6,728 research outputs found

    Unified Gas-kinetic Wave-Particle Methods III: Multiscale Photon Transport

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    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 SnS_n -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 SnS_n -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

    Well-balanced and asymptotic preserving schemes for kinetic models

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    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 ϵ\epsilon tends to zero, the density can be described by (advection) diffusion type (linear or nonlinear) macroscopic models; 2) When ϵ\epsilon = 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 particle micro-macro decomposition based numerical scheme for collisional kinetic equations in the diffusion scaling

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    In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step Δt\Delta t in a consistent way. This non-stiff reformulation of the micro-macro system allows the use of standard particle approximations for the transport part, and extends the work in [5] where a particle approximation has been applied using a micro-macro decomposition on kinetic equations in the fluid scaling. Beyond the so-called asymptotic-preserving property which is satisfied by our schemes, they significantly reduce the inherent noise of traditional particle methods, and they have a computational cost which decreases as the system approaches the diffusion limit

    Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium

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    In this work, we propose some numerical schemes for linear kinetic equations in the diffusion and anomalous diffusion limit. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion type equation. However, when a heavy-tailed distribution is considered, another time scale is required and the small mean free path limit leads to a fractional anomalous diffusion equation. Our aim is to develop numerical schemes for the original kinetic model which works for the different regimes, without being restricted by stability conditions of standard explicit time integrators. First, we propose some numerical schemes for the diffusion asymptotics; then, their extension to the anomalous diffusion limit is studied. In this case, it is crucial to capture the effect of the large velocities of the heavy-tailed equilibrium, so that some important transformations of the schemes derived for the diffusion asymptotics are needed. As a result, we obtain numerical schemes which enjoy the Asymptotic Preserving property in the anomalous diffusion limit, that is: they do not suffer from the restriction on the time step and they degenerate towards the fractional diffusion limit when the mean free path goes to zero. We also numerically investigate the uniform accuracy and construct a class of numerical schemes satisfying this property. Finally, the efficiency of the different numerical schemes is shown through numerical experiments

    A New Spherical Harmonics Scheme for Multi-Dimensional Radiation Transport I: Static Matter Configurations

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    Recent work by McClarren & Hauck [29] suggests that the filtered spherical harmonics method represents an efficient, robust, and accurate method for radiation transport, at least in the two-dimensional (2D) case. We extend their work to the three-dimensional (3D) case and find that all of the advantages of the filtering approach identified in 2D are present also in the 3D case. We reformulate the filter operation in a way that is independent of the timestep and of the spatial discretization. We also explore different second- and fourth-order filters and find that the second-order ones yield significantly better results. Overall, our findings suggest that the filtered spherical harmonics approach represents a very promising method for 3D radiation transport calculations.Comment: 29 pages, 13 figures. Version matching the one in Journal of Computational Physic
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