation, has been widely studied (see [ 2 , 14 , 18 , 21 ] **for** theoretical aspects and [ 1 , 4 , 17 , 20 ] **for** some numerical studies) and appears in several physical problems as, **for** instance, in non-newtonian fluids [ 9 ].
The aim of this work is to derive a **scheme** **for** ( 1.1 ) such that, in the diffusive regime, the approximate solution is consistent with the limit problem ( 1.6 ), or equivalently ( 1.7 )-( 1.8 ). This property to be satisfied by the numerical **scheme**, called **asymptotic** **preserving** (AP) [ 5 , 6 , 15 ], is not straightforward and the development of AP-schemes requires a particular attention. During the two last decades, numerous AP-schemes were proposed in the literature. **For** instance, in [ 3 , 6 , 7 ], AP numerical strategies have been introduced to accurately approximate the **asymptotic** diffusive regime issuing from the **radiative** transfer **models**. We also refer to [ 16 ] where **an** AP-**scheme** is derived to approximate the solutions of the isentropic gas dynamics in the Darcy law regime. Next, in [ 6 ], a generic formulation of AP-schemes is proposed by a suitable extension of the well-known HLL **scheme** [ 13 ]. At the wide discrepancy with the above mentioned works where the source term rescaling is governed by 1/ε, according to ( 1.2 ), in the present paper we have to deal with a rescaling prescribed by 1/ε 2 . High order source

En savoir plus
The angular moments **models** represent **an** alternative method situated between kinetic and fluid **models**. Their computational times are shorter than kinetics ones and provide results with a higher accuracy than fluid **models**. Originally, the moment closure hierarchy introduced by Grad [39] leads to hyperbolic set of equations **for** flows close to equilibrium but may suffer from closure breakdown and lead to unrealisable macroscopic moments. Grad hierarchy is derived from a truncated polynomial series expansion **for** the velocity distribution function near the Maxwellian equilibrium and does not ensure the postivity of the distribution function. Other moment closure approaches have been investigated based on entropy minimisation principles [57, 63, 64, 73, 2]. The distribution function derived, verifies a minimum entropy property and the consistency with the set of moments. Fundamental mathematical properties [41, 61] such as positivity of the distribution function, hyperbolicity and entropy dissipation can be exhibited. Levermore [57] proposed a hierarchy of minimum-entropy closure where the lowest order closure are the Maxwellian and Gaussian closure. In the present case, the aim is different. Here the energy of particles constitutes a free parameter. Then we integrate only the kinetic equation with respect to the angle variable and we return only the energy of particles as kinetic variable. By using a closure defined from a minimization entropy principle, we obtain the M 1 model

En savoir plus
∂ t ρ + ∂ x j = 0,
ε 2 ∂ t j − ∂ x ρ = −2j.
We easily recognize the initial model ( 1.7 ) with a linear pressure.
Recently, Lattanzio and Tzavaras [ 24 ] rigorously proved this convergence and exhib- ited the rate by adopting the well known relative entropy approach. Relative entropy is a usefull tool to compare the difference (in a sense to be prescribed) between two so- lutions. The notion of relative entropy **for** hyperbolic systems of conservation laws was introduced in the pioneer works of DiPerna [ 10 ] and Dafermos [ 8 ]. It was used to study rigorously the convergence from kinetic **models** to their hydrodynamic limit [ 14 , 33 ]. Later, in [ 9 ], Dafermos adopted this method to establish a stability result **for** classical solutions in the class of entropy weak solutions. Next, in [ 34 ], Tzavaras applied a similar relative entropy method to study the convergence of hyperbolic systems with stiff relax- ation towards the corresponding hyperbolic limit. Based on the same ideas, Lattanzio and Tzavaras considered in [ 24 ] the case of diffusive relaxation. They treated several hyperbolic systems with source term of type ( 1.7 ) which converge to a diffusive problem when ε goes to zero. In particular, they established the convergence of solutions to the p-system ( 1.7 ) towards solutions of the porous media equation ( 1.6 ).

