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Matrix form of the equations

The discretized group diffusion equations may be expressed in a block matrix form as stated below

$$[ \hspace{0.2ex}\underline{\underline{L}}{}\hspace{0.15ex}_0 ... ...{0.15ex} ]\hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex} = 0$$ (1.30)

where

$$\hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex} \;\hbox{$=$... ...{0.2ex}\underline{\phi}{}\hspace{0.15ex}^G \end{array} \right]$$ (1.31)

and $\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}^i$ contains the ith group flux at the points in the discretized domain.

$\hspace{0.2ex}\underline{\underline{L}}{}\hspace{0.15ex}_0$ is a diagonal matrix of block order G with elements which consist of (K-2) order matrices $(\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex} + \hspace{0.2ex}\underline{\underline{\cal D}}{}\hspace{0.15ex})_g$ as illustrated below

$$\hspace{0.2ex}\underline{\underline{L}}{}\hspace{0.15ex}_0 ... ...e{\underline{\cal D}}{}\hspace{0.15ex})_G \end{array} \right]$$ (1.32)

where $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_g$ is a diagonal matrix of order (K-2) describing the absorption of group g neutrons in region k, $\overline{\Sigma}{}_{a,k}^g$, and $\hspace{0.2ex}\underline{\underline{\cal D}}{}\hspace{0.15ex}_g$ is a matrix of order (K-2) consisting of the coefficient matrix which comes from the discretization of the leakage operator $-\nabla \cdot D_g \nabla$. One row in $\hspace{0.2ex}\underline{\underline{\cal D}}{}\hspace{0.15ex}_g$ describes the leakage of group g neutrons in region k. With the discretization procedure described in this text $\hspace{0.2ex}\underline{\underline{\cal D}}{}\hspace{0.15ex}_g$ becomes a tri-diagonal matrix.

The (block) diagonal matrix $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_s$ of block order G consists of matrix elements $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{s,g}$ of order (K-2) as shown below

$$\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_s ... ...ine{\underline{A}}{}\hspace{0.15ex}_{s,G} \end{array} \right]$$ (1.33)

The diagonal matrices $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{s,g}$ describe the total out-scattering (including self-scattering) of neutrons in group g in region k, ie we may define $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{s,g}$ as

$$\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{s... ...ne{\Sigma}{}_{s,(K-1)}^{g \rightarrow g'} \end{array} \right]$$ (1.34)

$\hspace{0.2ex}\underline{\underline{S}}{}\hspace{0.15ex}$ is a full block matrix of block order G as shown below

$$\hspace{0.2ex}\underline{\underline{S}}{}\hspace{0.15ex} \;... ...line{\underline{S}}{}\hspace{0.15ex}_{GG} \end{array} \right]$$ (1.35)

where the (K-2) x (K-2) diagonal matrix elements describe in-scattering (including self-scattering) from group g' to g in region k, ie the matrix element $\hspace{0.2ex}\underline{\underline{S}}{}\hspace{0.15ex}_{ij}$ is defined by

$$\hspace{0.2ex}\underline{\underline{S}}{}\hspace{0.15ex}_{i... ...ne{\Sigma}{}_{s,(K-1)}^{j \rightarrow i} \end{array} \right]$$ (1.36)

$\hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex}$ of block order G is defined by

$$\hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex} \;... ...underline{\underline{0}}{}\hspace{0.15ex} \end{array} \right]$$ (1.37)

where $\hspace{0.2ex}\underline{\underline{I}}{}\hspace{0.15ex}$ is the identity matrix of order (K-2) and $\hspace{0.2ex}\underline{\underline{\overline{\Sigma}{}}}{}\hspace{0.15ex}_f^g$ is a (K-2) x (K-2) diagonal matrix consisting of the fission cross sections in group g in region k, ie we may define

$$\hspace{0.2ex}\underline{\underline{\overline{\Sigma}{}}}{}... ... \\ & & & \overline{\Sigma}{}_{f,(K-1)}^g \end{array} \right]$$ (1.38)

One may write the matrix equation (1.48) in a more compact form with aid of the following definition

$$\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} \;... ...}_s - \hspace{0.2ex}\underline{\underline{S}}{}\hspace{0.15ex}$$ (1.39)

resulting in the matrix equation

$$\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} \h... ...hspace{0.15ex} \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$$ (1.40)

To elucidate the different matrix elements, $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ is written out below '

$$\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} = ... ...line{\underline{S}}{}\hspace{0.15ex}_{GG} \end{array} \right]$$ (1.41)

A fact of major importance is that $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ is column-wise diagonally dominant. This is seen by observing that the out-scattering matrix $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{s,g}$ may be defined by

$$\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{s... ...\hspace{0.2ex}\underline{\underline{S}}{}\hspace{0.15ex}_{g'g}$$ (1.42)

which actually states that the neutrons lost in group g by scattering is discovered again in groups g', ie no neutrons are lost by the scattering process.

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