Allan B. Wollaber

Impact of ejecta morphology and composition on the electromagnetic signatures of neutron star mergers

The electromagnetic transients accompanying compact binary mergers (gamma-ray bursts, after-glows and macronovae) are crucial to pinpoint the sky location of gravitational wave sources. Macronova ...

A Consistent, Moment-Based, Multiscale Solution Approach for Thermal Radiative Transfer Problems

We present an efficient numerical algorithm for solving the time-dependent grey thermal radiative transfer (TRT) equations. The algorithm utilizes the first two angular moments of the TRT equations (Quasi-diffusion (QD)) together with the material temperature equation to form a nonlinear low-order (LO) system. The LO system is solved via the Jacobian-free Newton-Krylov method. The combined high-order (HO) TRT and LO-QD system is used to bridge the diffusion and transport scales. In addition, a “consistency” term is introduced to make the truncation error in the LO system identical to the truncation error in the HO equation. The derivation of the consistency term is rather general; therefore, it can be extended to a variety of spatial and temporal discretizations.

An Efficient and Time Accurate, Moment-Based Scale-Bridging Algorithm for Thermal Radiative Transfer Problems

We present physics-based preconditioning and a time-stepping strategy for a moment-based scale-bridging algorithm applied to the thermal radiative transfer equation. Our goal is to obtain (asymptotically) second-order time accurate and consistent solutions without nonlinear iterations between the high-order (HO) transport equation and the low-order (LO) continuum system within a time step. Modified equation analysis shows that this can be achieved via a simple predictor-corrector time stepping that requires one inversion of the transport operator per time step. We propose a physics-based preconditioning based on a combination of the nonlinear elimination technique and Fleck--Cummings linearization. As a result, the LO system can be solved efficiently via a multigrid preconditioned Jacobian-free Newton--Krylov method. For a set of numerical test problems, the physics-based preconditioner reduces the number of GMRES iterations by a factor of 3

Four Decades of Implicit Monte Carlo

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Monte Carlo simulation methods in moment-based scale-bridging algorithms for thermal radiative-transfer problems

4 as compared to a standard preconditioner for advection-di...

A Discrete Maximum Principle for the Implicit Monte Carlo Equations

ABSTRACT In 1971, Fleck and Cummings derived a system of equations to enable robust Monte Carlo simulations of time-dependent, thermal radiative transfer problems. Denoted the “Implicit Monte Carlo” (IMC) equations, their solution remains the de facto standard of high-fidelity radiative transfer simulations. Over the course of 44 years, their numerical properties have become better understood, and accuracy enhancements, novel acceleration methods, and variance reduction techniques have been suggested. In this review, we rederive the IMC equations—explicitly highlighting assumptions as they are made—and outfit the equations with a Monte Carlo interpretation. We put the IMC equations in context with other approximate forms of the radiative transfer equations and present a new demonstration of their equivalence to another well-used linearization solved with deterministic transport methods for frequency-independent problems. We discuss physical and numerical limitations of the IMC equations for asymptotically small time steps, stability characteristics and the potential of maximum principle violations for large time steps, and solution behaviors in an asymptotically thick diffusive limit. We provide a new stability analysis for opacities with general monomial dependence on temperature. We consider spatial accuracy limitations of the IMC equations and discussion acceleration and variance reduction techniques.

A linear stability analysis for nonlinear, grey, thermal radiative transfer problems

We present a moment-based acceleration algorithm applied to Monte Carlo simulation of thermal radiative-transfer problems. Our acceleration algorithm employs a continuum system of moments to accelerate convergence of stiff absorption-emission physics. The combination of energy-conserving tallies and the use of an asymptotic approximation in optically thick regions remedy the difficulties of local energy conservation and mitigation of statistical noise in such regions. We demonstrate the efficiency and accuracy of the developed method. We also compare directly to the standard linearization-based method of Fleck and Cummings 1. A factor of 40 reduction in total computational time is achieved with the new algorithm for an equivalent (or more accurate) solution as compared with the Fleck-Cummings algorithm.

Moment-Based Acceleration of Monte Carlo Solution for Multifrequency Thermal Radiative Transfer Problems

Abstract It is well known that temperature solutions of the Implicit Monte Carlo (IMC) equations can exceed the external boundary temperatures, a violation of the “maximum principle.” Previous attempts to prescribe a maximum value of the time-step size Δt that is sufficient to eliminate these violations have recommended a Δt that is typically too small to be used in practice and that appeared to be much too conservative when compared to the actual Δt required to prevent maximum principle violations in numerical solutions of the IMC equations. In this paper we derive a new, approximate estimator for the maximum time-step size that includes the spatial-grid size Δx of the temperature field. We also provide exact necessary and sufficient conditions on the maximum time-step size that are easier to calculate. These explicitly demonstrate that the effect of coarsening Δx is to reduce the limitation on Δt. This helps explain the overly conservative nature of the earlier, grid-independent results. We demonstrate that the new time-step restriction is a much more accurate predictor of violations of the maximum principle. We discuss how the implications of the new, grid-dependent time-step restriction can affect IMC solution algorithms.

Implicit Monte Carlo with a linear discontinuous finite element material solution and piecewise non-constant opacity

We present a new linear stability analysis of three time discretizations and Monte Carlo interpretations of the nonlinear, grey thermal radiative transfer (TRT) equations: the widely used Implicit Monte Carlo (IMC) equations, the Carter Forest (CF) equations, and the Ahrens-Larsen or Semi-Analog Monte Carlo (SMC) equations. Using a spatial Fourier analysis of the 1-D Implicit Monte Carlo (IMC) equations that are linearized about an equilibrium solution, we show that the IMC equations are unconditionally stable (undamped perturbations do not exist) if α, the IMC time-discretization parameter, satisfies 0.5<α≤1. This is consistent with conventional wisdom. However, we also show that for sufficiently large time steps, unphysical damped oscillations can exist that correspond to the lowest-frequency Fourier modes. After numerically confirming this result, we develop a method to assess the stability of any time discretization of the 0-D, nonlinear, grey, thermal radiative transfer problem. Subsequent analyses of the CF and SMC methods then demonstrate that the CF method is unconditionally stable and monotonic, but the SMC method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium. This stability theory provides new conditions on the time step to guarantee monotonicity of the IMC solution, although they are likely too conservative to be used in practice. Theoretical predictions are tested and confirmed with numerical experiments.

An Analysis of Source Tilting and Sub-cell Opacity Sampling for IMC

We have extended a Monte Carlo-based, moment-based acceleration algorithm to the solution of multifrequency thermal radiative transfer problems. This study focuses on two aspects. First, we consider stability/accuracy issues for a predictor-corrector time-stepping. It is demonstrated that with a consistent Planckian-weighted opacity the predictor-corrector algorithm can run stably with a larger time-step size compared to a fixed (or lagged) opacity case. With this advancement, consistency improves by about two orders of magnitude, while additional computational cost is kept minimal (< 10%). We also extend the “asymptotic assistance” concept to multifrequency problems. This technique replaces the multifrequency Monte Carlo solution with an asymptotic solution of the O(ε) accurate solution of the equilibrium diffusion limit in optically thick regions. With this technique, the computational time is reduced about a factor of four in a one-dimensional problem. Furthermore, the solution with asymptotic assistance can become more accurate for a similar computational time because it enables a finer group structure in the high-order problem.