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This paper introduces and evaluates the Adding and Doubling Method (ADM) for solving the Bateman equations for depletion systems with varying numbers of nuclides and compares it to the Chebyshev Rational Approximation Method (CRAM), both implemented in the reactor physics analysis application Griffin. ADM, when applied to the Crank-Nicolson Finite Difference method, can produce results comparable in accuracy and precision to CRAM with comparable run times for systems with 35 or 297 nuclides. For systems with more than 300 nuclides, the matrix-matrix operations required by ADM are significantly more costly than the matrix-vector operations required by CRAM, making CRAM the more efficient method for systems with large numbers of nuclides. ADM is an accurate method that maintains other advantages over CRAM in that it does not depend on pre-generated coefficients or require complex number operations. ADM also manages to outperform CRAM by a factor of more than 250 in terms of run time for depletion systems that require multiple Bateman solves while the depletion matrix and time step size remain constant over all depletion intervals. [ABSTRACT FROM AUTHOR]

In recent years, the first author has developed three successful numerical methods to solve the 1D radiative transport equation yielding highly precise benchmarks. The second author has shown a keen interest in novel solution methodologies and an ability for their implementation. Here, we combine talents to generate yet another high precision solution, the Matrix Riccati Equation Method (MREM). MREM features the solution to two of the four matrix Riccati ODEs that arise from the interaction principle of particle transport. Through interaction coefficients, the interaction principle describes how particles reflect from- and transmit through- a single slab. On combination with Taylor series and doubling, a high-quality numerical benchmark, to nearly seven places, is established. [ABSTRACT FROM AUTHOR]

• Using adding and doubling. • High precision with simple methods. • Convergence acceleration. • Pade approximants. In the past, the method of adding and doubling has received considerable attention as an effective approach to solving the linear transport equation. Here, we consider doubling for the problem of thermal creep in Rarefied Gas Dynamics. The method, called the Discrete/Discrete Ordinates (D/DO) method is in contrast to the well-known Analytical/Discrete Ordinates (A/DO) method. The D/DO method solves the fully discretized transport equation to demonstrate that straightforward numerical approximation also yields high quality numerical solutions. The method combines the fundamental forward Euler evaluation of the matrix exponential with Richardson extrapolation to generate a high quality benchmark. We then generalize the method using Padé approximants for the matrix exponential and present a 7-place benchmark. [ABSTRACT FROM AUTHOR]

In the beautifully situated tenth-century castle, about 75 participants from all over the world presented and attended seven sessions during five days concerning such topics as radionuclide production, equations of state of a Fermi gas, the advection-diffusion equation, the Fokker-Planck equation, neutron multiplicity and branching, transport in a moving mesh, pressurized water reactor control rod motion, waves in gas bubbles, the Vlasov-Poisson-Landau kinetic equation, plasma transport, Monte Carlo, stochastic COVID-19 modeling, the BGK equation, charged particle transport, and more specialized topics in particle transport theory. Every odd year from 1969 through 2019 - and then in 2022 on account of the COVID-19 pandemic - a select group of mathematicians, physicists, engineers, and computational scientists calling themselves transport theorists gathered for one week at what was originally called the Blacksburg Conference (now known as the International Conference on Transport Theory [ICTT]). Originated by Paul F. Zweifel (deceased), Bob Erdman, and Chuck Siewert, the conference was intended to bring together theorists the world over to discuss common methods to solve the neutron transport equation with application to nuclear energy. [Extracted from the article]

The Sutherland approximation to the van der Waals forces is applied to the derivation of a self-consistent Vlasov-type field in a liquid filling a half space, bordering vacuum. The ensuing Vlasov equation is then derived, and solved to predict the behavior of the density at and in the vicinity of the liquid-vacuum interface. A numerical solution to the Vlasov equation is also produced and the density profile shown and discussed. [ABSTRACT FROM AUTHOR]

The discrete ordinates method (DOM) of solution to the 1D radiative transfer equation has been an effective method of solution for nearly 70 years. During that time, the method has experienced numerous improvements as numerical and computational techniques have become more powerful and efficient. Here, we again consider the analytical solution to the discrete radiative transfer equation in a homogeneous medium by proposing a new, and consistent, form of solution that improves upon previous forms. Aided by a Wynn-epsilon convergence acceleration, its numerical evaluation can achieve extreme precision as demonstrated by comparison with published benchmarks. Finally, we readily extend the solution to a heterogeneous medium through the star product formulation producing a novel benchmark for closed form Henyey–Greenstein scattering as an example. [ABSTRACT FROM AUTHOR]

