Two-moment Model Research Articles

DG-IMEX method for a two-moment model for radiation transport in the [formula omitted] limit

We consider neutral particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to O(v/c), expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. The model is conservative for the correct O(v/c) Eulerian-frame number density and is consistent, to O(v/c), with Eulerian-frame energy and momentum conservation. This model is closely related to the one promoted by Lowrie et al. [1] and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The proposed numerical method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical fluxes; (iii) a realizability-preserving implicit collision update; and (iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed global convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose a “spectral redistribution” scheme that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties.

Equivalent pin-forces or equivalent moments for the modelling of piezoelectric patches: a parametric study

Many piezoelectric (PZ) actuation or sensing systems consist of PZ patches bonded on elastic beams or blades. In order to optimise the design of such systems, Finite Element Analysis (FEA) can be used. However, this option is relatively time consuming and not necessarily appropriate to the first steps of the design process. The replacement of FEA by simple analytical tools is desirable in the early design stages in order to explore the optimal configurations for the device (beam dimensions, patch position and properties). Two main modelling approaches can be found in the literature, based on the Solid Mechanics beam theory. The first approach consists in replacing the PZ patch by two opposite forces positioned at the surface of the beam, each at one extremity of the PZ patch. The second approach consists in replacing the PZ patch by two opposite moments positioned at the neutral axis of the beam, each at one extremity of the PZ patch. The object of this paper is to detail these options, and to evaluate their range of validity. For this purpose, a parametric study is conducted on a cantilever beam structure to compare the different approaches for standard dimensions and material properties. The results of corresponding FEA simulations are taken as reference. It is shown that the validity of analytical models is restricted to a narrow range of material properties and dimensions. This range is chosen when the error between the normalised displacements obtained analytically and those obtained by numerical calculation does not exceed 6%. Within this range, the two-moment model is revealed a more precise choice than two pin-forces. As a consequence, its validity range is larger compared to other analytical approaches. This is due to the introduction of the flexural stiffness of the PZ patch and of a realistic strain profile across the section of the structure. These results can be used to obtain analytical expressions of stress and strains in PZ actuation and sensing devices.

Comparing different closure relations for cosmic ray hydrodynamics

ABSTRACT Cosmic ray (CR) hydrodynamics is a (re-)emerging field of high interest due to the importance of CRs for the dynamical evolution of the interstellar, the circumgalactic, and the intracluster medium. In these environments, CRs with GeV energies can influence large-scale dynamics by regulating star formation, driving galactic winds, or altering the pressure balance of galactic haloes. Recent efforts have moved the focus of the community from a one-moment description of CR transport towards a two-moment model as this allows for a more accurate description of the microphysics impacting the CR population. Like all hydrodynamical theories, these two-moment methods require a closure relation for a consistent and closed set of evolution equations. The goal of this paper is to quantify the impact of different closure relations on the resulting solutions. To this end, we review the common P1 and M1 closure relations, derive a new four-moment H1 description for CR transport, and describe how to incorporate CR scattering by Alfvén waves into these three hydrodynamical models. While there are significant differences in the transport properties of radiation in the P1 and M1 approximations in comparison to more accurate radiative transfer simulations using the discrete ordinates approximation, we only find small differences between the three hydrodynamical CR transport models in the free-streaming limit when we neglect CR scattering. Most importantly, for realistic applications in the interstellar, circumgalactic, or intracluster medium where CR scattering is frequent, these differences vanish and all presented hydrodynamical models produce the same results.

