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Eric Sonnendrücker - One of the best experts on this subject based on the ideXlab platform.
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solving the Vlasov Equation in complex geometries
Esaim: Proceedings, 2011Co-Authors: J Abiteboul, Eric Sonnendrücker, G Latu, V Grandgirard, Ahmed Ratnani, A StrugarekAbstract:This paper introduces an isoparametric analysis to solve the Vlasov Equation with a semi- Lagrangian scheme. A Vlasov-Poisson problem modeling a heavy ion beam in an axisymmetric congu- ration is considered. Numerical experiments are conducted on computational meshes targeting dierent geometries. The impact of the computational grid on the accuracy and the computational cost are shown. The use of analytical mapping or B ezier patches does not induce a too large computational overhead and is quite accurate. This approach successfully couples an isoparametric analysis with a semi-Lagrangian scheme, and we expect to apply it to a gyrokinetic Vlasov solver. R esum e. Nous pr esentons ici une analyse isoparam etrique pour r esoudre l' Equation de Vlasov
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A forward semi-Lagrangian method for the numerical solution of the Vlasov Equation
Computer Physics Communications, 2009Co-Authors: Nicolas Crouseilles, Thomas Respaud, Eric SonnendrückerAbstract:This work deals with the numerical solution of the Vlasov Equation, which provides a kinetic description of the evolution of a plasma, and is coupled with Poisson's Equation. A new forward semi-Lagrangian method is developed. The distribution function is updated on a Eulerian grid, and the pseudo-particles located on the mesh nodes follow the characteristics of the Equation forward for one time step, and are deposited on the 16 nearest nodes. This is an explicit way of solving the Vlasov Equation on a grid of the phase space, which makes it easier to develop high-order time schemes than the backward method.
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A Forward semi-Lagrangian Method for the Numerical Solution of the Vlasov Equation
Computer Physics Communications, 2009Co-Authors: Nicolas Crouseilles, Thomas Respaud, Eric SonnendrückerAbstract:This work deals with the numerical solution of the Vlasov Equation. This Equation gives a kinetic description of the evolution of a plasma, and is coupled with Poisson's Equation for the computation of the self-consistent electric field. The coupled model is non linear. A new semi-Lagrangian method, based on forward integration of the characteristics, is developed. The distribution function is updated on an eulerian grid, and the pseudo-particles located on the mesh's nodes follow the characteristics of the Equation forward for one time step, and are deposited on the 16 nearest nodes. This is an explicit way of solving the Vlasov Equation on a grid of the phase space, which makes it easier to develop high order time schemes than the backward method.
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Numerical methods for the Vlasov Equation
Numerical Mathematics and Advanced Applications, 2003Co-Authors: Francis Filbet, Eric SonnendrückerAbstract:In this paper, we give a fairly exhaustive review of the literature on numerical simulations of the Vlasov Equation.We first recall the range of applications of the Vlasov Equation and present the different approaches for the discretization.We briefly describe Lagrangian and Eulerian schemes and give a few numerical results comparing these methods.
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An adaptive numerical method for the Vlasov Equation based on a multiresolution analysis.
2003Co-Authors: Nicolas Besse, Francis Filbet, Michaël Gutnic, Ioana Paun, Eric SonnendrückerAbstract:In this paper, we present very first results for the adaptive solution on a grid of the phase space of the Vlasov Equation arising in particles accelarator and plasma physics. The numerical algorithm is based on a semi-Lagrangian method while adaptivity is obtained using multiresolution analysis.
Stefano Ruffo - One of the best experts on this subject based on the ideXlab platform.
