The subject of this paper is the application of the multi-grid method to the solution of -V. (D(x, y)VU(x, y))+(x, y)U(x, y)=f (x, y) in a bounded region f of R where D is positive and D, tr, and f are allowed to be discontinuous across internal boundaries F of fL The emphasis is on discontinuities of orders of magnitude in D, when special techniques must be applied to restore the multi-grid method to good efficiency. These techniques are based on the continuity of D VU across F. Two basic methods are derived, one in which the approximating finite difference operators on coarser grids are five point operators (assuming the finite difference operator on the finest grid is a five point one) and one in which they are nine point operators.

Los 5-5Version 3.0 iii 16-19Version 3.0 ix Table 5.2: Table 6.1: Table 6.2: Table 7.1: Table 7.2: -. The DANTSYS code package includes the following transport codes: ONEDANT, TWODANT, TWODANT/GQ, TWOHEX, and THREEDANT. This document is the central user, methods and programming documentation for the system of codes. LIST OF FIGURES LIST OF FIGURESThe DANTSYS code package is a modular computer program package designed to solve the time-independent, multigroup discrete ordinates form of the Boltzmann transport equation in several different geometries. The modular construction of the package separates the input processing, the transport equation solving, and the post processing (or edit) functions into distinct code modules: the Input Module, one or more Solver Modules, and the Edit Module, respectively. The Input and Edit Modules are very general in nature and are common to all the Solver Modules. The ONEDANT Solver Module contains a one-dimensional (slab, cylinder, and sphere), time-independent transport equation solver using the standard diamond-differencing method for space/angle discretization. It was previously documented in Ref.1. Also included in the package are Solver Modules named TWODANT, TWODANT/GQ, THREEDANT, and TWOHEX. The TWODANT Solver Module solves the time-independent two-dimensional transport equation using the diamond-differencing method for space/angle discretization and was previously documented in Ref. 2. We have also introduced an adaptive weighted diamond differencing (AWDD) method for the spatial and angular discretization into TWODANT as an option. The TWOHEX Solver Module solves the time-independent two-dimensional transport equation on an equilateral triangle spatial mesh. The user's guide for TWOHEX was previously documented in Ref. The THREEDANT SolverModule solves the time independent, three-dimensional transport equation for XYZ and RZO symmetries using both diamond differencing with set-to-zero fixup and the AWDD method. The TWODANT/GQ Solver Module solves the two-dimensional transport equation in X Y and RZ symmetries using a spatial mesh of arbitrary quadrilaterals. The spatial differencing method is based upon the diamond differencing method with set-tozero fixup with changes to accommodate the generalized spatial meshing.This manual describes the standardized Input and Edit Modules together with each of the Solvers in the package. Throughout this manual we will refer to this package as the DANTSYS code package.Some of the major features included in the DANTSYS code package are: a free-field format ASCII text input capability;standardized, data-and file-management techniques as defined and developed by the Module solves the time independent, three-dimensional transport equation for XYZ and RZO symmetries using both diamond differencing with set-to-zero fixup and the AWDD method. The TWODANTIGQ Solver Module solves the two-dimensional transport equation in X Y and RZ symmetries using a spatial mesh of arbitrary quadrilaterals. The spatial differencing meth...

We analyse the limits of the diffusion approximation to the time-independent equation of radiative transfer for homogeneous and heterogeneous biological media. Analytical calculations and finite-difference simulations based on diffusion theory are compared with discrete-ordinate, finite-difference transport calculations. The influence of the ratio of absorption and transport scattering coefficient (mu(a)/mu'(s)) on the accuracy of the diffusion approximation are quantified and different definitions for the diffusion coefficient, D, are discussed. We also address effects caused by void-like heterogeneities in which absorption and scattering are very small compared with the surrounding medium. Based on results for simple homogeneous and heterogeneous systems, we analyse diffusion and transport calculation of light propagation in the human brain. For these simulations we convert density maps obtained from magnetic resonance imaging (MRI) to optical-parameter maps (mu(a) and mu'(s)) of the brain. We show that diffusion theory fails to describe accurately light propagation in highly absorbing regions, such as haematoma, and void-like spaces, such as the ventricles and the subarachnoid space.

We investigate a class of acceleration schemes that resemble the conventional synthetic method in that they utillze the difiusion operator in the transport iteration schemes. The acctilerated iteration involves alternate dif~usion and transport solutj.onswhere coupling between the equations is achieved using a correction term applied to either (1) the diffusion coefficient, (2) the removal cross-section; or (3) the source of the diffusion equation. Tl,emethods involving the modification of the diffusion coefficient and of the removal term yield nonlinear acceleration schemes and are used in k calculations, eff while the source term modification approach is linear at-least before discretization, and used fur inhomogr.neous source problems. there is n preferred diffcrencing method observed instability of the preferred difference scheme at the same time stable and conventional A careful analysis shows that which eliminates the previously synthetic method. Using this results in m acceleration method which is efficient. This preferred difference approach renders the source correction scheme, which is linear in Its continuous * Work performed under the auspices of the U. S. Energy Research and Development Administration.

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In this study we analyze the limits of the diffusion approximation to the Boltzmann transport equation for photon propagation in the human brain. Two dimensional slices through the head are obtained with the method of magnetic resonance imaging (MRI). Based on these images we assign optical properties to different regions of the brain. A finitedifference transport/diffusion code is then used to calculate the fluence throughout the head. Differences between diffusion and transport calculations occur especially in void-like spaces and regions where the absorption coefficient is comparable to the reduced scattering coefficient.

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