Abstract
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.
Highlights
The transport equation has been the focus of many groups of researchers in recent years [1, 2, 3]
Integral Methods, such as the Nyström method applied in this work, are known to produce accurate results for the transport equation in various geometries, since that the discretization is careful enough to remove the singularities from the kernel
We present the results for two numerical quadrature schemes namely, Gauss-Legendre quadrature and Boole’s rule
Summary
The transport equation has been the focus of many groups of researchers in recent years [1, 2, 3]. Integral Methods, such as the Nyström method applied in this work, are known to produce accurate results for the transport equation in various geometries, since that the discretization is careful enough to remove the singularities from the kernel. These methods work very well when the transport equation must be solved thousands of times with the same set of parameters and different sources. This happens in evolutionary problems, where the phenomenon of heat transfer involves diffusion, convection and radiation.
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