The discrete ordinates method (DOM) of solution to the 1D radiative transfer equation has been an effective method of solution for nearly 70 years. During that time, the method has experienced numerous improvements as numerical and computational techniques have become more powerful and efficient. Here, we again consider the analytical solution to the discrete radiative transfer equation in a homogeneous medium by proposing a new, and consistent, form of solution that improves upon previous forms. Aided by a Wynn-epsilon convergence acceleration, its numerical evaluation can achieve extreme precision as demonstrated by comparison with published benchmarks. Finally, we readily extend the solution to a heterogeneous medium through the star product formulation producing a novel benchmark for closed form Henyey–Greenstein scattering as an example. [ABSTRACT FROM AUTHOR]

Abstract: Multidimensional semi-analytical particle transport benchmarks to provide highly accurate standards of assessment are few and far between. Because of a well-established 1D theory for the analytical solution of the transport equation, it is sometimes possible however, to “bootstrap” 1D solutions to give more comprehensive solution representations. Here, we propose the internal searchlight problem in a half space, designated ISLP/HS, as a multidimensional benchmark to be constructed from 1D solutions. This is a variation of the usual SLP/HS where a source emits within the half space rather than striking its surface. Our primary interest is in the exiting intensity at the free surface established through a new Fn formulation. The benchmark features true 2/3D particle transport through integration of a point kernel to simulate 2/3D source emission. In this way, we accommodate a solid or hollow cylindrical source and a general line source in addition to the standard point, ring and disk sources featured in previous investigations. [Copyright &y& Elsevier]

We present a new accurate approach to solving a class of problems in the theory of rarefied–gas dynamics using a Physics-Informed Neural Networks framework, where the solution of the problem is approximated by the constrained expressions introduced by the Theory of Functional Connections. The constrained expressions are made by a sum of a free function and a functional that always analytically satisfies the equation constraints. The free function used in this work is a Chebyshev neural network trained via the extreme learning machine algorithm. The method is designed to accurately and efficiently solve the linear one-point boundary value problem that arises from the Bhatnagar–Gross–Krook model of the Poiseuille flow between two parallel plates for a wide range of Knudsen numbers. The accuracy of our results is validated via the comparison with the published benchmarks. [ABSTRACT FROM AUTHOR]

This work aims at accurately solve a thermal creep flow in a plane channel problem, as a class of rarefied-gas dynamics problems, using Physics-Informed Neural Networks (PINNs). We develop a particular PINN framework where the solution of the problem is represented by the Constrained Expressions (CE) prescribed by the recently introduced Theory of Functional Connections (TFC). CEs are represented by a sum of a free-function and a functional (e.g., function of functions) that analytically satisfies the problem constraints regardless to the choice of the free-function. The latter is represented by a shallow Neural Network (NN). Here, the resulting PINN-TFC approach is employed to solve the Boltzmann equation in the Bhatnagar–Gross–Krook approximation modeling the Thermal Creep Flow in a plane channel. We test three different types of shallow NNs, i.e., standard shallow NN, Chebyshev NN (ChNN), and Legendre NN (LeNN). For all the three cases the unknown solutions are computed via the extreme learning machine algorithm. We show that with all these networks we can achieve accurate solutions with a fast training time. In particular, with ChNN and LeNN we are able to match all the available benchmarks. [ABSTRACT FROM AUTHOR]

The slowing down equation for elastic scattering of neutrons in an infinite homogeneous medium is solved analytically by decomposing the neutron energy spectrum into collision intervals. Since scattering physically smooths energy distributions by redistributing neutron energy uniformly, it is informative to observe how mathematics accommodates the scattering process, which increases entropy through disorder. [ABSTRACT FROM AUTHOR]

Our paper presents a comparison of the extrapolated-based acceleration of source iteration (SI) techniques against classical acceleration schemes. The basis of the SI method is reviewed before presenting its combination with non-linear extrapolations. Two acceleration strategies are discussed, one based on the extrapolation of the source term for the SIs and the other based on the acceleration of the solution itself. Their performances are compared with classical acceleration techniques (diffusion synthetic acceleration and transport synthetic acceleration, showing the effectiveness of the combination of extrapolation with standard acceleration schemes. [ABSTRACT FROM AUTHOR]

Propagation of sound waves through an ideal gas is studied in the framework of kinetic theory, starting from the integral representation of the Boltzmann Equation (BE). The distribution function is calculated and used to obtain the dispersion relation of density waves. The parallel existence of temperature waves (in some contexts referred to as second sound) is also part of this investigation. Temperature waves are found to exist; the phase difference between density and temperature waves is determined. [ABSTRACT FROM AUTHOR]

Starting from the modified Lennard–Jones potential, a method is proposed to calculate the internal energy in liquids, and from this the specific heat. The method is applied to liquid helium-4 and the well-known shape of the “lambda point” is reproduced. [ABSTRACT FROM PUBLISHER]

