# An application of Chebyshev polynomials to the solution of a two-dimensional elliptic boundary-value problem

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## Abstract

The goal of this dissertation is to investigate the feasibility of using a bivariate Chebyshev polynomial to approximate solutions to the two-dimensional neutron diffusion equation.

The two-dimensional two-group neutron diffusion equations are solved by expanding neutron fluxes in a finite series of Chebyshev polynomials over large regions of a fission reactor. All the equations for the expansion coefficients necessary to satisfy the appropriate boundary conditions for the flux and current for a typical region are developed. The resulting system of algebraic equations is solved, using the power iteration method. Since the system of equations is overdetermined, the Gram-Schmidt method of orthogonalization is used. Calculations are done with the aid of a computer code, CDP, developed as part of this dissertation.

Two different test problems are solved using a first order finite difference computer code, PDQ-7 as a standard for comparison, and CDP. The first problem is a water-reflected square core with homogeneous material properties in each region. This problem is selected to provide a rather severe test of the CDP method in the calculation of large thermal neutron flux peaking at the core- reflector interface. The second problem is an actual problem solved by a utility for on-line-fuel management in a Pressurized Water Reactor. The reactor core consists of four-different fuel assemblies arranged in a checkerboard pattern in the interior of the core.

For the first problem, both PDQ-7 and CDP give about the same fast neutron flux distributions. The eigenvalues calculated by both methods are identical to one part in 4800. Except for a small region near the core-reflector interface, the thermal neutron fluxes within the core differ by less than about 4%. At the core-reflector interface, the difference in thermal fluxes is about 20%. A significant reduction in computer time required for the solution is achieved (0.45 sec for CDP vs 32.6 sec for PDQ-7). It is important to note, however, that the time required to obtain the "standard" solution for comparison, using PDQ-7 is larger than would be needed for a large mesh spacing. Thus, the savings in computer time required to achieve a solution using a PDQ-7 type code giving results comparable to those obtained with the CDP code cannot be inferred directly from these results. In any event, note that the CDP method is indeed. economical in both preparation of input data and computer time.

In the second problem, four very large regions are used in the CDP method for the entire reactor. These regions are much larger than those used in a typical finite difference solution for the same problem. Both methods give about the same fast neutron flux distribution. The eigenvalues calculated by the two methods are identical to one part in 2140. Although the CDP method does not show the assembly-to-assembly variation in thermal neutron flux, it gives the Same average thermal neutron flux as calculated with PDQ-7 to within 2%, except near the core boundary. Again, a large reduction in computer time is achieved (0.69 sec vs 101 sec).

The feasibility of using Chebyshev polynomial expansions for two-dimensional multi-group diffusion calculations has been demonstrated for these two problems. The method gives, not only very accurate eigenvalues, but also reasonably accurate neutron flux distributions within the reactor core.