Simulation for non-homogeneous transport equation by Nyström method

In this work we solve numerically the one-dimensional transport equation with semi-reflective boundary conditions and non-homogeneous domain. The proposed methodology consists of applying the Nyström method in order to discretize the integral formulation of this problem which is an equation involving weakly singular integral operators. For this purpose, analytical and computational techniques were applied to deal with the singularities. The Nyström method is an integral method which approximates the integral operator by a numerical quadrature and turns the integral equation into a finite dimensional linear system. This formulation allows us to use any function to describe both scattering cross section and total cross section. The algorithm is implemented in C language with the use of routines of GNU scientific library and computational techniques for code optimization. The scalar flux was calculated for two numerical quadrature, namely Gauss-Legendre quadrature and Boole's rule. The numerical results were determined for transport problem with homogeneous and non-homogeneous domains. In order to validate the proposed method-ology, our numerical results were compared with those from the literature and presented with several correct significant digits.