CONCENTRATIONS IN AGE-STRUCTURED MODELS ARISING IN
ADAPTIVE DYNAMICS ∗
LUIS ALMEIDA † , BENOIT PERTHAME † , AND XINRAN RUAN †
Abstract. We propose anasymptoticpreserving (A-P) schemefor 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.
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 asymptoticpreserving (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.
The purpose of this paper is to extend Godunov-type asymptoticpreserving 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 anasymptoticpreservingschemefor the "diffusive" part (nodal asymptoticpreservingfor example [ BDF11 ]) and a classical hyperbolic schemefor the "non diffusive" part we obtain anasymptoticpreserving discretization for the complete system. After this, we show how angular discretizations such as P N and S N models fall within this framework.
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 schemefor ( 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 asymptoticpreserving (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
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  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  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
∂ 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 ).
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  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.
Figure 23: Angle for different pressure ratios calculated with Euler and BGK models.
In recent years, the notion of AsymptoticPreserving schemes  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 asymptoticpreserving 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.
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 asymptoticpreserving 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 schemefor 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 asymptoticpreserving, 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.
Asymptotic-Preserving schemes have been recently used for numerous applications in the context of strong magnetic fields ([13, 4]) for the gyro-fluid limit as well as in fluid mechanics for the hydrodynamic limit ([23, 24]). Other applications can be found for example in [21, 16, 11, 5].
Kinetic descriptions are accurate but can be too numerically expensive to be used for many real physical applications. An alternative way could be to consider a fluid description based on average quantities. Nevertheless, macroscopic descriptions are often not accurate enough. For example, in the context of inertial confinement fusion, the distribution functions considered can be far from equilibrium and in this case the fluid description is not adapted. Moreover kinetic effects like non local transport ([3, 28]) or the development of some instabilities () can be important on long collisional time scales and are not captured by fluid simulations. At the same time, kinetic codes are usually limited to short time scales and cannot reach time scales studied by fluid simulations. It is therefore an important challenge to describe kinetic effects using reduced kinetic codes on fluid time scales. Then angular moments models represent intermediate models between the kinetic and fluid levels. They are less numerically expensive than kinetic models and more accurate than fluid models. They are constructed by using an angular moments extraction ([25, 29]) from the kinetic equations. But, there exists several moment models whose differences come from the choice of the closure. For example, the very popular P N closure ()
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.
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 ].
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.
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  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  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
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  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.
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