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Vlasov Equation for long-range interactions on a lattice
Physical Review E : Statistical Nonlinear and Soft Matter Physics, 2011Co-Authors: Romain Bachelard, Thierry Dauxois, F. Staniscia, Giovanni De Ninno, Stefano RuffoAbstract:We show that, in the continuum limit, the dynamics of Hamiltonian systems defined on a lattice with long-range couplings is well described by the Vlasov Equation. This Equation can be linearized around the homogeneous state and a dispersion relation, that depends explicitly on the Fourier modes of the lattice, can be derived. This allows one to compute the stability thresholds of the homogeneous state, which turn out to depend on the mode number. When this state is unstable, the growth rates are also function of the mode number. Explicit calculations are performed for the $\alpha$-HMF model with $0 \leq \alpha
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stability criteria of the Vlasov Equation and quasi stationary states of the hmf model
Physica A-statistical Mechanics and Its Applications, 2004Co-Authors: Julien Barré, Yoshiyuki Y. Yamaguchi, Freddy Bouchet, Thierry Dauxois, Stefano RuffoAbstract:We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field model, a prototype for long-range interactions in N-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated N→∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite N, dynamics. We then propose, and verify numerically, a scenario for the relaxation process, relying on the Vlasov Equation. When starting from a nonstationary or a Vlasov unstable stationary state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov Equation via nonstationary states: we characterize numerically this dynamical instability in the finite N system by introducing appropriate indicators. This first step of the evolution towards Boltzmann–Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov Equation. If the finite N system is initialized in a Vlasov stable homogeneous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N1.7. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the q-exponential distributions derived from Tsallis statistics.
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Stability criteria of the Vlasov Equation and quasi-stationary states of the HMF model
Physica A, 2004Co-Authors: Y. Y. Yamaguchi, Julien Barré, Freddy Bouchet, Thierry Dauxois, Stefano RuffoAbstract:We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in $N$-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated $N ~$ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite $N$, dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov Equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov Equation via non stationary states: we characterize numerically this dynamical instability in the finite $N$ system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov Equation. If the finite $N$ system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law $N^{1.7}$. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the $q$-exponential distributions derived from Tsallis statistics.
Shun Ogawa - One of the best experts on this subject based on the ideXlab platform.
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Stability criterion of spatially inhomogeneous solutions to the Vlasov Equation
Journal of Physics A: Mathematical and Theoretical, 2017Co-Authors: Shun OgawaAbstract:Stability of spatially inhomogeneous stationary solutions to the Vlasov Equation for multi-dimensional systems is investigated. The formal stability criterion taking into account all of the Casimir invariants is derived in an explicit form without use of the angle-action variables of a single particle's Hamiltonian associated with a stationary solutions to the Vlasov Equation, when biorthogonal basis can be defined.
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spectral and formal stability criteria of spatially inhomogeneous stationary solutions to the Vlasov Equation for the hamiltonian mean field model
Physical Review E, 2013Co-Authors: Shun OgawaAbstract:Stability of spatially inhomogeneous solutions to the Vlasov Equation is investigated for the Hamiltonian mean-field model to provide the spectral and formal stability criteria in the form of necessary and sufficient conditions. These criteria determine stability of spatially inhomogeneous solutions whose stability has not been decided correctly by using a less refined formal stability criterion. It is shown that some of such solutions can be found in a family of stationary solutions to the Vlasov Equation, which is parametrized with macroscopic quantities and has a two-phase coexistence region in the parameter space.
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Linear response theory in the Vlasov Equation for homogeneous and for inhomogeneous quasistationary states.
Physical Review E, 2012Co-Authors: Shun Ogawa, Yoshiyuki Y. YamaguchiAbstract:Response to small external forces is investigated in quasistationary states of Hamiltonian systems having long-range interactions. Quasistationary states are recognized as stable stationary solutions to the Vlasov Equation, and, hence, the linear response theory to the Vlasov Equation is proposed for spatially one-dimensional systems with periodic boundary condition. The proposed theory is applicable both to homogeneous and to inhomogeneous quasistationary states and is demonstrated in the Hamiltonian mean-field model. In the homogeneous case magnetic susceptibility is explicitly obtained, and the Curie-Weiss like law is suggested in a high-energy region. The linear response is also computed in the inhomogeneous case, and resonance absorption is investigated to extract nonequilibrium dynamics in the unforced system. Theoretical predictions are examined by direct numerical simulations of the Vlasov Equation.
