Multigroup diffusion

The radiation-hydrodynamic simulation includes the Quotidien EOS [29] and Ion EOS based on the Cowan model [30], For the electron component, a set of fitting formulae derived from the numerical results from the Thomas-Fermi model and a semi-empirical bonding correction [31] are adopted. The effective Z-number of the partially ionized plasma is obtained from the average atom model. Radiation transport is treated by multigroup diffusion. 

Derivation of the Fermi age equation. The Boltzmann equation is an integro-dififerential equation involving distance, energy and directional variables. By making suitable approximations we shall now proceed to reduce this equation to a set of coupled differential equations in which the spatial coordinates are the only independent variables. These are the so-called multigroup diffusion equations. But first we discuss the so-caUed age theory. 

The interface conditions associated with the multigroup diffusion equations are generally written in the form... 

Here djdn denotes the derivative in the direction normal to the interface.) To the best of our knowledge these interface conditions have not been derived from the fundamental interface condition (2.3) and the hypotheses already made in deriving the multigroup diffusion equations (except in the very important special case of one-dimensional geometry). To illustrate the difficulties we substitute the condition (3.1) into equation (2.3) and find that... 

In order to solve a particular reactor problem it is generally necessary to adjoin to the multigroup diffusion equations (4.2) another equation which describes the thermal neutron fiux, since this is generally required for a determination of the fission sources 8j, This extra equation is taken to be of... 

Continuous models. The extension from the finite models of 2 to multigroup diffusion and transport theory models involving a continuum of possible positions and velocities, involves some very careful technical analysis. 

This extension has been achieved, in the usual multigroup diffusion approximation, by Drs. Martino and Habetler [8]. Their principal tool is the theorem of Jentzsch, which is just the analog for integral linear operators of the theorem of Perron and Frobenius for non-negative matrices. As Drs. Martino and Habetler will themselves describe this work, which closely parallels [3], I will say no more about it. 

G. J. Habetler and M. A. Martino, The multigroup diffusion equations of reactor physics. Report KAPL-1886, General Electric Co., Knolls Atomic Power Lab., 1958. 

EXISTENCE THEOREMS AND SPECTRAL THEORY FOR THE MULTIGROUP DIFFUSION MODEL... 

In the case of the one-dimensional multigroup diffusion operator, Q, it is shown that the spectral subspaces generated by the generalized eigenfunctions of Q are complete, and biorthogonal expansions in these generalized... 

D = Djh jc). Also, define two matrices of scalar functions hy A = (dj]c(r)) F = fjk T)). The differential equations of reactor kinetics for the multigroup diffusion model become (neglecting delayed neutrons)... 

G. Goertzel, M. M. Shapiro and H. S. Wilf, The numerical integration of the multigroup diffusion equation, NDA Document (no date). 

NUMERICAL METHODS FOR SOLVING MULTI-DIMENSIONAL MULTIGROUP DIFFUSION EQUATIONS... 

We shall concentrate on surveying the available numerical methods for solving the multi-dimensional multigroup diffusion equations. Since Dr. Ehrlich has already sketched the numerical methods available for treating the case of one space variable, we shall therefore concentrate on the cases of several space variables, although in general our theoretical discussions will be phrased independently of the number of space variables. In all cases we shall attempt to discuss both the rigorous mathematical features and the practical applications of these various numerical methods to both the time independent and time dependent diffusion equations. 

For the time independent multigroup diffusion equations, we have... 

MULTI-DIMENSIONAL MULTIGROUP DIFFUSION EQUATIONS 181 and p = cosh i (2/5 — 1). To start the process, and... 

Iterative methods for time dependent multigroup diffusion equations. 

The numerical solution of time dependent multigroup diffusion equations has not yet received as much attention as the number of numerical techniques, which are presently available for such problems, would suggest. We now consider the time dependent multigroup diffusion equations with an external source... 

R. S. Varga and M. A. Martino, The theory for the numerical solution of time-dependent and time-independent multigroup diffusion equations. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, vol. 16, pp. 570-577. 

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