3,521 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
Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures
This paper describes a methodology that enables NURBS (Non-Uniform Rational B-spline) based Isogeometric Analysis (IGA) to be locally refined. The methodology is applied to continuous Bubnov-Galerkin IGA spatial discretisations of second-order forms of the neutron transport equation. In particular this paper focuses on the self-adjoint angular flux (SAAF) and weighted least squares (WLS) equations. Local refinement is achieved by constraining degrees of freedom on interfaces between NURBS patches that have different levels of spatial refinement. In order to effectively utilise constraint based local refinement, adaptive mesh refinement (AMR) algorithms driven by a heuristic error measure or forward error indicator (FEI) and a dual weighted residual (DWR) or goal-based error measure (WEI) are derived. These utilise projection operators between different NURBS meshes to reduce the amount of computational effort required to calculate the error indicators. In order to apply the WEI to the SAAF and WLS second-order forms of the neutron transport equation the adjoint of these equations are required. The physical adjoint formulations are derived and the process of selecting source terms for the adjoint neutron transport equation in order to calculate the error in a given quantity of interest (QoI) is discussed. Several numerical verification benchmark test cases are utilised to investigate how the constraint based local refinement affects the numerical accuracy and the rate of convergence of the NURBS based IGA spatial discretisation. The nuclear reactor physics verification benchmark test cases show that both AMR algorithms are superior to uniform refinement with respect to accuracy per degree of freedom. Furthermore, it is demonstrated that for global QoI the FEI driven AMR and WEI driven AMR produce similar results. However, if local QoI are desired then WEI driven AMR algorithm is more computationally efficient and accurate per degree of freedom
Diffusion limits of the Lorentz model: Asymptotic preserving schemes
This paper deals with the diffusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diffusive limit, the right discrete diffusion equation with the same value of the diffusion coefficient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation
Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering
In highly diffusion regimes when the mean free path tends to
zero, the radiative transfer equation has an asymptotic behavior which is
governed by a diffusion equation and the corresponding boundary condition.
Generally, a numerical scheme for solving this problem has the truncation error
containing an contribution, that leads to a nonuniform
convergence for small . Such phenomenons require high resolutions
of discretizations, which degrades the performance of the numerical scheme in
the diffusion limit. In this paper, we first provide a--priori estimates for
the scaled spherical harmonic () radiative transfer equation. Then we
present an error analysis for the spherical harmonic discontinuous Galerkin
(DG) method of the scaled radiative transfer equation showing that, under some
mild assumptions, its solutions converge uniformly in to the
solution of the scaled radiative transfer equation. We further present an
optimal convergence result for the DG method with the upwind flux on Cartesian
grids. Error estimates of
(where is the maximum element length) are obtained when tensor product
polynomials of degree at most are used
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