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Ordering of flux components

In this section we discuss the two different ways of ordering the unknown flux components. As discussed in section 1.6.3 we are free to choose an appropriate ordering.

According to equation (1.43) we identify that $\phi_k^g$ is coupled to the other $\phi$'s in two distinct ways

$\parbox{16cm}{ \begin{eqnarray*} \parbox[t]{0.6cm}{a)}\parbox[t]{14... ...D^g(\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}) \nabla]$.} \end{eqnarray*}}$
The first way of ordering the flux components is to gather point fluxes with the same group number. This ordering is described extensively in section 1.6.1. Using this ordering it is easy to show that $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$-matrix in the matrix formulation (1.58) has the form

$$\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} = ... ...line{\underline{B}}{}\hspace{0.15ex}_{GG} \end{array} \right]$$ (1.53)

where all block matrices have order (K-2), matrices $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}_{ii}$ are symmetric tri-diagonal and matrices $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}_{ij} , i \ne j$ are diagonal matrices. The matrix $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}$ defined by (1.65) has the form

$$\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex} = ... ...line{\underline{C}}{}\hspace{0.15ex}_{GG} \end{array} \right]$$ (1.54)

with all matrix elements being diagonal matrices of order (K-2).

The second way of ordering the unknowns is to order by node k, ie to gather all the group fluxes at node k. With this ordering $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ has the form

$$\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} = ... ...ine{\underline{B}}{}\hspace{0.15ex}_{K-1} \end{array} \right]$$ (1.55)

where $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}_i$ is in general a full (G x G) matrix (see, however, footnote 25 on page ) and the matrices $\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_i$ and $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}_i$ are diagonal matrices of order G with the property

$$\hspace{0.2ex}\underline{\underline{A}}{}\hspace{0.15ex}_{i... ...space{0.15ex}_i \hspace{3cm} \forall i \in \{2,\ldots,(K-2)\}$$ (1.56)

The (block) order of $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ corresponds to the number of internal grid points (K-2). The form of $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ may be stated as symmetric block-tri-diagonal. The $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}$ matrix defined by (1.65) now shows the following form

$$\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex} = ... ...nderline{\hat{C}}}{}\hspace{0.15ex}_{K-1} \end{array} \right]$$ (1.57)

ie a square block-diagonal matrix of order (K-2) with in general full (G x G) matrices as elements.

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