A DG-IMEX Method for Two-moment Neutrino Transport: Nonlinear Solvers for Neutrino–Matter Coupling*

Abstract Neutrino–matter interactions play an important role in core-collapse supernova (CCSN) explosions, as they contribute to both lepton number and/or four-momentum exchange between neutrinos and matter and thus act as the agent for neutrino-driven explosions. Due to the multiscale nature of neutrino transport in CCSN simulations, an implicit treatment of neutrino–matter interactions is desired, which requires solutions of coupled nonlinear systems in each step of the time integration scheme. In this paper, we design and compare nonlinear iterative solvers for implicit systems with energy-coupling neutrino–matter interactions commonly used in CCSN simulations. Specifically, we consider electron neutrinos and antineutrinos, which interact with static matter configurations through the Bruenn 85 opacity set. The implicit systems arise from the discretization of a nonrelativistic two-moment model for neutrino transport, which employs the discontinuous Galerkin (DG) method for phase-space discretization and an implicit–explicit (IMEX) time integration scheme. In the context of this DG-IMEX scheme, we propose two approaches to formulate the nonlinear systems: a coupled approach and a nested approach. For each approach, the resulting systems are solved with Anderson-accelerated fixed-point iteration and Newton’s method. The performance of these four iterative solvers has been compared on relaxation problems with various degrees of collisionality, as well as proto–neutron star deleptonization problems with several matter profiles adopted from spherically symmetric CCSN simulations. Numerical results suggest that the nested Anderson-accelerated fixed-point solver is more efficient than other tested solvers for solving implicit nonlinear systems with energy-coupling neutrino–matter interactions.

A high-performance and portable asymptotic preserving radiation hydrodynamics code with the M1 model

Aims. We present a new radiation hydrodynamics code called ARK-RT which uses a two-moment model with the M1 closure relation for radiative transfer. This code was designed to be ready for high-performance computing on exascale architectures. Methods. The two-moment model is solved using a finite-volume scheme. The scheme is designed to be asymptotic preserving in order to accurately capture both optically thick and thin regimes. We also propose a well-balanced discretization of the radiative flux source term which allows users to capture constant flux steady states with discontinuities in opacity. We use the library Trilinos for linear algebra and the package Kokkos allows us to reach high-performance computing and portability across different architectures, such as multi-core, many-core, and GP-GPU. Results. ARK-RT is able to reproduce standard tests in both free-streaming and diffusive limits, including purely radiative tests and radiation hydrodynamics ones. Using a time-implicit solver is profitable as soon as the time-step given by the hydrodynamics is between 50 and 100 times larger than the explicit time-step for radiative transfer, depending on the preconditioner and the architecture. Nevertheless, more work is needed to ensure stability in all circumstances. Using ARK-RT, we study the propagation of an ionization front in convective dense cores. We show that the ionization front is strongly stable against perturbations even with destabilizing convective motions. As a result, the presence of instabilities should be interpreted with caution. Overall, ARK-RT is well-suited to studying many astrophysical problems involving convection and radiative transfer such as the dynamics of H II regions in massive pre-stellar dense cores and future applications could include planetary atmospheres.

Thornado-transport: Anderson- and GPU-accelerated nonlinear solvers for neutrino-matter coupling1 1This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published

Algorithms for neutrino-matter coupling in core-collapse supernovae (CCSNe) are investigated in the context of a spectral two-moment model, which is discretized in space with the discontinuous Galerkin method, integrated in time with implicit-explicit (IMEX) methods, and implemented in the toolkit for high-order neutrino-radiation hydrodynamics (thornado). The model considers electron neutrinos and antineutrinos and tabulated opacities from Bruenn (1985), which includes neutrino-electron scattering and pair processes. The nonlinear system arising from implicit time discretization of the equations governing neutrino-matter coupling is iterated to convergence using Anderson-accelerated fixed-point methods, which avoid formation of Jacobians and inversion of dense linear systems. Numerical experiments show that, for a given tolerance, a nested iteration scheme which aims to reduce opacity evaluations can lower the computational cost. Our initial port to GPUs, using both OpenMP and OpenACC, shows an overall speedup of up to ∼ 100× when compared to results using a single CPU core. These results indicate that the algorithms implemented in thornado are well-suited to GPU acceleration.