En savoir plus
where the last term on the right hand side is just the numerical viscosity induced by the upwind **scheme**. So, when the function u is near to the equilibrium, and the second
derivative of u is large, the space step h needs to be very small, actually hu xx = o(1)
is needed, in order to make the function v constant. Therefore, if the **asymptotic** state is constant, the upwind **scheme** will be perfectly adapted, whereas **for** a non-constant **asymptotic** state, with a large second derivative, a new **scheme** has to be found. This is a typical situation when dealing with a problem which tends toward **an** equilibrium state where the flux in the conservation equation has to vanish in the time **asymptotic** limit. **For** example, this is the case of kinetic problems (**radiative** transfer [13] or Boltzmann equation near the equilibrium state). Notice that here the problem does not arise from the stiffness of the source term, so implicit schemes are not expected to give better performances. Moreover, higher order schemes (TVD, WENO) give some limited results, but to radically improve the behavior of the schemes it is necessary to take into account the qualitative behavior of the solutions.

En savoir plus
Figure 23: Angle **for** different pressure ratios calculated with Euler and BGK **models**.
7 Conclusion
In recent years, the notion of **Asymptotic** **Preserving** schemes [14] has been introduced answering the need **for** numerical methods that automatically converge to discretizations of appropriate reduced **models**, as the Knudsen number changes within the flow. This work shows that such methods are not AP in presence of interior boundaries, unless a proper treatment of the boundary condition is introduced at the price of coslty higher order interpolations in the velocity space. Here we propose **an** efficient boundary condition which ensures that **an** AP **scheme** remains AP up to the boundary. We illustrate this result by comparing several numerical schemes to model the impermeability condition **for** the BGK model with emphasis on **asymptotic** **preserving** properties in the Euler limit. We have also shown how to recover second order accuracy on Cartesian meshes using this new wall condition and simulated non-trivial rarefied regime test cases. In future work, we will concentrate on the **asymptotic** limit of wall **models** towards Navier-Stokes equations using **an** ES-BGK model.

En savoir plus
in region where ε = O(1) or a limit model in regions where ε → 0. The coupling of these **models** is however not straightforward and brings many other technical difficulties related to the interfaces reconnection. To avoid such difficulties, we employ the so called **asymptotic** **preserving** approach introduced by Jin [ 16 , 17 ] so as to be able to simulate both regimes ε = O(1) and ε → 0 with a single model. The efficiency of this approach has led to significant development both in plasma physics and fluid mechanics [ 7 , 8 , 10 , 26 , 28 , 31 ]. However efficient in practice, rigorous stability analysis and proof of convergence in the limit ε → 0 **for** complete physical **models** is often a difficult task. In this respect, the present work is devoted to analyse some properties of a **scheme** **for** the quasi-neutral Euler–Boltzmann system. The motivation of this work comes from some of the questions raised in a previous work of Deluzet et al. [ 7 ] on the Euler–Lorentz model. Notably, besides the ability of the **scheme** to be numerically **asymptotic** **preserving**, some questions around the ability of the **scheme** to preserve the invariants (physical energy, positivity) and computing the correct shock speed in a non conservative form of the Euler–Lorentz equations was formulated.

En savoir plus
Abstract.
In the present work, the Eulerian Large Eddy Simulation of dilute disperse phase flows is in- vestigated. By highlighting the main advantages and drawbacks of the available approaches in the literature, a choice is made in terms of modelling: a Fokker-Planck-like filtered kinetic equation pro- posed by Zaichik et al. 2009 and a Kinetic-Based Moment Method (KBMM) based on a Gaussian closure **for** the NDF proposed by Vie et al. 2014. The resulting Euler-like system of equations is able to reproduce the dynamics of particles **for** small to moderate Stokes number flows, given a LES model **for** the gaseous phase, and is representative of the generic difficulties of such **models**. Indeed, it encounters strong constraints in terms of numerics in the small Stokes number limit, which can lead to a degeneracy of the accuracy of standard numerical methods. These constraints are: 1/as the resulting sound speed is inversely proportional to the Stokes number, it is highly CFL-constraining, and 2/the system tends to **an** advection-diffusion limit equation on the number density that has to be properly approximated by the designed **scheme** used **for** the whole range of Stokes numbers. Then, the present work proposes a numerical **scheme** that is able to handle both. Relying on the ideas introduced in a different context by Chalons et al. 2013: a Lagrange-Projection, a relaxation formulation and a HLLC **scheme** with source terms, we extend the approach to a singular flux as well as properly handle the energy equation. The final **scheme** is proven to be **Asymptotic**-**Preserving** on 1D cases comparing to either converged or analytical solutions and can easily be extended to multidimensional configurations, thus setting the path **for** realistic applications.

