Benchmark solutions for the galactic ion transport equations with spatial and energy coupling

In order to anticipate future space shielding requirements, NASA has initiated an effort to formulate computational methods to simulate radiation effects in space. As part of the program, numerical transport algorithms have been developed for the deterministic Boltzman equation describing galactic cosmic ray (GCR) interactions with matter. It thus becomes necessary to assess the accuracy of proposed deterministic algorithms. For this reason, analytical benchmark solutions to mathematically tractable galactic cosmic ray equations have recently been obtained. Even though these problems involve simplifying assumptions of the associated physics, they still contain the essential features of the basic transport processes. The solutions obtained are features of the basic transport processes. The solutions obtained are compared to results from numerical algorithms in order to ensure proper coding and to provide a measure of the accuracy of the numerical methods used in the algorithm. For the first time, mathematical methods have been applied to the galactic ion transport (GIT) equations in the straight ahead approximation with constant nuclear properties. The approach utilizes a Laplace transforms inversion yielding a closed form benchmark solution which is also computationally efficient

Benchmark solutions for the galactic ion transport equations: Energy and spatially dependent problems

Nontrivial benchmark solutions are developed for the galactic ion transport (GIT) equations in the straight-ahead approximation. These equations are used to predict potential radiation hazards in the upper atmosphere and in space. Two levels of difficulty are considered: (1) energy independent, and (2) spatially independent. The analysis emphasizes analytical methods never before applied to the GIT equations. Most of the representations derived have been numerically implemented and compared to more approximate calculations. Accurate ion fluxes are obtained (3 to 5 digits) for nontrivial sources. For monoenergetic beams, both accurate doses and fluxes are found. The benchmarks presented are useful in assessing the accuracy of transport algorithms designed to accommodate more complex radiation protection problems. In addition, these solutions can provide fast and accurate assessments of relatively simple shield configurations

Time-Dependent Neutral Particle Transport Benchmarks in Two and Three Dimensions

The main objective of NEER grant was to generate highly accurate 2D and 3D time-dependent neutral particle intensity maps from 3D pulsed wire sources through integration of the analytical representation of a time-dependent point source

Benchmark solutions for the galactic heavy-ion transport equations with energy and spatial coupling

Nontrivial benchmark solutions are developed for the galactic heavy ion transport equations in the straightahead approximation with energy and spatial coupling. Analytical representations of the ion fluxes are obtained for a variety of sources with the assumption that the nuclear interaction parameters are energy independent. The method utilizes an analytical LaPlace transform inversion to yield a closed form representation that is computationally efficient. The flux profiles are then used to predict ion dose profiles, which are important for shield design studies

A Taylor series solution of the reactor point kinetics equations

The method of Taylor series expansion is used to develop a numerical solution to the reactor point kinetics equations. It is shown that taking a first order expansion of the neutron density and precursor concentrations at each time step gives results that are comparable to those obtained using other popular and more complicated methods. The algorithm developed using a Taylor series expansion is simple, completely transparent, and highly accurate. The procedure is tested using a variety of initial conditions and input data, including step reactivity, ramp reactivity, sinusoidal, and zigzag reactivity. These results are compared to those obtained using other methods.Comment: 13 pages, added 3 new figures, and 3 new reactivity conditions. Corrected data in table for sin reactivity cas

A Hierarchy of Transport Approximations for High Energy Heavy (HZE) Ions

The transport of high energy heavy (HZE) ions through bulk materials is studied neglecting energy dependence of the nuclear cross sections. A three term perturbation expansion appears to be adequate for most practical applications for which penetration depths are less than 30 g per sq cm of material. The differential energy flux is found for monoenergetic beams and for realistic ion beam spectral distributions. An approximate formalism is given to estimate higher-order terms

A 2D Benchmark for the Verification of the PEBBED Code

A new benchmarking concept is presented for verifying the PEBBED 3D multigroup finite difference/nodal diffusion code with application to pebble bed modular reactors (PBMRs). The key idea is to perform convergence acceleration, also called extrapolation to zero discretization, of a basic finite difference numerical algorithm to give extremely high accuracy. The method is first demonstrated on a 1D cylindrical shell and then on an r,8 wedge where the order of the second order finite difference scheme is confirmed to four places