A comparison of iterative methods for the solution of elliptic partial differential equations, particularly the neutron diffusion equation Public Deposited

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http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/8623j110h

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  • Two new concepts have been explored in solving the neutron diffusion equation in one and two dimensions. At the present time, the diffusion equation is solved using source iterations. These iterations are performed in a mathematical form which has a great deal of physical significance. Specifically, the neutron production term is on the right-hand side, while the absorption and leakage terms are on the left side. In performing a single source iteration, a distribution for the neutron flux is assumed so that the production term can be calculated. This provides a "known" right-hand side. Solving the difference equation for the flux, which corresponds to this assumed source distribution, gives the next estimate for the flux distribution. This type of iteration has the physically significant characteristic of finding directly, for each iteration, a flux which corresponds to an assumed source distribution. In this thesis it was found that by subtracting the absorption term from both sides of the diffusion equation, and performing "source iterations" with both absorption and production terms on the right-hand side (and only the leakage term on the left-hand side), improved convergence rates were attained in many cases. In one neutron energy group, this new idea of putting the absorption term on the right-hand side worked best with only one region, and where reactor dimensions were large compared to the thermal neutron diffusion length (a>>L). In small reactors, where a=L, convergence behavior was similar for both forms of iteration. This new idea was also found to work quite well in one-group multiregion problems. However, due to problems with numerics (inherent asymmetric treatment of the scattering terms), the method does not work at all in a multi-energy group formulation. Secondly, in two dimensions, a closed-form solution to a single source iteration has been found. At this time, the standard method of solution for a two-dimensional source iteration is to perform "inner iterations" to approximately solve for the flux that corresponds to an assumed source. The alternative, up until now, was to solve a giant matrix of the order (N² x N²). This is a sparse matrix, but it has always been considered as highly undesirable to work with a solution (even though it may be closed-form) where the matrix to be solved increases in order roughly as the fourth power of the number of mesh intervals. The new algebraic form for this closed-form solution involves a matrix of order (N x N), not (N² x N²). The matrix is, however, a full matrix. What is done, essentially, is to solve simultaneously for all the flux values along the vertical centerline of the two-dimensional problem, and then use a reflective boundary condition across the core centerline, and then the difference equation itself (in vector form) as a set of flux-vector generating equations to generate the entire flux field, line by line. In solving for the first flux vector (at the x = o, or z = o, core centerline), the right-hand side of the matrix problem incorporates all of the source values in the entire problem space. The initial inversion of the full (N x N) matrix algebraically guarantees that the (M+1)th flux vector (on the problem space boundary) will go to zero. This matrix method for two-dimensional neutronic analysis was shown to work well in both cartesian and cylindrical coordinates.
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