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On the stability of spectral methods for the homogeneous Boltzmann equation
 Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI
"... In this paper we introduce a new smoothed scheme derived from the spectral Fourier method for the homogeneous Boltzmann equation recently introduced in [14, 15]. More precisely we show that using suitable smoothing lters the method can be designed in such a way that the spectral solution remains pos ..."
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Cited by 45 (14 self)
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In this paper we introduce a new smoothed scheme derived from the spectral Fourier method for the homogeneous Boltzmann equation recently introduced in [14, 15]. More precisely we show that using suitable smoothing lters the method can be designed in such a way that the spectral solution remains positive in time and preserves the total mass. Several numerical examples are given to illustrate the previous analysis. Key words: Boltzmann equation, spectral methods, smoothing tecniques. 1 Introduction The numerical solution of the Boltzmann equation has been a challenge for several decades. The main diculties that are encountered concern the high dimensionality of the problem (the density function f depends in general on seven independent variables, namely position, velocity, and time), and the evaluation of the collisional integral. Even when considering space homogeneous problems, it is not easy to construct a scheme that is accurate and maintains most of the qualitative properties of ...
Fast Spectral Methods for the FokkerPlanckLandau Collision Operator of Plasma Physics
 J. Comput. Phys
, 1999
"... In this paper we present a new spectral method for the fast evaluation of the FokkerPlanckLandau collision operator. The method allows to obtain spectrally accurate numerical solutions with simply O(n log 2 n) operations in contrast with the usual O(n 2 ) cost of a deterministic scheme. We show ..."
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Cited by 36 (10 self)
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In this paper we present a new spectral method for the fast evaluation of the FokkerPlanckLandau collision operator. The method allows to obtain spectrally accurate numerical solutions with simply O(n log 2 n) operations in contrast with the usual O(n 2 ) cost of a deterministic scheme. We show that the method preserves the total mass whereas momentum and energy are approximated with spectral accuracy. Numerical results for both the Maxwellian and the Coulombian case in 2D and 3D velocity space are also given. Key words: FokkerPlanckLandau equation, spectral methods, fast Fourier transform. 1 Introduction. This paper is devoted to the development of numerical schemes for the accurate computation of the solution of the FokkerPlanckLandau equation. This work was partially supported by TMR project \Asymptotic Methods in Kinetic Theory", Contract Number ERB FMRX CT97 0157. y Department of Mathematics, University of Ferrara, via Machiavelli 35, I44100 Ferrara, ITALY. (pares...
High order numerical methods for the space non homogeneous Boltzmann equation.
 J. Comput. Phys
, 2003
"... In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rareed gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) Positive and Flux conservative (PFC) method. The collision step is treate ..."
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Cited by 26 (7 self)
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In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rareed gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) Positive and Flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrator in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods. Keywords: Boltzmann equation, Rareed gas dynamics, spectral methods, splitting algorithms 1
Fast algorithms for computing the Boltzmann collision operator
 Math. Comp
, 2004
"... Abstract. The development of accurate and fast numerical schemes for the five fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the socalled hard spheres model in dimension three, we are able to derive spect ..."
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Cited by 26 (11 self)
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Abstract. The development of accurate and fast numerical schemes for the five fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the socalled hard spheres model in dimension three, we are able to derive spectral methods and discrete velocity methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and, in particular, for the spectral method spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed. Keywords. algorithms.
Spectral Methods For The Non CutOff Boltzmann Equation And Numerical Grazing Collision Limit
"... . In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cuto# to the FokkerPlanckLandau equation in the socalled grazing collision limit. To this aim we derive a Fourier spectral method for the non cuto# Boltzmann equation in the spirit of [21, 2 ..."
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Cited by 17 (5 self)
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. In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cuto# to the FokkerPlanckLandau equation in the socalled grazing collision limit. To this aim we derive a Fourier spectral method for the non cuto# Boltzmann equation in the spirit of [21, 23]. We show that the kernel modes that define the spectral method have the correct grazing collision limit providing a consistent spectral method for the limiting FokkerPlanckLandau equation. In particular, for small values of the scattering angle, we derive an approximate formula for the kernel modes of the non cuto# Boltzmann equation which, similarly to the FokkerPlanckLandau case, can be computed with a fast algorithm. The uniform spectral accuracy of the method with respect to the grazing collision parameter is also proved. Key words. Spectral methods, Boltzmann equation, cuto# assumption, FokkerPlanck Landau equation, grazing collision limit. 1. Introduction In this paper we ar...
