Development and Verification of Nuclear Calculation Methods for ...

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4. 3. The TWODIM Programme
The fundamental methods outlined above form the basic principles in
the two-dimensional (XY, RZ, R8), multi-group (maximum 20 groups
for 25 x 25 meshes) finite difference equation computer programme
TWODIM. The programme was especially designed for the detailed
determination of flux and power distributions in light-water fuel element
assemblies with a detailed treatment of control rods and water gaps as
shown in fig. 13.
A flux representation where the fluxes were assigned to the midpoints
instead of the intersections of the mesh lines was chosen. This is
enormously advantageous for particular multi-group problems, where the
scattering terms in equation (4. 6) have smaller storage requirements.
Reaction rates are determined quite exactly by means of this English method,
whereas the leakage seems to be less precisely determined than with the
American one. Furthermore, the leakage constitutes only part of the total
neutron balance.
Boundary conditions either in the shape of full gamma matrices or the
conventional diagonal terms may be obtained from the HECS programme
(ref. 15).
A successive overrelaxation method, SOR, or a successive line overrelaxation
method, SLOR, is applied for the inner iterations with the best
relaxation factor found. In the SLOR method blocks of horizontal or vertical
lines of mesh points may be solved simultaneously in a numerically
stable way by means of a tridiagonal forward elimination and backward substitution
method. This direct method for the subproblem implies no appreciable
further arithmetic efforts compared with the usual point relaxation
scheme and it generally gains a factor of V"2~ in convergence rate. More
impresrive gains can be achieved, especially in problems with non-uniform
mesh spacings.
The extension of the method to greater blocks of subproblem« is described
in ref. 16 . The spectral radius for the block Jacobi iterative matrix
does indeed decrease as more points are improved simultaneously,
which gives a greater (asymptotic) convergence rate (per iteration). However,
the increased computational efforts per iteration and associated numerical
difficulties due to dangerous accumulation of rounding-off errors
seem to make the method leas advantageous.
As the spectral radius ji(B) for the block Jacobi matrix approaches
unity, the accurate determination of «_ becomes more important as can