Thierry Dauxois - One of the best experts on this subject based on the ideXlab platform.
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Vlasov Equation for long-range interactions on a lattice
Physical Review E : Statistical Nonlinear and Soft Matter Physics, 2011Co-Authors: Romain Bachelard, Thierry Dauxois, F. Staniscia, Giovanni De Ninno, Stefano RuffoAbstract:We show that, in the continuum limit, the dynamics of Hamiltonian systems defined on a lattice with long-range couplings is well described by the Vlasov Equation. This Equation can be linearized around the homogeneous state and a dispersion relation, that depends explicitly on the Fourier modes of the lattice, can be derived. This allows one to compute the stability thresholds of the homogeneous state, which turn out to depend on the mode number. When this state is unstable, the growth rates are also function of the mode number. Explicit calculations are performed for the $\alpha$-HMF model with $0 \leq \alpha
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stability criteria of the Vlasov Equation and quasi stationary states of the hmf model
Physica A-statistical Mechanics and Its Applications, 2004Co-Authors: Julien Barré, Yoshiyuki Y. Yamaguchi, Freddy Bouchet, Thierry Dauxois, Stefano RuffoAbstract:We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field model, a prototype for long-range interactions in N-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated N→∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite N, dynamics. We then propose, and verify numerically, a scenario for the relaxation process, relying on the Vlasov Equation. When starting from a nonstationary or a Vlasov unstable stationary state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov Equation via nonstationary states: we characterize numerically this dynamical instability in the finite N system by introducing appropriate indicators. This first step of the evolution towards Boltzmann–Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov Equation. If the finite N system is initialized in a Vlasov stable homogeneous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N1.7. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the q-exponential distributions derived from Tsallis statistics.
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Stability criteria of the Vlasov Equation and quasi-stationary states of the HMF model
Physica A, 2004Co-Authors: Y. Y. Yamaguchi, Julien Barré, Freddy Bouchet, Thierry Dauxois, Stefano RuffoAbstract:We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in $N$-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated $N ~$ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite $N$, dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov Equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov Equation via non stationary states: we characterize numerically this dynamical instability in the finite $N$ system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov Equation. If the finite $N$ system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law $N^{1.7}$. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the $q$-exponential distributions derived from Tsallis statistics.
Pierre Bertrand - One of the best experts on this subject based on the ideXlab platform.
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conservative numerical schemes for the Vlasov Equation
Journal of Computational Physics, 2001Co-Authors: Francis Filbet, Eric Sonnendrücker, Pierre BertrandAbstract:Abstract A new scheme for solving the Vlasov Equation using a phase space grid is proposed. The algorithm is based on the conservation of the flux of particles, and the distribution function is reconstructed using various techniques that allow control of spurious oscillations or preservation of the positivity. Several numerical results are presented in two- and four-dimensional phase space and the scheme is compared with the semiLagrangian method. This method is almost as accurate as the semiLagrangian one, and the local reconstruction technique is well suited for parallel computation.
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The numerical integration of the Vlasov Equation possessing an invariant
Journal of Computational Physics, 1995Co-Authors: Giovanni Manfredi, Magdi M. Shoucri, Eric Fijalkow, Marc R. Feix, Pierre Bertrand, Alain GhizzoAbstract:A new method for the numerical integration of the Vlasov Equation is presented, which can be applied whenever its characteristics possess an exact invariant. It consists in expressing the distribution function in terms of the invariant itself. The dimensionality of the phase space is thus reduced of one unity, since the invariant only appears as a label of the Vlasov Equation and can be coarsely discretized. This technique is applied to the study of the Kelvin-Helmoltz instability, with a very limited number of invariants. Subsequently an example of ion-temperature-gradient instability is analyzed. Although a larger number of invariants are required to describe the temperature profile, qualitatively correct results can be obtained with fewer invariants. Test particles are used to illustrate stochastic diffusion in the phase space and to calculate the diffusion coefficients.
