588 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
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
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
Spatial second-order positive and asymptotic preserving filtered schemes for nonlinear radiative transfer equations
A spatial second-order scheme for the nonlinear radiative transfer equations
is introduced in this paper. The discretization scheme is based on the filtered
spherical harmonics () method for the angular variable and the unified
gas kinetic scheme (UGKS) framework for the spatial and temporal variables
respectively. In order to keep the scheme positive and second-order accuracy,
firstly, we use the implicit Monte Carlo linearization method [6] in the
construction of the UGKS numerical boundary fluxes. Then, by carefully
analyzing the constructed second-order fluxes involved in the macro-micro
decomposition, which is induced by the angular discretization, we
establish the sufficient conditions that guarantee the positivity of the
radiative energy density and material temperature. Finally, we employ linear
scaling limiters for the angular variable in the reconstruction and for
the spatial variable in the piecewise linear slopes reconstruction
respectively, which are shown to be realizable and reasonable to enforce the
sufficient conditions holding. Thus, the desired scheme, called the
-based UGKS, is obtained. Furthermore, in the regime
and the regime , a simplified spatial second-order scheme,
called the -based SUGKS, is presented, which possesses all the
properties of the non-simplified one. Inheriting the merit of UGKS, the
proposed schemes are asymptotic preserving. By employing the method for
the angular variable, the proposed schemes are almost free of ray effects. To
our best knowledge, this is the first time that spatial second-order, positive,
asymptotic preserving and almost free of ray effects schemes are constructed
for the nonlinear radiative transfer equations without operator splitting.
Various numerical experiments are included to validate the properties of the
proposed schemes
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