Wavelet theory for solution of the neutron diffusion equation

Abstract

In this thesis, we solve the neutron diffusion equation by Wavelet Galerkin(WG) scheme. Wavelet functions are generated by dilation and translation operation and are localized in space. We can construct the wavelet functions from the scaling function which has recursive property. So these properties may be utilized to solve a differential equation which has severe "stiffness". The WG method represents the solution as a summation of Daubechies`` scaling functions, which are also used as the weighting function. The Daubechies`` scaling functions have the properties of orthogonality and high smoothness. Unlike the finite element method, the weighting function is the Daubechies`` scaling function and the unknowns determined are not the fluxes of the node but the coefficients of the scaling functions. The scaling functions are overlapping in the nodes and require special treatment at interfaces between nodes and at the boundaries. The WG method is applied to one-dimensional fixed-source and eigen-value problems and to simple homogeneous two-dimensional fixed-source and eigenvalue problems in reactor physics. In constructing the elements of matrices, numerical integrations such as Gaussian quadrature and trapezoidal method are required. These integration procedures consume most of the computing time. The resulting matrix equation is solved by Gaussian elimination. In extending to the two-dimensional problem, the basic idea used is that the solution can be expressed in the product of two scaling functions. The form of the resulting matrix is more complicated than that of one-dimensional case. We tested the method to several problems. In a one-dimensional fixed-source problem, the solution is very accurate with increasing Daubechies`` order and dilation order. The boundary conditions are also satisfied very well. In particular, the WG method is a good solver for heterogeneous problems. We can choose higher order in a stiff node to get more accurate solution. In a two-g...