2 CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France.
(Received xx; revised xx; accepted xx)
We present an **Asymptotic**-**Preserving** **method** to solve numerically **the** two-dimensional **vorticity**-Poisson (Navier-Stokes) system. **The** main focus is **the** validation of **the** numer- ical scheme. As test cases we consider **the** unforced evolution of Taylor-Green vortices and **the** forced Kolmogorov flow with a sinusoidal source term. **The** scheme is validated by comparing **the** results with those obtained with an explicit spectral code and with an analytic result about **the** linear instability regime. We show that **the** AP-properties of **the** **method** allow one to deal eﬃciently with **the** multi-scale nature of **the** problem by tuning **the** time step to **the** physics one aims to study and not by stability constraints. As a side result, we investigate **the** long time scale evolution of **the** Kolmogorov flow, observing that it evolves into a final stable stationary state characterised by a seemingly universal relation between stream-function and **vorticity**.

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In this paper, we propose a new **Asymptotic** **Preserving** Monte Carlo **method** which solves **the** Boltzmann **equation** of gas dynamics. It is specifically designed to address **the** complexity of **the** underlying kinetic **equation**, to reduce **the** numerical noise of classical MC methods and to overcome **the** stiffness of **the** **equation** close to **the** fluid limit. **The** scheme proposed here is inspired by some recent papers on **the** same subject [24, 25, 21, 17, 60] while improving **the** results obtained. In details, we focus on **the** space homogeneous problem and we design **the** Monte Carlo **method** by rewriting **the** **equation** in term of **the** time evolution of **the** perturbation from equilibrium. Then, we use exponential Runge-Kutta methods to discretize **the** resulting **equation**. Particles are successively used to describe only **the** perturbation from **the** equilib- rium and a Monte Carlo interpretation of **the** resulting **equation** is furnished. One of **the** major problem when this kind of MC approach is used is that **the** total number of particles increases with time due to collisions with particles sampled from **the** equilibrium state [59, 60, 42]. Here, we solve this problem by using a subset of samples to estimate **the** distribution function shape through kernel density reconstruction techniques [6] and then we use this estimate as a proba- bility **for** discarding or keeping particles through an acceptance-rejection algorithm [53]. This approach permits to eliminate samples which give redundant information at a cost proportional to **the** number of samples which are present at a fixed time of **the** simulation in **the** domain. In this way, **the** **method** enjoys both **the** unconditional stability property and **the** complexity reduction one as **the** solution approaches **the** thermodynamic equilibrium. In fact, **the** parti- cles are used to describe only **the** perturbation which goes to zero exponentially fast and thus disappear exponentially fastly. Thus, **the** statistical error due to **the** MC **method** decreases as **the** number of interactions increases, realizing a variance reduction **method** which effectiveness depends on **the** regime studied. Far from equilibrium **the** same variance of classical MC methods is obtained, while close to equilibrium **the** variance is lower than that of a classical MC. **The** approach presented here can then be incorporated in a solver **for** **the** spatially non homogeneous case by coupling it with deterministic methods **for** **the** equilibrium part of **the** solution. We do not discuss this issue here and we refer to a future work to extend **the** present **method** to **the** non homogeneous case.

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March 10, 2016
Abstract
This paper presents a hybrid numerical **method** to solve efficiently a class of highly anisotropic elliptic problems. **The** anisotropy is aligned with one coordinate- axis and its strength is described by a parameter ε ∈ (0, 1], which can largely vary in **the** study domain. Our hybrid model is based on **asymptotic** techniques and couples (spatially) an **Asymptotic**-**Preserving** model with its **asymptotic** Limit model, **the** latter being used in regions where **the** anisotropy parameter ε is small. Adequate coupling conditions link **the** two models. Aim of this hybrid procedure is to reduce **the** computational time **for** problems where **the** region of small ε- values extends over a significant part of **the** domain, and this due to **the** reduced complexity of **the** limit model.

