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The convergence of Boltzmann Fokker Planck solution can become arbitrarily slow with iterative procedures like source iteration. This paper derives and investigates a nonlinear diffusion acceleration scheme for the solution of the Boltzmann Fokker Planck equation in slab geometry. This method is a conventional high order low order scheme with a traditional diffusion-plus-drift low-order system. The method, however, differs from the earlier variants as the definition of the low order equation, which is adjusted according to the zeroth and first moments of the Boltzmann Fokker Planck equation. For the problems considered, we observe that the NDA-accelerated solution follows the unaccelerated well and provides roughly an order of magnitude savings in iteration count and runtime compared to source iteration., Comment: This draft has been submitted to Physor 2024 for review

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., Comment: 31 pages

We develop a multiphysics-based model to predict the response of localized tumors to combined-hyperthermia-radiotherapy (CHR) treatment. This procedure combines hyperthermia (tumor heating) with standard radiotherapy to improve efficacy of the overall treatment. In addition to directly killing tumor cells, tumor heating amends several parameters within the tumor microenvironment. This leads to radiosensitization, which improves the performance of radiotherapy while reducing the side-effects of excess radiation in the surrounding normal tissue. Existing tools to model this kind of treatment consider each of the physics separately. The model presented in this paper accounts for the synergy between hyperthermia and radiotherapy providing a more realistic and holistic approach to simulate CHR treatment. Our model couples radiation transport and heat-transfer with cell population dynamics., Comment: 10 pages, 3 figures, submitted to ANS Topical Meeting

Particle transport through a duct by Lambertian reflection from duct walls is again considered. This popular transport example has been solved by most numerical transport methods except notably one- the method of doubling. We shall show that the method of doubling provides every bit as, or more, accurate reflectances and transmittances as the numerical discrete ordinates (NDO) and analytical discrete ordinates (ADO) methods with less mathematical and numerical effort., Comment: 39 pages

Nearly 45 years ago, Ken Case published his seminal paper on the singular eigenfunction solution for the Green's function of the monoenergetic neutron transport equation with isotropic scattering. Previously, the solution had been obtained by Fourier transform. While it is apparent the two had to be equivalent, a convincing equivalence proof for general anisotropic scattering remained a challenge until now., Comment: 34 pages,2 figs, to be submitted to either Nucl. sci & Eng or J. Math Phys

Proteinaceous aggregation occurs through self-assembly-- a process not entirely understood. In a recent article [1], an analytical theory for amyloid fibril growth via secondary rather than primary nucleation was presented. Remarkably, with only a single kinetic parameter, the authors were able to unify growth characteristics for a variety of experimental data. In essence, they seem to have uncovered the underlying allometric laws governing the evolution of filament elongation simply from two coupled non-linear ordinary differential equations (ODEs) stemming from a master equation. While this work adds significantly to our understanding of filament self-assembly, it required an approximate analytical solution representation. Here, we show that the same results are found by purely numerical means once a straightforward and reliable numerical solution to the set of ODEs has been established., Comment: 28 pages,8 figures

To explicate the sources of uncertainty in the prediction of biophysical variables over space, consider the general equation: where z is a variable with values on some nominal, ordinal, interval or ratio scale; y is a vector of input variables; u is the spatial support of y and z ; x and u are the spatial locations of y and z , respectively; f is a model and B is the vector of the parameters of this model. Any y or z has a value and a spatial extent which is called its support. Viewed in this way, categories of uncertainty are from variable (e.g. measurement), parameter, positional. support and model (e.g. structural) sources. The prediction of Leaf Area Index (LAI) and the fraction of absorbed photosynthetically active radiation (fPAR) are examples of z variables predicted using model(s) as a function of y variables and spatially constant parameters. The MOD15 algorithm is an example of f, called f(sub 1), with parameters including those defined by one of six biome types and solar and view angles. The Leaf Canopy Model (LCM)2, a nested model that combines leaf radiative transfer with a full canopy reflectance model through the phase function, is a simpler though similar radiative transfer approach to f(sub 1). In a previous study, MOD15 and LCM2 gave similar results for the broadleaf forest biome. Differences between these two models can be used to consider the structural uncertainty in prediction results. In an effort to quantify each of the five sources of uncertainty and rank their relative importance for the LAI/fPAR prediction problem, we used recent data for an EOS Core Validation Site in the broadleaf biome with coincident surface reflectance, vegetation index, fPAR and LAI products from the Moderate Resolution Imaging Spectrometer (MODIS). Uncertainty due to support on the input reflectance variable was characterized using Landsat ETM+ data. Input uncertainties were propagated through the LCM2 model and compared with published uncertainties from the MOD15 algorithm.

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 formalism for photon transport in leaf canopies with finite-dimensional scattering centers that mutually cross-shade was recently proposed resulting in an equation set which is rather complicated. In this paper, we propose a simplification of the correction factors that describe the correlation in photon fates. These simplified correction factors permit the derivation of a modified integro-differential transfer equation with two additional terms that arise from considerations of finite size scatterers. A numerical procedure for the solution of this transfer equation in the framework of a successive collisions approach is also presented.

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.

The development of an interaction data base and a numerical solution to the transport of baryons through an arbitrary shield material based on a straight ahead approximation of the Boltzmann equation are described. The code is most accurate for continuous energy boundary values, but gives reasonable results for discrete spectra at the boundary using even a relatively coarse energy grid (30 points) and large spatial increments (1 cm in H2O). The resulting computer code is self-contained, efficient and ready to use. The code requires only a very small fraction of the computer resources required for Monte Carlo codes.

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.

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.