Conditional relation between return and co-moments – an empirical study for emerging Indian stock market

Due to many theoretical and practical shortcomings of the traditional CAPM model, this study aims at analyzing the CAPM with possible extensions. The analysis aims to know the empirical soundness of Conditional Higher Moment CAPM in emerging India’s capital market. The sample consists of 69 company’s daily stock price data from April 2004 to March 2019 from NSE 100. Panel data analysis is used on 21 cross-sections. The overall results show that when both up and down markets are incorporated separately, all three moments, namely, co-variance, co-skewness, and co-kurtosis, are priced during the normal Indian economy phase. Further, this study states that including higher moments (co-skewness and co-kurtosis) in the two-moment model provides symmetry in both the up and down markets. This is one of the first studies in the Indian Stock market explaining the variation in portfolio returns through panel data analysis by extending CAPM with conditional higher-order co-moments. The portfolio managers should consider skewness and kurtosis along with variance in constructing the optimal portfolios.

Thornado-hydro: towards discontinuous Galerkin methods for supernova hydrodynamics11This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of

The toolkit for high-order neutrino-radiation hydrodynamics (thornado) is being developed for simulations of core-collapse supernovae (CCSNe) and related problems. Current capabilities in thornado include solvers for the Euler equations — in non-relativistic and special relativistic limits — and the two-moment model of neutrino transport. The spatial discretization in thornado is based on the discontinuous Galerkin (DG) method, which is receiving increased attention from the computational astrophysics community. In this paper, we provide an overview of the numerical methods for the Euler equations in thornado, and present some encouraging preliminary numerical results from a set of basic tests in one and two spatial dimensions.

Thornado-transport: IMEX schemes for two-moment neutrino transport respecting Fermi-Dirac statistics

We develop implicit-explicit (IMEX) schemes for neutrino transport in a background material in the context of a two-moment model that evolves the angular moments of a neutrino phase-space distribution function. Considering the upper and lower bounds that are introduced by Pauli’s exclusion principle on the moments, an algebraic moment closure based on Fermi-Dirac statistics and a convex-invariant time integrator both are demanded. A finite-volume/first-order discontinuous Galerkin(DG) method is used to illustrate how an algebraic moment closure based on Fermi-Dirac statistics is needed to satisfy the bounds. Several algebraic closures are compared with these bounds in mind, and the Cernohorsky and Bludman closure, which satisfies the bounds, is chosen for our IMEX schemes. For the convex-invariant time integrator, two IMEX schemes named PD-ARS have been proposed. PD-ARS denotes a convex-invariant IMEX Runge-Kutta scheme that is high-order accurate in the streaming limit, and works well in the diffusion limit. Our two PD-ARS schemes use second-and third-order, explicit, strong-stability-preserving Runge-Kutta methods as their explicit part, respectively, and therefore are second-and third-order accurate in the streaming limit, respectively. The accuracy and convex-invariance of our PD-ARS schemes are demonstrated in the numerical tests with a third-order DG method for spatial discretization and a simple Lax-Friedrichs flux. The method has been implemented in our high-order neutrino-radiation hydrodynamics (thornado) toolkit. We show preliminary results employing tabulated neutrino opacities.

Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Building on the framework of Zhang & Shu [1,2], we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function f. The two-moment model is closed using algebraic moment closures; e.g., as proposed by Cernohorsky & Bludman [3] and Banach & Larecki [4]. Variations of this model have recently been used to simulate neutrino transport in nuclear astrophysics applications, including core-collapse supernovae and compact binary mergers. We employ the discontinuous Galerkin (DG) method for spatial discretization (in part to capture the asymptotic diffusion limit of the model) combined with implicit-explicit (IMEX) time integration to stably bypass short timescales induced by frequent interactions between particles and the background. Appropriate care is taken to ensure the method preserves strict algebraic bounds on the evolved moments (particle density and flux) as dictated by Pauli's exclusion principle, which demands a bounded distribution function (i.e., f∈[0,1]). This realizability-preserving scheme combines a suitable CFL condition, a realizability-enforcing limiter, a closure procedure based on Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a convex combination of forward Euler steps combined with a backward Euler step. The IMEX scheme is formally only first-order accurate, but works well in the diffusion limit, and — without interactions with the background — reduces to the optimal second-order strong stability-preserving explicit Runge-Kutta scheme of Shu & Osher [5]. Numerical results demonstrate the realizability-preserving properties of the scheme. We also demonstrate that the use of algebraic moment closures not based on Fermi-Dirac statistics can lead to unphysical moments in the context of fermion transport.

