Wavelet-Based Algorithms for Solving Neutron Diffusion Equations

Abstract

This work develops efficient algorithms for numerically solving the neutron diffusion equation by a wavelet Galerkin method (WGM). One of the main problems encountered in solving neutron diffusion equation using WGM is the treatment of the boundary and interface conditions. In one-dimensional problems, the boundaries of the wavelet series expansions are assumed to be the analytical boundaries of the problem, and the boundary condition equations are replaced by end equations in Galerkin system. In two-dimensional problems, in order to maintain the integrity of the system, the boundaries of the wavelet series are shifted until the end is independent on any expansion coefficients of the scaling function that affect the solution within the real boundaries.Since the scaling functions are compactly supported, only a finite number of the connection coefficients are nonzero. The resultant matrix has a block diagonal structure, which can be inverted easily, therefore, enables us to extend the solution to two-dimensional heterogeneous cases and make the inner iteration efficient in the eigenvalue and multigroup problems. We tested our WGM algorithm with several diffusion problems including two-dimensional heterogeneous problems and multi-group problems. The solutions are very accurate with a proper selection of Daubechies' order and dilation order. Our WGM algorithm provides very accurate solutions for one-dimensional and two-dimensional heterogeneous problems in which the flux exhibits very steep gradients.

An economic prediction of the finer resolution level wavelet coefficients in electronic structure calculations

“…The most widespread use of wavelets is still in the image compression techniques [2,3], and the other applications are also mainly analyzers. Wavelets can build a basis for differential equation discretisation [4] and solving [5,6,7], and the solvers have been developed and tested in various fields of science from diffusions to electromagnetic waves [8,9,10,11]. In electron structure calculations wavelet basis has been present since the early nineties [12,13,14,15], and in the previous decade both a wavelet based [13,14,16] and a multiwavelet based [17,18,19] solver have been developed with chemical accuracy and massively parallel computation possibility.…”

On Wavelet Based Modeling of the Nitrogen Oxides Emission and Concentration due to Road Traffic in Urban Environment

“…The goal is not to solve the diffusion equation in wavelet basis, it can be found in many applications, like [12][13][14], but to give a solution for modeling the urban environment -mainly the position of the buildings and roads, the various wind speed, -in formulating the discretized matrix equations and calculating the matrix elements. For a proper modeling of the obstacles locally different diffusion coefficients would be necessary, and these coefficients can be well approximated as step functions of the position.…”