8,727 research outputs found
Well-balanced and asymptotic preserving schemes for kinetic models
In this paper, we propose a general framework for designing numerical schemes
that have both well-balanced (WB) and asymptotic preserving (AP) properties,
for various kinds of kinetic models. We are interested in two different
parameter regimes, 1) When the ratio between the mean free path and the
characteristic macroscopic length tends to zero, the density can be
described by (advection) diffusion type (linear or nonlinear) macroscopic
models; 2) When = O(1), the models behave like hyperbolic equations
with source terms and we are interested in their steady states. We apply the
framework to three different kinetic models: neutron transport equation and its
diffusion limit, the transport equation for chemotaxis and its Keller-Segel
limit, and grey radiative transfer equation and its nonlinear diffusion limit.
Numerical examples are given to demonstrate the properties of the schemes
A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation
In this paper we consider the development of Implicit-Explicit (IMEX)
Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such
systems the scaling depends on an additional parameter which modifies the
nature of the asymptotic behavior which can be either hyperbolic or parabolic.
Because of the multiple scalings, standard IMEX Runge-Kutta methods for
hyperbolic systems with relaxation loose their efficiency and a different
approach should be adopted to guarantee asymptotic preservation in stiff
regimes. We show that the proposed approach is capable to capture the correct
asymptotic limit of the system independently of the scaling used. Several
numerical examples confirm our theoretical analysis
A Unified Gas-kinetic Scheme for Continuum and Rarefied Flows IV: full Boltzmann and Model Equations
Fluid dynamic equations are valid in their respective modeling scales. With a
variation of the modeling scales, theoretically there should have a continuous
spectrum of fluid dynamic equations. In order to study multiscale flow
evolution efficiently, the dynamics in the computational fluid has to be
changed with the scales. A direct modeling of flow physics with a changeable
scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS)
is a direct modeling method in the mesh size scale, and its underlying flow
physics depends on the resolution of the cell size relative to the particle
mean free path. The cell size of UGKS is not limited by the particle mean free
path. With the variation of the ratio between the numerical cell size and local
particle mean free path, the UGKS recovers the flow dynamics from the particle
transport and collision in the kinetic scale to the wave propagation in the
hydrodynamic scale.
The previous UGKS is mostly constructed from the evolution solution of
kinetic model equations. This work is about the further development of the UGKS
with the implementation of the full Boltzmann collision term in the region
where it is needed. The central ingredient of the UGKS is the coupled treatment
of particle transport and collision in the flux evaluation across a cell
interface, where a continuous flow dynamics from kinetic to hydrodynamic scales
is modeled. The newly developed UGKS has the asymptotic preserving (AP)
property of recovering the NS solutions in the continuum flow regime, and the
full Boltzmann solution in the rarefied regime. In the mostly unexplored
transition regime, the UGKS itself provides a valuable tool for the flow study
in this regime. The mathematical properties of the scheme, such as stability,
accuracy, and the asymptotic preserving, will be analyzed in this paper as
well
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
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