3.8 The Algebraic Equation System
57
large that the diagonal-storage scheme may allow the problem to be kept in
main memory when the full-array scheme does not.
If the nodal values are referenced using the grid indices, say 4i,j in 2D,
they look like matrix elements or components of a tensor. Since they are
actually components of a vector 4, they should have only the single index
indicated in Table 3.2.
The linearized algebraic equations in two dimensions can now be written
in the form:
Fig. 3.5. Structure of the matrix for a five-point computational molecule (nonzero entries in the coefficient matrix on five diagonals are shaded; each horizontal
set of boxes corresponds to one grid line)
As noted above, it makes little sense to store the matrix as an array. If,
instead, the diagonals are kept in separate arrays, it is better to give each
diagonal a separate name. Since each diagonal represents the connection to
the variable at a node that lies in a particular direction with respect to the
central node, we shall call them Aw, As, Ap, AN and AE; their locations in
the matrix for a grid with 5 x 5 internal nodes are shown in Fig. 3.5. With this
57
large that the diagonal-storage scheme may allow the problem to be kept in
main memory when the full-array scheme does not.
If the nodal values are referenced using the grid indices, say 4i,j in 2D,
they look like matrix elements or components of a tensor. Since they are
actually components of a vector 4, they should have only the single index
indicated in Table 3.2.
The linearized algebraic equations in two dimensions can now be written
in the form:
Fig. 3.5. Structure of the matrix for a five-point computational molecule (nonzero entries in the coefficient matrix on five diagonals are shaded; each horizontal
set of boxes corresponds to one grid line)
As noted above, it makes little sense to store the matrix as an array. If,
instead, the diagonals are kept in separate arrays, it is better to give each
diagonal a separate name. Since each diagonal represents the connection to
the variable at a node that lies in a particular direction with respect to the
central node, we shall call them Aw, As, Ap, AN and AE; their locations in
the matrix for a grid with 5 x 5 internal nodes are shown in Fig. 3.5. With this