About the influence on the accuracy of cubature formulas on the integral characteristics of solutions of the transport equation

E. N. Aristova ; G. O. Astafurov

Matematičeskoe modelirovanie, Tome 32 (2020) no. 1, pp. 15-30

Résumé

For technical applications, it is important to be able to solve the radiation transfer equation not for free boundary conditions, but for reflection conditions. The reflection conditions leads to the situation when all angular directions for which the transport equation is calculated are dependent on each other even in the case of scattering absence. Angular directions are taken from a discrete set of nodes of a cubature formula on the unit sphere, implementation of reflection conditions leads to necessity to remain within this discrete set of angular directions. One of the variants of the algorithm, based on the implementation of a discrete analogue of the radiation flux conservation at the boundary, is presented in this paper. The use of the interpolation-characteristic scheme entails the need to construct a correct reflection condition not only in faces where it is simple, but also in vertices and edges where it requires additional definitions due to the lack of the concept of normal. The radiation density as an integral value depends not only on the circuit error of the solution of the transport equation, but also on the error of the cubature formulas used. For smooth solutions, it is usually quite a small number of nodes on the sphere, so that the effect of errors of cubature formulas is small. In the case of an undifferentiated solution, there is a threshold value for the fineness of the spatial grid partitioning so that, at steps below this value, the error of the cubature formula is the main contributor to the error.

Détail

Mots-clés : transport equation
Keywords: reflection conditions, unstructured meshes, cubature formulas.