1,544 research outputs found
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
A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit
We propose a multilevel Monte Carlo method for a particle-based
asymptotic-preserving scheme for kinetic equations. Kinetic equations model
transport and collision of particles in a position-velocity phase-space. With a
diffusive scaling, the kinetic equation converges to an advection-diffusion
equation in the limit of zero mean free path. Classical particle-based
techniques suffer from a strict time-step restriction to maintain stability in
this limit. Asymptotic-preserving schemes provide a solution to this time step
restriction, but introduce a first-order error in the time step size. We
demonstrate how the multilevel Monte Carlo method can be used as a bias
reduction technique to perform accurate simulations in the diffusive regime,
while leveraging the reduced simulation cost given by the asymptotic-preserving
scheme. We describe how to achieve the necessary correlation between simulation
paths at different levels and demonstrate the potential of the approach via
numerical experiments.Comment: 20 pages, 7 figures, published in Monte Carlo and Quasi-Monte Carlo
Methods 2018, correction of minor typographical error
Unified Gas-kinetic Wave-Particle Methods III: Multiscale Photon Transport
In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) method
to the multiscale photon transport. In this method, the photon free streaming
and scattering processes are treated in an un-splitting way. The duality
descriptions, namely the simulation particle and distribution function, are
utilized to describe the photon. By accurately recovering the governing
equations of the unified gas-kinetic scheme (UGKS), the UGKWP preserves the
multiscale dynamics of photon transport from optically thin to optically thick
regime. In the optically thin regime, the UGKWP becomes a Monte Carlo type
particle tracking method, while in the optically thick regime, the UGKWP
becomes a diffusion equation solver. The local photon dynamics of the UGKWP, as
well as the proportion of wave-described and particle-described photons are
automatically adapted according to the numerical resolution and transport
regime. Compared to the -type UGKS, the UGKWP requires less memory cost
and does not suffer ray effect. Compared to the implicit Monte Carlo (IMC)
method, the statistical noise of UGKWP is greatly reduced and computational
efficiency is significantly improved in the optically thick regime. Several
numerical examples covering all transport regimes from the optically thin to
optically thick are computed to validate the accuracy and efficiency of the
UGKWP method. In comparison to the -type UGKS and IMC method, the UGKWP
method may have several-order-of-magnitude reduction in computational cost and
memory requirement in solving some multsicale transport problems.Comment: 27 pages, 15 figures. arXiv admin note: text overlap with
arXiv:1810.0598
The Moment Guided Monte Carlo method for the Boltzmann equation
In this work we propose a generalization of the Moment Guided Monte Carlo
method developed in [11]. This approach permits to reduce the variance of the
particle methods through a matching with a set of suitable macroscopic moment
equations. In order to guarantee that the moment equations provide the correct
solutions, they are coupled to the kinetic equation through a non equilibrium
term. Here, at the contrary to the previous work in which we considered the
simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we
introduce an hybrid setting which permits to entirely remove the resolution of
the kinetic equation in the limit of infinite number of collisions and to
consider only the solution of the compressible Euler equation. This
modification additionally reduce the statistical error with respect to our
previous work and permits to perform simulations of non equilibrium gases using
only a few number of particles. We show at the end of the paper several
numerical tests which prove the efficiency and the low level of numerical noise
of the method.Comment: arXiv admin note: text overlap with arXiv:0908.026
A unified gas-kinetic particle method for frequency-dependent radiative transfer equations with isotropic scattering process on unstructured mesh
In this paper, we extend the unified kinetic particle (UGKP) method to the
frequency-dependent radiative transfer equation with both absorption-emission
and scattering processes. The extended UGKP method could not only capture the
diffusion and free transport limit, but also provide a smooth transition in the
physical and frequency space in the regime between the above two limits. The
proposed scheme has the properties of asymptotic-preserving, regime-adaptive,
and entropy-preserving, which make it an accurate and efficient scheme in the
simulation of multiscale photon transport problems. The methodology of scheme
construction is a coupled evolution of macroscopic energy equation and the
microscopic radiant intensity equation, where the numerical flux in macroscopic
energy equation and the closure in microscopic radiant intensity equation are
constructed based on the integral solution. Both numerical dissipation and
computational complexity are well controlled especially in the optical thick
regime. A 2D multi-thread code on a general unstructured mesh has been
developed. Several numerical tests have been simulated to verify the numerical
scheme and code, covering a wide range of flow regimes. The numerical scheme
and code that we developed are highly demanded and widely applicable in the
high energy density engineering applications
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