[Display omitted] • PINNs and TFC to obtain highly accurate solutions for the Point Kinetics Equations. • The boundary-free solution is expanded in a Shallow Neural Network. • The proposed method is tested against a number of benchmarks from literature. • Accuracy of up to ten digits is achieved even for the most challenging problems. The paper presents a novel approach based on Physics-Informed Neural Networks (PINNs) for the solution of Point Kinetics Equations (PKEs) with temperature feedback. The approach is based on a new framework developed by the authors, which combines PINNs with Theory of Functional Connections and Extreme Learning Machines in the so called Extreme Theory of Functional Connections (X-TFC). The accuracy of X-TFC is tested against a number of published benchmarks (including for non-linear PKEs), showing its performance both in terms of accuracy and computational time. One of the main advantages of the proposed framework is in its flexibility to adapt to a variety of problems with minimal changes in coding and, after the training of the network, in its ability to offer an analytical representation (by Neural Networks) of the solution at any desired time instant outside the initial discretization. [ABSTRACT FROM AUTHOR]

Highlights: [•] The paper presents a novel semi-analytical technique to solve reactor kinetics equations. [•] The solution of kinetics equations is sought starting from a piecewise constant approximation. [•] The performance of the technique is improved via a corrective source term and iterative cycles. [•] The high accuracy of the method is tested against literature for both linear and non-linear problems. [Copyright &y& Elsevier]

This article presents a review of the special features exhibited by the transport of photons in dense vegetation canopies. The conservation of photons for particles moving in canopy media yields a linear Boltzmann Equation similar to conventional participating media. However, because of the turbid medium approximation where the atomic leaves are modeled as a gas cloud of infinitesimal plates with specified orientation, the overall system is not rotationally invariant. The latter affects the mathematical nature of the canopy equation and requires special numerical treatment to generate accurate numerical solutions. Here, the most common solution techniques are reviewed with special emphasis on two highly accurate methodologies, namely the Singular Eigenfunction Expansion, which is the basis of the F N method for canopy transport and the Analytical Discrete Ordinate (ADO) method. It is shown that the conventional ADO method can be easily extended to canopy transport problems by taking advantage of special symmetries exhibited by the intercept function and the area scattering function employed to describe the canopy optical properties. [ABSTRACT FROM PUBLISHER]

The article presents a novel kinetic method for the solution of time-dependent source-driven linear transport problems. As will be discussed, the method is based on a multigeneration representation of linear Boltzmann equations and is derived from physically grounded considerations. The technique is said to be hybrid since it combines together collisionless transport solutions and factorization techniques. The methodology is tested on systems with different scattering properties and shows very promising performance both in terms of accuracy and computational times. [ABSTRACT FROM PUBLISHER]

Abstract: This paper describes a novel hybrid technique for the solution of the time-dependent neutron transport equation in source switch-on transients. The Hybrid Transport Point Kinetics (HTPK) method is derived from a multi-collision description of the transport equation and is based on the observation that the external source mainly influences the first few neutron generations and, through collisions and multiplications, an ensemble behavior statistically emerges at higher order. The HTPK is designed to separately describe the propagation phenomenon from the ensemble response of the system, using both the rigorous transport equation and simplified kinetic models. Numerical simulations performed on a typical acceleration driven system configuration confirm the basic physical intuition of HTPK and show a significant gain in accuracy when compared to classical point kinetics. The proposed HTPK technique is also shown to be attractive because of its flexibility and competitive computational effort. [Copyright &y& Elsevier]

Computing in the energy dimension is one of the greatest challenges confronting present-day deterministic neutron transport solvers. Accurately resolving the neutron flux as neutrons downscatter across resonances in the nuclear cross sections currently requires considerable computing power and suffers from approximation errors. Flux uncertainty resulting from the uncertainty of the resonance structure is the single-largest cause of reactivity uncertainty. Any additional reference solution for the critical neutron downscattering problem with resonance phenomena would be a boon to verification and validation of neutronics codes. This paper establishes a benchmark to verify the accuracy of neutron transport criticality solvers along the energy dimension. For the first time, the analytic solution of the flux amplitude is derived in the particular case of an infinite homogeneous medium with isotropic scattering in the center of mass and an arbitrary number of no-threshold, neutral particle reaction resonances (e.g., radiative capture, fission, and resonance scattering). Original analytic expressions are established to quantify the discrepancy between the ψ k (E) and ψ α (E) flux amplitudes, respective solutions of the multiplication factor k , or the exponential time-evolution frequency α eigenproblems. The physical study of these relations led to analysis of their first-order relative difference near the criticality condition α = 0. Finally, numerical solutions are provided to a benchmark problem constituted of the first resonance of 239Pu, the 6.67-eV resonance of 238U, and a scattering isotope with a flat cross section, allowing for the computational verification of the energy resolution of current neutron transport criticality codes. Through these novel results, this analytic benchmark can serve as a reference to verify the energy resolution and sensitivity analysis of neutron transport criticality calculations. [ABSTRACT FROM AUTHOR]

