8,333 research outputs found
Adaptive and Recursive Time Relaxed Monte Carlo methods for rarefied gas dynamics
Recently a new class of Monte Carlo methods, called Time Relaxed Monte Carlo
(TRMC), designed for the simulation of the Boltzmann equation close to fluid
regimes have been introduced. A generalized Wild sum expansion of the solution
is at the basis of the simulation schemes. After a splitting of the equation
the time discretization of the collision step is obtained from the Wild sum
expansion of the solution by replacing high order terms in the expansion with
the equilibrium Maxwellian distribution; in this way speed up of the methods
close to fluid regimes is obtained by efficiently thermalizing particles close
to the equilibrium state. In this work we present an improvement of such
methods which allows to obtain an effective uniform accuracy in time without
any restriction on the time step and subsequent increase of the computational
cost. The main ingredient of the new algorithms is recursivity. Several
techniques can be used to truncate the recursive trees generated by the schemes
without deteriorating the accuracy of the numerical solution. Techniques based
on adaptive strategies are presented. Numerical results emphasize the gain of
efficiency of the present simulation schemes with respect to standard DSMC
methods
Exponential Runge-Kutta methods for stiff kinetic equations
We introduce a class of exponential Runge-Kutta integration methods for
kinetic equations. The methods are based on a decomposition of the collision
operator into an equilibrium and a non equilibrium part and are exact for
relaxation operators of BGK type. For Boltzmann type kinetic equations they
work uniformly for a wide range of relaxation times and avoid the solution of
nonlinear systems of equations even in stiff regimes. We give sufficient
conditions in order that such methods are unconditionally asymptotically stable
and asymptotic preserving. Such stability properties are essential to guarantee
the correct asymptotic behavior for small relaxation times. The methods also
offer favorable properties such as nonnegativity of the solution and entropy
inequality. For this reason, as we will show, the methods are suitable both for
deterministic as well as probabilistic numerical techniques
Fluid Solver Independent Hybrid Methods for Multiscale Kinetic equations
In some recent works [G. Dimarco, L. Pareschi, Hybrid multiscale methods I.
Hyperbolic Relaxation Problems, Comm. Math. Sci., 1, (2006), pp. 155-177], [G.
Dimarco, L. Pareschi, Hybrid multiscale methods II. Kinetic equations, SIAM
Multiscale Modeling and Simulation Vol 6., No 4,pp. 1169-1197, (2008)] 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 form-fitting 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 Boltzmann-BGK equation are presented to show the
performance of the new methods in comparison with classical approaches used in
the simulation of kinetic equations.Comment: 31 page
Direct simulation Monte Carlo schemes for Coulomb interactions in plasmas
We consider the development of Monte Carlo schemes for molecules with Coulomb
interactions. We generalize the classic algorithms of Bird and Nanbu-Babovsky
for rarefied gas dynamics to the Coulomb case thanks to the approximation
introduced by Bobylev and Nanbu (Theory of collision algorithms for gases and
plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation,
Physical Review E, Vol. 61, 2000). Thus, instead of considering the original
Boltzmann collision operator, the schemes are constructed through the use of an
approximated Boltzmann operator. With the above choice larger time steps are
possible in simulations; moreover the expensive acceptance-rejection procedure
for collisions is avoided and every particle collides. Error analysis and
comparisons with the original Bobylev-Nanbu (BN) scheme are performed. The
numerical results show agreement with the theoretical convergence rate of the
approximated Boltzmann operator and the better performance of Bird-type schemes
with respect to the original scheme
Extending fragment-based free energy calculations with library Monte Carlo simulation: Annealing in interaction space
Pre-calculated libraries of molecular fragment configurations have previously
been used as a basis for both equilibrium sampling (via "library-based Monte
Carlo") and for obtaining absolute free energies using a polymer-growth
formalism. Here, we combine the two approaches to extend the size of systems
for which free energies can be calculated. We study a series of all-atom
poly-alanine systems in a simple dielectric "solvent" and find that precise
free energies can be obtained rapidly. For instance, for 12 residues, less than
an hour of single-processor is required. The combined approach is formally
equivalent to the "annealed importance sampling" algorithm; instead of
annealing by decreasing temperature, however, interactions among fragments are
gradually added as the molecule is "grown." We discuss implications for future
binding affinity calculations in which a ligand is grown into a binding site
Stochastic classical field model for polariton condensates
We use the truncated Wigner approximation to derive stochastic classical
field equations for the description of polariton condensates. Our equations are
shown to reduce to the Boltzmann equation in the limit of low polariton
density. Monte Carlo simulations are performed to analyze the momentum
distribution and the first and second order coherence when the particle density
is varied across the condensation threshold.Comment: 10 pages, 7 figure
Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit
We develop a new Monte Carlo method that solves hyperbolic transport
equations with stiff terms, characterized by a (small) scaling parameter. In
particular, we focus on systems which lead to a reduced problem of parabolic
type in the limit when the scaling parameter tends to zero. Classical Monte
Carlo methods suffer of severe time step limitations in these situations, due
to the fact that the characteristic speeds go to infinity in the diffusion
limit. This makes the problem a real challenge, since the scaling parameter may
differ by several orders of magnitude in the domain. To circumvent these time
step limitations, we construct a new, asymptotic-preserving Monte Carlo method
that is stable independently of the scaling parameter and degenerates to a
standard probabilistic approach for solving the limiting equation in the
diffusion limit. The method uses an implicit time discretization to formulate a
modified equation in which the characteristic speeds do not grow indefinitely
when the scaling factor tends to zero. The resulting modified equation can
readily be discretized by a Monte Carlo scheme, in which the particles combine
a finite propagation speed with a time-step dependent diffusion term. We show
the performance of the method by comparing it with standard (deterministic)
approaches in the literature
- …