En savoir plus
It is considered that angular moments **models** provide higher accuracy than ﬂuid model because the veloc- ity modulus (denoted ζ in this work) is kept as a variable. The integration of the kinetic equation to obtain such **models** is performed only in angle (integration on the unit sphere). Angular moments represent angular average quantities of the distribution function. Therefore, they can be seen as intermediate **models** between kinetic and classic ﬂuid **models**. Originally, the moment closure hierarchy introduced by Grad [ 40 ] leads to a hyperbolic set of equations **for** ﬂows close to equilibrium but may suﬀer from closure breakdown and lead to unrealisable macroscopic moments. Grad hierarchy is derived from a truncated polynomial series expansion **for** the velocity distribution function near the Maxwellian equilibrium and does not ensure the positivity of the distribution function. Other moment closure approaches have been investigated based on entropy minimisa- tion principles [ 2 , 61 , 68 , 69 , 77 ]. The distribution function derived, veriﬁes a minimum entropy property and the consistency with the set of moments. Fundamental mathematical properties [ 42 , 66 ] such as positivity of the dis- tribution function, hyperbolicity and entropy dissipation can be exhibited. Levermore [ 61 ] proposed a hierarchy of minimum-entropy closure where the lowest order closure are the Maxwellian and Gaussian closure. In the present case, the aim is diﬀerent. Here the energy of particles constitutes a free parameter. Then we integrate only the kinetic equation with respect to the angle variable and we return only the energy of particles as kinetic variable. By using a closure deﬁned from a minimisation entropy principle, we obtain the M 1 model [ 33 , 34 , 63 ]. The M 1 model is largely used in various applications such as **radiative** transfer [ 7 , 24 , 35 , 71 , 72 , 79 , 80 ] or electronic transport [ 33 , 63 ]. The M 1 model is known to satisfy fundamental properties such as the positivity of the ﬁrst angular moment, the ﬂux limitation and conservation of total energy. Also, it correctly recovers the **asymptotic** diﬀusion equation in the limit of long time behaviour with important collisions [ 34 ].

En savoir plus
There is a large number of applications based on transport equations having a diffusive **asymptotic**. Let us mention, **for** instance, neutron transport, **radiative** transfer in the “optically thick limit” (see [1, 31, 32, 38, 39] and the references therein) and semiconductor modeling (see [25, 29, 37]). However, many of the previous works deal with the so-called telegrapher equation, or equivalently Goldstein–Taylor equation which is a kinetic equation where the distribution function is localized on two opposite velocities (see [4, 20, 21, 26, 27, 40]). On the other hand, in this paper, we shall not separate the particle density function with respect to the sign of the velocity, usually called parity method or even-odd decomposition. We refer more precisely to the introduction of Section 4.4 **for** a detailed explanation of this fact and a comparison with previous works. Let us finally mention that the space discretization **scheme** we propose could be also apply to the Boltzmann–Lorentz operator with **an** arbitrary cross-section.

