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Characteristics and accuracy

Despite the simplification in comparison with the continuous energy equation, the solution of the multi-group diffusion equations is still a formidable problem since we have to solve an eigenvalue problem of G coupled (in general partial) differential equations in space to obtain the flux distribution and the fundamental eigenvalue $\lambda_0$. The coupling of the G equations arises partly from the scattering terms and partly from the fission source terms. To solve these equations numerically we first need to discretize the domain corresponding to the reactor core and use this discrete domain to derive the discretized equations suitable for solution by means of a computer.

It is important to recognize the assumptions and the limitations of accuracy in connection with the multi-group treatment. The diffusion approximation is based upon use of Fick's law. It is known ([3, p. 134] & [1, p. 123]) that Fick's law breaks down in neighborhood of material interfaces and within strong absorbers such as control rods in a reactor core. In general, use of the diffusion approximation is restricted to areas lying not closer than 2 to 3 scattering mean free paths^1.9 to material interfaces. Consequently, we have to be aware of the poor accuracy of the diffusion equation near the interface between the reactor core and the reflector. Another thing of importance affecting the accuracy is the procedure for the evaluation of the group constants (see p. and section 1.4).

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