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 $S_n$ -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 $S_n$ -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

A High-Order Low-Order Algorithm with Exponentially-Convergent Monte Carlo for Thermal Radiative Transfer Problems

We have implemented a new high-order low-order (HOLO) algorithm for solving thermal radiative transfer (TRT) problems. Within each discrete time step, fixed-point iterations are performed between a high-order (HO) exponentially-convergent Monte Carlo (ECMC) solver and a low-order (LO) system of equations. The LO system is based on spatial and angular moments of the transport equation and a linear-discontinuous finite-element (LDFE) spatial representation, producing equations similar to the standard S2 equations. The LO solver is fully implicit in time and efficiently converges the non-linear temperature dependence with Newton's method. The HO solver provides a globally accurate solution for the angular intensity to a fixed-source, pure absorber transport problem. This global solution is used to compute consistency terms in the LO equations that require the HO and LO solutions to converge towards the same solution. The use of ECMC allows for the efficient reduction of statistical noise in the solution. We investigated several extensions of this algorithm. A parametric closure of the LO system was used for the spatial variable, based on local relations computed with the HO solver. The spatial closure improves consistency between the two solvers compared to a standard LDFE spatial discretization of the LO system. The ECMC algorithm has been extended to integrate the angular intensity in time, with a consistent time closure of the LO radiation equations. The time closure increases accuracy in optically-thin problems compared to a backward Euler discretization. Finally, we have applied standard source iteration and Krylov procedures to iteratively solve the LO equations, with linear diffusion synthetic acceleration. Herein, we present results for one-dimensional, gray test problems. Results demonstrate several desirable properties of this algorithm: the HOLO method preserves the equilibrium diffusion limit, prevents violation of the maximum principle, and can provide high-fidelity MC solutions to the TRT equations with minimal statistical noise. We have compared results with an implicit Monte Carlo (IMC) code and compared the efficiency of ECMC to standard Monte Carlo in this HOLO algorithm. Our HOLO algorithm is more accurate and more efficient than standard IMC. The extent to which this is so is problem-dependent

A New Monte Carlo Method for Time-Dependent Neutrino Radiation Transport

Monte Carlo approaches to radiation transport have several attractive properties compared to deterministic methods. These include simplicity of implementation, high accuracy, and good parallel scaling. Moreover, Monte Carlo methods can handle complicated geometries and are relatively easy to extend to multiple spatial dimensions, which makes them particularly interesting in modeling complex multi-dimensional astrophysical phenomena such as core-collapse supernovae. The aim of this paper is to explore Monte Carlo methods for modeling neutrino transport in core-collapse supernovae. We generalize the implicit Monte Carlo photon transport scheme of Fleck & Cummings and gray discrete-diffusion scheme of Densmore et al. to energy-, time-, and velocity-dependent neutrino transport. Using our 1D spherically-symmetric implementation, we show that, similar to the photon transport case, the implicit scheme enables significantly larger timesteps compared with explicit time discretization, without sacrificing accuracy, while the discrete-diffusion method leads to significant speed-ups at high optical depth. Our results suggest that a combination of spectral, velocity-dependent, implicit Monte Carlo and discrete-diffusion Monte Carlo methods represents an attractive approach for use in neutrino radiation-hydrodynamics simulations of core-collapse supernovae. Our velocity-dependent scheme can easily be adapted to photon transport

Anisotropic Diffusion Approximations for Time-dependent Particle Transport.

In this thesis, we develop and numerically test new approximations to time-dependent radiation transport with the goal of obtaining more accurate solutions than the diffusion approximation can generate, yet requiring less computational effort than full transport. The first method is the nascent anisotropic diffusion (AD) approximation, which we extend to time-dependent problems in finite domains; the second is a novel anisotropic P_1-like (AP_1) approximation. These methods are ``anisotropic'' in that, rather than operating under the assumption of linearly anisotropic radiation, they incorporate an arbitrary amount of anisotropy via a transport-calculated diffusion coefficient. This anisotropic diffusion tensor is the second angular moment of a simple, purely absorbing transport problem. In this thesis, much of the computational testing of the new methods is performed in ``flatland'' geometry, a fictional two-dimensional universe that provides a realistic but computationally inexpensive testbed. As work ancillary to anisotropic diffusion and the numerical experiments, a complete description of flatland diffusion, including boundary conditions, is developed. Also, implementation details for both Monte Carlo and S_N transport in flatland are provided. The two new anisotropic methods, along with a ``flux limited'' modification to anisotropic diffusion, are tested in a variety of problems. Some aspects of the theory, including the newly formulated boundary conditions, are tested first with diffusive, steady-state problems. The new methods are compared against existing ones in linear, time-dependent radiation transport problems. Finally, the efficacy and performance of the anisotropic methods are investigated in several thermal radiative transfer (TRT) computational experiments. Our results demonstrate that for many multi-dimensional problems, the new anisotropic methods perform much better than their conventional counterparts. In every time-dependent test, the flux-limited anisotropic diffusion approach produced the most accurate solutions of the new methods. Based on our numerical testing, we believe this method to be a strong contender for accurate, inexpensive simulations of time-dependent transport and thermal radiative transfer problems.Ph.D.Nuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91465/1/sethrj_1.pd