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International audience; In this contribution, we recall the derivation of the anti bounce back boundary condition for the D2Q9 lattice Boltzmann scheme. We recall various elements of the state of the art for anti bounce back applied to linear heat and acoustics equations and in particular the possibility to take into account curved boundaries. We present an asymptotic analysis that allows an expansion of all the fields in the boundary cells. This analysis based on the Taylor expansion method confirms the well known behaviour of anti bounce back boundary for the heat equation. The analysis puts also in evidence a hidden differential boundary condition in the case of linear acoustics. Indeed, we observe discrepancies in the first layers near the boundary. To reduce these discrepancies, we propose a new boundary condition mixing bounce back for the oblique links and anti bounce back for the normal link. This boundary condition is able to enforce both pressure and tangential velocity on the boundary. Numerical tests for the Poiseuille flow illustrate our theoretical analysis and show improvements in the quality of the flow.

International audience; In a previous work, we have proposed a method for the analysis of the bounce back boundary condition with the Taylor expansion method in the linear case. In this work two new schemes of modified bounce back are proposed. The first one is based on the expansion of the iteration of the internal scheme of the lattice Boltzmann method. The analysis puts in evidence some defects and a generalized version is proposed with a set of essentially four possible parameters to adjust. We propose to reduce this number to two with the elimination of spurious density first order terms. Thus a new scheme for bounce back is found exact up to second order and allows an accurate simulation of the Poiseuille flow for a specific combination of the relaxation and boundary coefficients. We have validated the general expansion of the value in the first cell in terms of given values on the boundary for a stationary ''accordion'' test case.

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0, Comment: 34 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1511.01192

International audience; The aim of this paper is to derive an efficient scheme for solving one-dimensional time-fractional nonlinear Schrödinger equations set in unbounded domains. We first derive some absorbing boundary conditions for the fractional system by using the unified approach introduced in [57,58] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized and the error estimate O(h 2 + τ) is stated. To accelerate the scheme in time, the fractional derivative is approximated through a linearized L1-scheme. Finally, we end the paper by some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme. In addition, we illustrate the behavior of the solution by reporting a few simulations for various parameter values of the fractional order 0 < α < 1, nonlinearities and potentials.

International audience; This paper is concerned with inverse source problems for the time-dependent Lamé system and the recovery of initial data in an unbounded domain corresponding to the exterior of a bounded cavity or the full space R 3. If the time and spatial variables of the source term can be separated with compact support, we prove that the vector valued spatial source term can be uniquely determined by boundary Dirichlet data in the exterior of a given cavity. If the cavity is absent, uniqueness and stability for recovering source terms depending on the time variable and two spatial variables in the whole space are also obtained using partial Dirichlet boundary data.

We are concerned with time-dependent inverse source problems in elastodynamics. The source term is supposed to be the product of a spatial function and a temporal function with compact support. We present frequency-domain and time-domain approaches to show uniqueness in determining the spatial function from wave fields on a large sphere over a finite interval. Stability estimate of the temporal function from the data of one receiver and uniqueness result using partial boundary data are proved. Our arguments rely heavily on the use of the Fourier transform, which motivated inversion schemes that can be easily implemented. A Landweber iterative algorithm for recovering the spatial function and a non-iterative inversion scheme based on the uniqueness proof for recovering the temporal function are proposed. Numerical examples are demonstrated in both two and three dimensions., 28 pages, 14 figures

International audience; We show that the asymptotic properties of the link-wise artificial com-pressibility method are not compatible with a correct approximation of fluid properties. We propose to adapt the previous method through a framework suggested by the Tay-lor expansion method and to replace first order terms in the expansion by appropriate three or five points finite differences and to add non linear terms. The " FD-LBM " scheme obtained by this method is tested in two dimensions for shear wave, Stokes modes and Poiseuille flow. The results are compared with the usual lattice Boltzmann method in the framework of multiple relaxation times.

International audience; The purpose of this paper is to discuss some recent developments concerning the numerical simulation of space and time fractional Schrödinger and Gross-Pitaevskii equations. In particular, we address some questions related to the discretization of the models (order of accuracy and fast implementation) and clarify some of their dynamical properties. Some numerical simulations illustrate these points.