FLUID SOLVER INDEPENDENT HYBRID METHODS FOR MULTISCALE KINETIC EQUATIONS
, 2009
"... In some recent works [11, 12] we developed a general framework for the construction of hybrid algorithms which are able to face efficiently the multiscale nature of some hyperbolic and kinetic problems. Here, at variance with respect to the previous methods, we construct a method formfitting to an ..."
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Cited by 9 (6 self)
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In some recent works [11, 12] we developed a general framework for the construction of hybrid algorithms which are able to face efficiently the multiscale nature of some hyperbolic and kinetic problems. Here, at variance with respect to the previous methods, we construct a method formfitting to any type of finite volume or finite difference scheme for the reduced equilibrium system. Thanks to the coupling of Monte Carlo techniques for the solution of the kinetic equations with macroscopic methods for the limiting fluid equations, we show how it is possible to solve multiscale fluid dynamic phenomena faster with respect to traditional deterministic/stochastic methods for the full kinetic equations. In addition, due to the hybrid nature of the schemes, the numerical solution is affected by less fluctuations when compared to standard Monte Carlo schemes. Applications to the BoltzmannBGK equation are presented to show the performance of the new methods in comparison with classical approaches used in the simulation of kinetic equations.
SOLVING THE BOLTZMANN EQUATION IN N log 2 N
"... Abstract. In [28, 27], fast deterministic algorithms based on spectral methods were derived for the Boltzmann collision operator for a class of interactions including the hard spheres model in dimension 3. These algorithms are implemented for the solution of the Boltzmann equation in dimension 2 and ..."
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Cited by 7 (3 self)
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Abstract. In [28, 27], fast deterministic algorithms based on spectral methods were derived for the Boltzmann collision operator for a class of interactions including the hard spheres model in dimension 3. These algorithms are implemented for the solution of the Boltzmann equation in dimension 2 and 3, first for homogeneous solutions, then for general nonhomogeneous solutions. The results are compared to explicit solutions, when available, and to MonteCarlo methods. In particular, the computational cost and accuracy are compared to those of Monte Carlo methods as well as to those of previous spectral methods. Finally, for inhomogeneous solutions, we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium, which was suggested in the work [14].
Spectral methods for onedimensional kinetic models of granular flows and numerical quasi elastic limit
 ESAIM RAIRO Math. Model. Numer. Anal
, 2003
"... Abstract. In this paper we introduce numerical schemes for a onedimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction ..."
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Cited by 7 (3 self)
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Abstract. In this paper we introduce numerical schemes for a onedimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the socalled quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [25,26] and show that the kernel modes that define the spectral method have the correct quasi elastic limit providing a consistent spectral method for the limiting nonlinear friction equation.
Spectral Lagrangian methods for Collisional Models of Non Equilibrium Statistical States
, 710
"... We propose a new spectral Lagrangian based deterministic solver for the nonlinear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or nonconservative binary interactions. The method is based on symmetries of the Fourier transform of the collision i ..."
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Cited by 5 (0 self)
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We propose a new spectral Lagrangian based deterministic solver for the nonlinear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or nonconservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computing is reduced to a separate integral over the unit sphere S 2. In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space is very versatile and adjusts in a very simple manner, to several cases that involve energy dissipation due to local microreversibility (inelastic interactions) or elastic model of slowing down process. Our simulations are benchmarked with the available exact selfsimilar solutions, exact moment equations and analytical estimates for homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the simulations involves the selection of a time selfsimilar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in [13] and generalized to a wide range of related models in [12]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hardspheres (inelastic collisions under thermal bath), where overpopulated nonGaussian exponential tails have been conjectured in computations by stochastic methods in [49; 26; 46; 35] and rigourously proven in [34] and [15].