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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]

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]

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]

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]

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]

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]