Statistical validity and derivation of balance equations for the two-level model of a production line

A method of constructing a system of multi-moment balance equations, based on the statistical description of a production system is proposed. The need for research is determined by current trends in the development of production systems. The system of equations, which simulates the behavior of the production line for the transient conditions is obtained. It is shown that the resulting balance equations are not closed. The methods of closure of the self-linking chain of balance equations: the small-parameter method and the method of setting the equations of states for higher-order moments are considered. The known models using various methods of closure of the system of equations are analyzed. The model of the production line for the assembly-line production method is considered. The limitations and constraint equations, which enable the transition to single-moment PDE model of description of the assembly line and two-moment PDE model of the production line using the Burgers' equation are shown. The model of the production line for the company with the flow production method is considered. One-, two- and three-moment systems of equations for the two-level model of the production line are obtained. A general system of balance equations for the flow parameters of the production line is constructed.

Multi-fluid modelling of pulsed discharges for flow control applications

Experimental evidence suggests that short-pulse dielectric barrier discharge actuators are effective for speeds corresponding to take-off and approach of large aircraft, and thus are a fruitful direction for flow control technology development. Large-eddy simulations have reproduced some of the main fluid dynamic effects. The plasma models used in such simulations are semi-empirical, however, and need to be tuned for each flowfield under consideration. In this paper, the discharge physics is examined in more detail with multi-fluid modelling, comparing a five-moment model (continuity, momentum, and energy equations) to a two-moment model (continuity and energy equations). A steady-state, one-dimensional discharge was considered first, and the five-moment model was found to predict significantly lower ionisation rates and number densities than the two-moment model. A two-dimensional, transient discharge problem with an elliptical cathode was studied next. Relative to the two-moment model, the five-moment model predicted a slower response to the activation of the cathode, and lower electron velocities and temperatures as the simulation approached steady-state. The primary reason for the differences in the predictions of the two models can be attributed to the effects of particle inertia, particularly electron inertia in the cathode layer. The computational cost of the five-moment model is only about twice that of the simpler variant, suggesting that it may be feasible to use the more sophisticated model in practical calculations for flow control actuator design.

Effect of the effective anisotropy field of soft layer on magnetic properties of soft/hard stacked media

Magnetic properties of soft/hard stacked perpendicular magnetic recording media were investigated using a two-moment model with taking account of demagnetizing energy. Demagnetizing energy of the soft layer may improve the uniformity of magnetic properties of the composite media and also decrease the interlayer thickness, although weaken the thermal stability a little. The application of high saturation magnetization ( M s) soft magnetic material as the soft layer may compensate for the undesirable contribution while maintaining its benefits.

Comparison of collisionless macroscopic models and application to the ion–electron instability

In a first part, different macroscopic models of linear Landau damping are compared using a concise one-dimensional (1-D) collisionless formulation. The three-moment model of Chang and Callen (CC) [Phys. Fluids B 4, 1167 (1992)] with two closure relations (complex in the Fourier space) for the viscous stress and the heat conduction is found to be equivalent to the two-moment model of Stubbe–Sukhorukov (SS) [Phys. Plasmas 6, 2976 (1999)], which uses a single (complex) closure relation for the pressure. The comparison of the respective closure relations favors clearly the SS pressure law, which associates an anomalous resistivity to the Landau damping. In a second part, a macroscopic interpretation, with the SS model, of the ion–electron instability shows its resistive character for low and intermediate drift velocities, and the transition to the reactive Buneman limit. The pressure law for the electrons is found to verify a simple law, whereas approximate laws are discussed for the ion pressure. These laws are used to close a macroscopic model for stability analyses of nonhomogeneous plasma structures, where SS and CC models are not applicable easily.