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We now come to our main concern in **the** present article and seek after a numerical **method** that is able to capture these expected **asymptotic** properties, even when numerical discretization parameters are kept independent of ε hence are not adapted to **the** stiffness degree of **the** space interactions. Our objective enters in **the** general framework of so-called **Asymptotic** **Preserving** (AP) schemes, first introduced and widely studied **for** dissipative systems as in [ 22 ], [ 24 ]. Yet, in opposition with collisional kinetic equations in hydrodynamic or diffusion limits, transport equations like ( 1.3 ) involve of course some stiffness in time but it is also crucial to take care of **the** space discretization in order to capture **the** correction terms of **the** non-local operator K ε [f ε ]. By many respects this makes **the** identification of suitable schemes much more

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Grenoble Alpes, LCIS, F-26902, Valence, France e-mail: laurent.lefevre@lcis.grenoble-inp.fr Abstract: Discretizing open systems of conservation laws while preserving the power-balance at[r]

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dt kε∇a
ε (t) + ia ε (t)∇φ ε (t)k 2
L 2 + ka ε (t)k 4 L 4 = 0. (1.6)
Let us now describe **the** limit ε → 0 **for** (1.1). **The** toolbox **for** studying semi- classical Schrödinger equations contains a variety of methods, depending on **the** quantities we are interested in. If one is only interested in a description of **the** dynamics of quadratic observables, such as **the** mass, current or energy densities, **the** Wigner transform is well adapted and has been applied to linear Schrödinger equations and Schrödinger-Poisson systems [27, 41, 55]; on **the** other hand, it is not adapted to **the** study of NLS [14]. Still **for** a description of quadratic observables, **the** modulated energy **method** [10, 54, 40, 2] enables to treat **the** nonlinear **equation** (1.1). Yet, **for** a pointwise description of **the** wavefunction u ε , WKB techniques are more convenient. We refer to [12] **for** a presentation of WKB analysis of Schrödinger equations.

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In this work, we have constructed an **Asymptotic**-**Preserving** scheme **for** **the** full Fokker-Planck- Landau-Maxwell system which handles **the** quasi-neutral limit without any contraction of time and space steps. Remark that this model is considered with **the** real collisional operators. This fact is important in plasma physics because **the** model is relevant **for** Coulombian interactions. We have first established a reformulated Fokker-Planck-Landau-Maxwell system then used it to construct **the** **Asymptotic**-**Preserving** scheme. **The** **method** has been extended to **the** general case of collisional plasmas in electromagnetic fields **for** multi-dimensions problems. An M 1 -**Asymptotic**-**Preserving** scheme has been derived. Next, **the** (M 1 -

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A second track consist in focusing on **the** pressure **equation**. To this aim, a natural idea is to adapt classical incompressible schemes to **the** compressible case. **The** pressure- correction **method** SIMPLE [27, 47] solves an elliptic pressure correction **equation** obtained via **the** mass conservation **equation** and **the** **equation** of state. In [44], **the** elliptic pressure correction **equation** is obtained by introducing **the** pressure **equation** (derived from **the** energy **equation**) in **the** momentum **equation**. These methods respect **the** divergence constraint on **the** velocity but **the** formulation is not always conservative. In **the** ICE (Implicit Continuous Eulerian) **method** introduced by Harlow and Amsden and followers [4, 25], a splitting **method** is introduced between **the** explicit convective part and implicit acoustic part. However, **the** ICE **method** is not conservative and inaccurate shock speeds are observed. Klein [30] proposes a semi-implicit scheme which solves explicitly **the** leading order contribution of **the** pressure and **the** lower orders, implicitly. Other ways generating elliptic equations on **the** pressure can be found in [33, 43, 46, 48, 56, 59].

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however that **the** P1 system is not positive at continuous level, contrary to **the** diffusion **equation**. Hence positivity is not a relevant property of **the** scheme in this case. We also obtain a second order convergence **for** both polygonal/conical schemes **for** regular data and solution (see Figure 38).
In **the** transport regime case ( ∼ 1), **for** Dirac mass datum, we still get cross-stencil **for** and only **for** polygonal scheme (Figure 24 and 33). Here, we obtain a first order scheme both **for** polygonal and conical schemes see Figure 37. **For** complex test case Figure 39, **the** polygonal scheme exhibits both cross stencil and numerical instabilities (see Figure 41). On **the** contrary, **the** conical scheme on Figure 42 does not show such pathologies.