• CRAM's relative error at a fixed end time approaches 1/∞ for even/odd orders as Δ t → 0. • CRAM is inconsistent; its local truncation error does not reduce to zero as time step tends to zero. • Guidelines for using CRAM are provided to avoid deterioration of accuracy. The Chebyshev rational approximation method (CRAM) has become a widely adopted method for solving nuclear depletion problems. Therefore, understanding CRAM's accuracy is important for the safe operation of nuclear power plants. This article performs a global error analysis of CRAM and finds that, as the length of the time step approaches zero, the relative error measured between the exact and CRAM solutions at a fixed end time approaches one and infinity for even and odd orders, respectively; for intermediate time step sizes, a minimum in relative error is observed. We show that the reason for CRAM's behavior is that the method is inconsistent. Two best practices for using CRAM, derived from these results, are: (1) use CRAM order 16 or higher, (2) if necessary, increase the CRAM order when multiphysics coupling requires smaller time steps. [ABSTRACT FROM AUTHOR]

In the classical case, the hydrostatic pressure of an ideal gas is defined as being two-thirds of the specific kinetic energy of the gas: , where is the average kinetic energy of the particles. However, this is no longer the case when relativistic cases are considered. Further concerns may be raised by quantum considerations. In the present paper, both pressure and kinetic energy are calculated taking the appropriate averages weighted with the equilibrium distribution function of the gas. For the purpose of this work the quantum relativistic Fermi-Dirac and Bose-Einstein distribution functions, will be considered for the cases of bosons and weakly degenerate fermions. Integration yields results in a closed form containing modified Bessel functions. A large-argument approximation is then taken, leading to an equation of state composed of the classic part plus correction terms. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with finding accurate solutions for photon transport problems in highly heterogeneous media fastly, efficiently and with modest memory resources. We propose an extended version of the analytical discrete ordinates method, coupled with domain decomposition-derived algorithms and non-linear convergence acceleration techniques. Numerical performances are evaluated using a challenging case study available in the literature. A study of accuracy versus computational time and memory requirements is reported for transport calculations that are relevant for remote sensing applications. [Copyright &y& Elsevier]

Process models are widely used tools, both for studying fundamental processes themselves and as elements of larger system studies. A radiative transfer model (RTM) simulates the interaction of light with a medium. We are interested in RTMs that model light reflected from a vegetated region. Such an RTM takes as input various biospheric and illumination parameters and computes the upwelling radiation at the top of the canopy. The question we address is as follows: Which of the inputs to the RTM has the greatest impact on the computed observation? We study the leaf canopy model (LCM) RTM, which was designed to study the feasibility of observing leaf chemistry remotely. Its inputs are leaf chemistry variables (chlorophyll, water, lignin, and cellulose) and canopy structural parameters (leaf area index, leaf angle distribution, soil reflectance, and sun angle). We present a statistical approach to the sensitivity analysis of RTMs to answer the question previously posed. The focus is on global sensitivity analysis, studying how the RTM output changes as the inputs vary continuously according to a probability distribution over the input space. The influence of each input variable is captured through the "main effects" and "sensitivity indices." Direct computation requires extensive computationally expensive runs of-the RTM. We develop a Gaussian process approximation to the RTM output to enable efficient computation. We illustrate how the approach can effectively determine the inputs that are vital for accurate prediction. The methods are applied to the LCM with seven inputs and output obtained at eight wavelengths associated with Moderate-resolution Imaging Spectroradiometer bands that are sensitive to vegetation. [ABSTRACT FROM AUTHOR]

• A highly accurate solution for the Chandrasekhar's basic problem in radiative transfer is proposed. • The forward and backward scattered photon fluxes are computed via theory of functional connections. • The boundary conditions are always analytically satisfied via appropriate constrained expressions. • The boundary-free solution is expanded in Chebyshev polynomials and their coefficients are computed via Least-Squares method. • Seven digits of accuracy are achieved when compared to the available benchmarks. We present a novel approach to solving Chandrasekhar's problem in radiative transfer using the recently developed Theory of Functional Connections. The method is designed to elegantly and accurately solve the Linear Boundary Value Problem from the angular discretization of the integrodifferential Boltzmann equation for Radiative Transfer. The proposed algorithm falls under the category of numerical methods for the solution of radiative transfer equations. This new method's accuracy is tested via comparison with the published benchmarks for Mie and Haze L scattering laws. [ABSTRACT FROM AUTHOR]