En savoir plus
The angular moments **models** represent **an** alternative method situated between kinetic and fluid **models**. Their computational times are shorter than kinetics ones and provide results with a higher accuracy than fluid **models**. Originally, the moment closure hierarchy introduced by Grad [39] leads to hyperbolic set of equations **for** flows close to equilibrium but may suffer from closure breakdown and lead to unrealisable macroscopic moments. Grad hierarchy is derived from a truncated polynomial series expansion **for** the velocity distribution function near the Maxwellian equilibrium and does not ensure the postivity of the distribution function. Other moment closure approaches have been investigated based on entropy minimisation principles [2,57,63,64,73]. The distribution function derived, verifies a minimum entropy property and the consistency with the set of moments. Fundamental mathematical properties [41, 61] such as positivity of the distribution function, hyperbolicity and entropy dissipation can be exhibited. Levermore [57] proposed a hierarchy of minimum-entropy closure where the lowest order closure are the Maxwellian and Gaussian closure. In the present case, the aim is different. Here the energy of particles constitutes a free parameter. Then we integrate only the kinetic equation with respect to the angle variable and we return only the energy of particles as kinetic variable. By using a closure defined from a minimization entropy principle, we obtain the M 1 model [32, 33, 58]. The M 1 model is largely used in

En savoir plus
One of the main challenges **for** this objective is to control the wall-plasma inter- actions. Indeed, the magnetic confinement is not perfect and the plasma is in contact with the wall. In a tokamak such as TORE SUPRA, **an** obstacle called the limiter, is settled at the bottom of the machine. Due to the strong magnetic confinement, the plasma transport essentially occurs along the magnetic field lines. Thus, the parallel resistivity η is very small (typically, η = 10 −6 ), generating a strong anisotropy in

The new conical schemes obtained by adding a control point located on each mid edge (shoulder point) are strictly different from those of the original polygonal case, especially **for** the planar degenerate conical case (all ω are set to 0). The (solved) singularity coming from the flux at shoulder point is that only the normal component is well defined. We have a degree of freedom to define the tangential component, here we choose to take the average of nodal fluxes in this tangential direction (cf also [6, 3] **for** hydrodynamical **scheme**).

ε = 1. From the PIC point of view, our hybrid **scheme** can be seen as a δ f method (see [4] **for** example). We thus take the same advantages: the noise due to the prob- abilistic character of the particles discretization is reduced since it affects only the perturbation g, and not the whole function f . This noise appears on the represen- tation of ρ when N is too small, and **for** example in the black curve obtained with Full-PIC and N = 5 × 10 4 . With the same order of N, the black line labeled with

6. Conclusion. In this work, we have presented a new numerical method **for** the BGK equation which enjoys the following properties: (i) its statistical noise is smaller than the one of standard Monte Carlo methods; (ii) it is asymptotically stable with respect to the Knudsen number; (iii) its computational cost as well as its variance diminish as the equilibrium is approached; (iv) no artificial criteria is required. The method is based on a micro-macro decomposition (Euler-kinetic or Navier-Stokes-kinetic) **for** which the macro part is solved using a finite volume method whereas the micro part uses a Monte-Carlo method. This enables to derive a low variance Monte Carlo based method **for** which additionally, the number of particles used to sample the micro unknown diminishes automatically as the fluid regime is approached. The numerical results illustrate the efficiency of the proposed method compared to the standard Monte Carlo approach.

En savoir plus
Attribute-based Signatures (ABS **for** short) is a cryptographic primitive that enables users to sign data with fine-grained control over the required identifying information [14]. To use ABS, a user shall possess a set of attributes and a secret signing key per attribute. The signing key must be provided by a trusted au- thority. The user can sign, e.g., a document, with respect to a predicate satisfied by the set of attributes. Common settings **for** ABS must include a Signature Trustee (ST ), **an** Attribute Authority (AA), and several signers and verifiers. The ST acts as a global entity that generates valid global system parameters. The AA issues the signing keys **for** the set of attributes of the users (e.g., the signers). The role of the ST and the AA can be provided by the same entity. The AA can hold knowledge about the signing keys and the attributes of the users. However, the AA should not be capable to identifying which attributes have been used in a given valid signature. This way, the AA will not be able to link the signature to the source user. The AA should not be able to link back the signatures to the signers. This is a fundamental requirement from ABS, in order to fulfill common privacy requirements.

En savoir plus