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This article is cited in 2 scientific papers (total in 2 papers)
About the influence on the accuracy of cubature formulas on the integral characteristics of solutions of the transport equation
E. N. Aristovaab, G. O. Astafurova a Keldysh Institute of Applied Mathematics RAS
b Moscow Institute of Physics and Technology
Abstract:
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.
Keywords:
transport equation, reflection conditions, unstructured meshes, cubature formulas.
Received: 01.07.2019 Revised: 01.07.2019 Accepted: 09.09.2019
Citation:
E. N. Aristova, G. O. Astafurov, “About the influence on the accuracy of cubature formulas on the integral characteristics of solutions of the transport equation”, Matem. Mod., 32:1 (2020), 15–30; Math. Models Comput. Simul., 12:5 (2020), 685–695
Linking options:
https://www.mathnet.ru/eng/mm4145 https://www.mathnet.ru/eng/mm/v32/i1/p15
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Abstract page: | 360 | Full-text PDF : | 69 | References: | 25 | First page: | 12 |
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