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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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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 ].

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are there handled too. Their **method** differs in many respects with **the** ones above and relies on a combination of variational and dynamical arguments in **the** form of localized monotonicity formulas. Our work was strongly motivated by **the** possibility to obtain such an extension **for** **the** Gross-Pitaevskii **equation**. Even though **the** later is also a nonlinear Schrödinger **equation**, **the** non-vanishing boundary conditions at spatial infinity modifies **the** dispersive properties. More precisely, **the** dispersion relation **for** **the** linearization of (GP) around **the** constant 1 is given by

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Abstract. We present a new scheme **for** **the** simulation of **the** barotropic Euler **equation** in low Mach regimes. **The** **method** uses two main ingredients. First, **the** system is treated with a suitable time splitting strategy, directly inspired from [J. Haack, S. Jin, J.-G. Liu, Comm. Comput. Phys., 12 (2012) 955–980], that separates low and fast waves. Second, we adapt a numerical scheme where **the** discrete densities and velocities are stored on staggered grids, in **the** spirit of MAC methods, and with numerical fluxes derived form **the** kinetic approach. We bring out **the** main properties of **the** scheme in terms of consistency, stability, and **asymptotic** behaviour, and we present a series of numerical experiments to validate **the** **method**.

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Burgers **equation** provides a simple, yet challenging **equation**, which admits discontinuous solutions and it provides a simplified setup **for** **the** design and validation of shock-capturing numerical methods. Burgers **equation** and its generalizations to a curved manifold have been widely used in **the** physical and mathematical literature. In [ 3 ], we have used a class of Burgers-type equations on **the** sphere and adopted **the** methodology first proposed by Ben-Artzi, Falcovitz, and LeFloch [ 5 ], which uses second–order approximations based on generalized Riemann problems. In [ 3 ], a scheme was proposed which uses piecewise linear reconstructions based on solution values at **the** center of **the** computational cells and on values of Riemann solutions at **the** cell interfaces. A second-order approximation based on a generalized Riemann solver was then proposed, together with a total variation diminishing Runge-Kutta **method** (TVDRK3) with operator splitting **for** **the** temporal integration.

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third order AHO is made using a fourth order discretization of Φ x and **the** final scheme
is really third order in our numerical tests. Please notice that **the** computational costs of all these schemes do not increase with respect of **the** standard ones.
Let us now mention that our approach is somewhat related to **the** well-balanced philosophy, see **for** instance [16] or **for** **the** specific case of hyperbolic chemotaxis prob- lems, see [10]. It is interesting to compare **the** **Asymptotic** High Order schemes we obtain in this paper with **the** results of some very recent works on Well-Balanced schemes [14, 15] **for** **the** same system considered here, which actually appeared only after **the** first submission of **the** present paper. First, both methods start with **the** problem of balancing **the** flux term **for** **the** conservation **equation**. **The** AHO schemes use **the** upwinding approach to approximate **the** stationary solutions by using a Tay- lor expansion in **the** truncation error, whereas **the** well-balanced approach **for** this

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Beyond free energy and entropy methods, **the** study of **the** large time asymptotics of **the** Poisson-Nernst-Planck system involves various tools of nonlinear analysis. Proving an exponential rate of convergence is in- teresting **for** studies of Poisson-Nernst-Planck systems by methods of sci- entific computing. Specific methods are needed **for** **the** numerical com- putation of **the** solutions, see [ 4 , 23 ]. In [ 21 ], Liu and Wang implement at **the** level of **the** free energy a finite difference **method** to compute **the** nu- merical solution in a bounded domain. Concerning rates of convergence from a more theoretical point of view, let us mention that **the** existence of special solutions and self-similar solutions is considered in [ 5 , 6 , 17 ]. We refer to [ 26 ] **for** a discussion of **the** evolution problem from **the** point of view of physics.

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