Top PDF Asymptotic-preserving method for the vorticity equation

Asymptotic-preserving method for the vorticity equation

Asymptotic-preserving method for the vorticity equation

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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A new deviational Asymptotic Preserving Monte Carlo method for the homogeneous Boltzmann equation

A new deviational Asymptotic Preserving Monte Carlo method for the homogeneous Boltzmann equation

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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A HYBRID METHOD FOR ANISOTROPIC ELLIPTIC PROBLEMS BASED ON THE COUPLING OF AN ASYMPTOTIC-PRESERVING
METHOD WITH THE ASYMPTOTIC LIMIT MODEL

A HYBRID METHOD FOR ANISOTROPIC ELLIPTIC PROBLEMS BASED ON THE COUPLING OF AN ASYMPTOTIC-PRESERVING METHOD WITH THE ASYMPTOTIC LIMIT MODEL

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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Asymptotic preserving schemes for the FitzHugh-Nagumo transport equation with strong local interactions

Asymptotic preserving schemes for the FitzHugh-Nagumo transport equation with strong local interactions

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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A structure-preserving Partitioned Finite Element Method for the 2D wave equation

A structure-preserving Partitioned Finite Element Method for the 2D wave equation

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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Design of asymptotic preserving schemes for the hyperbolic heat equation on unstructured meshes

Design of asymptotic preserving schemes for the hyperbolic heat equation on unstructured meshes

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]

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Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: particular cases.

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: particular cases.

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
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An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit

An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit

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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ASYMPTOTIC-PRESERVING SCHEME FOR THE FOKKER-PLANCK-LANDAU-MAXWELL SYSTEM IN THE QUASI-NEUTRAL REGIME.

ASYMPTOTIC-PRESERVING SCHEME FOR THE FOKKER-PLANCK-LANDAU-MAXWELL SYSTEM IN THE QUASI-NEUTRAL REGIME.

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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An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

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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Asymptotic preserving schemes on conical unstructured 2D meshes

Asymptotic preserving schemes on conical unstructured 2D meshes

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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Singular asymptotic expansion of the exact control for the perturbed wave equation

Singular asymptotic expansion of the exact control for the perturbed wave equation

For any fixed function v ε , the asymptotic analysis of solutions of singular systems like (1) can be performed using the matched asymptotic method [13, 10]. Precisely, under additional regularity and compatibility assumptions on the data (y0, y1) and on the function v ε , one may construct explicitly strong convergent approximations of y ε . We refer for instance to [1] for an advection- diffusion equation. When the exact controllability issue comes into play, such analysis requires more care, since the regularity/compatibility conditions mentioned above become additional constraints on the control set C. Typically, L 2 regularity for the control is in general not sufficient to get strong convergence results for the corresponding controlled solution, at any order. It is then necessary to enrich the set C and to modify the optimality system accordingly. This is probably the reason for which there exists in the literature only very few asymptotic analysis for controllability problem, a fortiori for singular partial differential equations. We mention [19], [1] following [7]. We also mention the book [14] and the review [9] in the close context of optimal control problems.
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Asymptotic preserving and time diminishing schemes for rarefied gas dynamic

Asymptotic preserving and time diminishing schemes for rarefied gas dynamic

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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Asymptotic-preserving well-balanced scheme  for the electronic M1 model in the diffusive limit: Particular cases

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: Particular cases

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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Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation

Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation

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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An asymptotic preserving scheme on staggered grids for the
barotropic Euler system in low Mach regimes

An asymptotic preserving scheme on staggered grids for the barotropic Euler system in low Mach regimes

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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A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere

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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Asymptotic high order mass-preserving schemes for a hyperbolic model of chemotaxis

Asymptotic high order mass-preserving schemes for a hyperbolic model of chemotaxis

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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Asymptotic behavior of Nernst-Planck equation

Asymptotic behavior of Nernst-Planck equation

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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ASYMPTOTIC-PRESERVING WELL-BALANCED SCHEME FOR THE ELECTRONIC M 1 MODEL IN THE DIFFUSIVE LIMIT: PARTICULAR CASES.

ASYMPTOTIC-PRESERVING WELL-BALANCED SCHEME FOR THE ELECTRONIC M 1 MODEL IN THE DIFFUSIVE LIMIT: PARTICULAR CASES.

the space and energy dependencies of the angular moments, lead to a very complexe diffusion equation in the asymptotic limit with mixed derivatives. In this paper, the case without electric field and the homogeneous case are studied. The extension to the general case is beyond the scope of this paper and postponed to another paper. However, the generalisation to the general problem requires a deep understanding of the two configurations studied here. The approach retained is noticeably different with [6, 7, 43]. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modified to enable the derivation of a scheme which also satisfies the admissibility conditions (4) and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and correctly handles the diffusive limit recovering the good diffusion equation.
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