We propose a stochastic method to generate exactly the overdamped Langevin dynamics of semi-flexible Gaussian chains, conditioned to evolve between given initial and final conformations in a preassigned time. The initial and final conformations have no restrictions, and hence can be in any knotted state. Our method allows the generation of statistically independent paths in a computationally efficient manner. We show that these conditioned paths can be exactly generated by a set of local stochastic differential equations. The method is used to analyze the transition routes between various knots in crossable filamentous structures, thus mimicking topological reconnections occurring in soft matter systems or those introduced in DNA by topoisomerase enzymes. We find that the average number of crossings, writhe and unknotting number are not necessarily monotonic in time and that more complex topologies than the initial and final ones can be visited along the route., 12 pages, 3 figures

In a seminal paper Ginzburg and Adler analyzed the bounce-back boundary conditions for the lattice Boltzmann scheme and showed that it could be made exact to second order for the Poiseuille flow if some expressions depending upon the parameters of the method were satisfied, thus defining so-called "magic parameters". Using the Taylor expansion method that one of us developed, we analyze a series of simple situations (1D and 2D) for diffusion and for linear fluid problems using bounce-back and "anti bounce-back" numerical boundary conditions. The result is that "magic parameters" depend upon the detailed choice of the moments and of their equilibrium values. They may also depend upon the way the flow is driven., Comment: 14 pages

We study in detail the hydrodynamic theories describing the transition to collective motion in polar active matter, exemplified by the Vicsek and active Ising models. Using a simple phenomenological theory, we show the existence of an infinity of propagative solutions, describing both phase and microphase separation, that we fully characterize. We also show that the same results hold specifically in the hydrodynamic equations derived in the literature for the active Ising model and for a simplified version of the Vicsek model. We then study numerically the linear stability of these solutions. We show that stable ones constitute only a small fraction of them, which however includes all existing types. We further argue that in practice, a coarsening mechanism leads towards phase-separated solutions. Finally, we construct the phase diagrams of the hydrodynamic equations proposed to qualitatively describe the Vicsek and active Ising models and connect our results to the phenomenology of the corresponding microscopic models., Comment: 20 pages, 20 figures

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0, Comment: 25 pages, 1 figure

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size $N$, we study two ways of sampling the intrinsic frequencies according to the {\it same} given unimodal distribution $g(\omega)$. In the `{\em random}' case, frequencies are generated independently in accordance with $g(\omega)$, which gives rise to oscillator number fluctuation within any given frequency interval. In the `{\em regular}' case, the $N$ frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasi-uniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude.

In this contribution, we propose a new method of analysis for the lattice Boltzmann scheme's boundary conditions. We develop this approach for bounce-back with a given velocity between two mesh points. We develop a "accordion" test case for linear stationary Stokes problem with an analytic expression. Numerical experiment show a good agreement at first and second order accuracy. This study has been also applied to the computation of momentum transfer. Resume. Dans cette contribution, nous proposons une nouvelle methode d'analyse des conditions aux limites pour les schemas de Boltzmann sur reseau. Nous developpons cette approche pour une condition de rebond avec une donnee de vitesse imposee. Nous developpons un cas test "accordeon" pour le probleme de Stokes stationnaire et en explicitons la solution analytique. Nos experiences numeriques montrent un bon accord entre solution approchee numerique et solution analytique, a l'ordre un et a l'ordre deux. L'etude precedente est egalement appliquee pour le calcul du transfert d'impulsion a la paroi.

This paper is dedicated to a detailed analysis of the symmetry properties of the cubical Stokes eigenmodes and to the numerical verification of its predictions. These modes are computed numerically by using two different Stokes solvers, the Projection-Diffusion Chebyshev spectral method and a lattice Boltzmann approach. They distribute themselves into 10 symmetry families, 2 families of singlets, 2 families of doublets and 6 families of triplets. The existence of the singlets, doublets and triplets is directly related to the fact that two different bases can be constructed for describing the cyclic-permutation state operator. The singlets and doublets are associated with one of these bases, and they are therefore generated together, the triplets being associated with the other basis.

We propose the derivation of acoustic-type isotropic partial differential equations that are equivalent to linear lattice Boltzmann schemes with a density scalar field and a momentum vector field as conserved moments. The corresponding linear equivalent partial differential equations are generated with a new "Berliner version" of the Taylor expansion method. The details of the implementation are presented. These ideas are applied for the D2Q9, D2Q13, D3Q19 and D3Q27 lattice Boltzmann schemes. Some limitations associated with necessary stability conditions are also presented., 24 pages

We propose to extend the d'Humi\'eres version of the lattice Boltzmann scheme to triangular meshes. We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant. On such meshes, it is possible to define the lattice Boltzmann scheme as a discrete particle method, without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice. We test this idea for the heat equation and perform an asymptotic analysis with the Taylor expansion method for two schemes named D2T4 and D2T7. The results show a convergence up to second order accuracy and set new questions concerning a possible super-convergence., Comment: 23 pages

Using the Taylor expansion method, we show that it is possible to improve the lattice Boltzmann method for acoustic applications. We derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy. The corresponding discrete equations are solved with the help of symbolic manipulation. The solutions are explicited in the case of D3Q27 lattice Boltzmann scheme. Various numerical tests support the coherence of this approach., Comment: 23 pages