Modeling of vacancy cluster formation in ion implanted silicon

Ion implantation of silicon introduces excess point defects that quickly recombine during annealing leaving net interstitial and vacancy populations. For higher energy implants, the separation between interstitials and vacancies is larger, leading to a vacancy rich region towards the surface and an interstitial rich region deeper in the bulk. The high supersaturation of vacancies in the near surface region can lead to their aggregation into vacancy clusters or voids. In this work we have developed a continuum model for vacancy clusters using discrete cluster sizes. Results from atomistic calculations [Bongiorno et al., Europhys. Lett. 43, 695 (1998)] are used for the energetics of the cluster growth/dissolution. The model is compared to data from Venezia et al. [Appl. Phys. Lett. 73, 2980 (1998)] for Au indiffusion subsequent to Si high energy implants. We found good agreement with experimental data using this model without any tuning of the parameters. However, this model is too complex and computationally expensive to be effectively incorporated into continuum process simulation tools. Hence we reduced this system of discrete rate equations into a two-moment model by carefully considering the behavior of the full model under a range of conditions. The parameters of the moment-based model follows from the full model, which in turn is based on atomistic calculations. The resulting simple and computationally efficient model is found to accurately reproduce the Au labeling experiments.

Simulations of sulfate aerosol dynamics part II: Model intercomparison

Comparisons between a two-moment, bimodal aerosol model and single-moment sectional models of varying resolution are presented. Predictions from the two-moment model are in good agreement with those from the highest resolution sectional model, whereas lower-resolution single-moment sectional models with a comparable number of variables perform more poorly. The sensitivity of model initializations and predictions to the choice of sectional boundaries is demonstrated, particularly the inability of low-resolution single-moment models to completely capture important features of the distribution. The mean sizes represented in the two-moment model vary in time, responding to changes in the distribution, and thus, this formulation offers greater flexibility in applications to large-scale models.

Two-moment models for the performance evaluation of B-ISDN networks

We consider the performance evaluation of circuit-switched B-ISDN networks with two traffic types of different bandwidth and holding times. Recent results have indicated that the Poisson assumption produces an optimistic evaluation of the network performance. Also, two-moment link models have been developed for use in network analysis. We investigate here the effect of two-moment methods on network performance evaluation methods based on the link decomposition technique. We extend the equivalent method of Butto to the multiservice case. We finally present some numerical studies on small networks, showing that the inclusion of two moments produces conservative evaluations, and quantify the change in performance measure brought about by the use of two-moment methods.

Turbulent diffusional flames in the presence of an external pressure gradient

The pressure gradient arising because of the heat liberation in the combustion zone and gravitational effects lead to the appearance of additional fluxes of scalar quantities in turbulent flames. In this work, on the basis of a two-moment closure model, a calculation scheme is constructed for turbulent diffusional flames in the presence of a fixed mean-pressure gradient and numerical modeling of the scalar quantities associated with the heat liberation and large density fluctuations is undertaken.

Asset Pricing in the Australian Industrial Equity Market

The two-moment, mean-variance model of asset pricing is tested against data from the Melbourne stock exchange. The model appears to describe the data quite well, though there are problems in experimental design which are yet to be cleared up. Neither variance nor skewness appears to explain additional price behaviour to that explained by covariance, as is predicted by the two-moment model.

An Empirical Comparison of Stochastic Dominance and Mean-Variance Portfolio Choice Criteria

An important issue in the financial literature concerns the conflict between the stochastic dominance (SD) and the mean-variance (EV) methods of choosing optimal portfolios of risky assets. Much of the recent theoretical and empirical work in portfolio analysis has been devoted to the extension and testing of the Markowitz two-moment model, in which it is assumed that either (a) decision makers have quadratic utility functions with negative second derivatives or (b) the probability functions are from some appropriate two-parameter family and the investor is risk averse.