# Characteristic Scheme for the Solution of the Transport Equation on an Unstructured Grid with Barycentric Interpolation

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### **Abstract**—

An interpolative-characteristic method is constructed with the approximation not lower than the second order for solving the transport equation on an unstructured grid composed of tetrahedra. The problem of finding a numerical solution by this method, further referred to as the method of short characteristics, is subdivided into two subproblems. The first concerns the resolution of a single simplicial cell. A set of grid values must be specified, which when set on the illuminated faces is mathematically sufficient for obtaining all the remaining grid values in the cell. Depending on the location of the cell and the propagation direction of the radiation there are three different types of illumination. It is proposed to use interpolation of a cell in barycentric coordinates with 14 free coefficients, which enables taking into account the values of the radiation intensity at the nodes and the average integral values of the intensity over the edges and faces without adding new stencil points. This interpolation ensures at least the second order of approximation with additional allowance for the terms with the third-order approximation. Moreover, the method takes into account a conservative redistribution of the outbound flow over the edges of the cell. The second subproblem is associated with choosing the order of tracing the cell and can be solved using methods of graph theory. The numerical calculations confirmed approximately the second order of convergence.

## Keywords:

transport equation method of short characteristics interpolative-characteristic method second order of approximation barycentric coordinates## Notes

### ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 18-01-00857.

## REFERENCES

- 1.B. N. Chetverushkin,
*Mathematical Modeling of Problems of Radiating Gas Dynamics*(Nauka, Moscow, 1985) [in Russian].zbMATHGoogle Scholar - 2.V. B. Rozanov, D. V. Barishpol’tsev, G. A. Vergunova, et al., “Interaction of laser radiation with a low density structured absorber,” J. Exp. Theor. Phys.
**122**, 256–276 (2016).CrossRefGoogle Scholar - 3.E. N. Aristova, M. N. Gertsev, and A. V. Shilkov, “Lebesgue averaging method in serial computations of atmospheric radiation,” Comput. Math. Math. Phys.
**57**, 1022–1035 (2017).MathSciNetCrossRefzbMATHGoogle Scholar - 4.E. N. Aristova, G. O. Astafurov, and A. V. Shilkov, “Calculation of radiation in shockwave layer of a space vehicle taking into account details of photon spectrum,” Komp’yut. Issled. Model.
**9**, 579–594 (2017).Google Scholar - 5.S. T. Surzhikov, “Radiative-convective heat transfer of a spherically shaped space vehicle in carbon dioxide,” High Temp.
**49**, 92–107 (2011).CrossRefGoogle Scholar - 6.D. A. Andrienko and S. T. Surzhikov, “The unstructured two-dimensional grid-based computation of selective thermal radiation in CO
_{2}-N_{2}mixture flows,” High Temp.**50**, 545–555 (2012).CrossRefGoogle Scholar - 7.A. L. Zheleznyakova and S. T. Surzhikov, “Calculation of a hypersonic flow over bodies of complex configuration on unstructured tetrahedral meshes using the AUSM scheme,” High Temp.
**52**, 271–281 (2014).CrossRefGoogle Scholar - 8.K. M. Magomedov and A. S. Kholodov,
*Grid-Characteristic Numerical Methods*(Nauka, Moscow, 1986) [in RUssian].zbMATHGoogle Scholar - 9.O. V. NIkolaeva, “Nodal scheme to the radiation transport equation on unstructed tetrahedron mesh,” Math. Models Comput. Simul.
**7**, 581–592 (2015).Google Scholar - 10.E. N. Aristova and G. O. Astafurov, “Second-order short characteristic method for solving the transport equation on a tetrahedron mesh,” Math. Models Comput. Simul.
**9**, 40–47 (2017).MathSciNetCrossRefGoogle Scholar - 11.E. P. Sychugova and E. F. Seleznev, “The finite element method for solving the transport equation on unstructured tetrahedral meshes,” Preprint No. ISDAE-2014-03 (Inst. Safe Develop. At. Energy RAS, Moscow, 2014).Google Scholar
- 12.G. O. Astafurov, “Cell traversal algorithm in characteristic methods for solving the transport equation,” Preprint KIAM (in press)Google Scholar
- 13.E. I. Yakovlev and L. A. Igumnov,
*Methods of Computing on Supercomputers the Topological Characteristics of Triangulated Objects*(Nizhegor. Gos. Univ., Nizh. Novgorod, 2014) [in Russian].Google Scholar - 14.S. Dasgupta, C. Papadimitriou, and U. Vazirani,
*Algorithms*(McGraw-Hill, New York, 2006).Google Scholar - 15.Y. I. Skalko, R. N. Karasev, A. V. Akopyan, I. V. Tsybulin, and M. A. Mendel, “Space-marching algorithm for solving radiative transfer problem basedon short-characteristics method,” Kompyut. Issled. Model.
**6**, 203–215 (2014).Google Scholar - 16.M. I. Bakirova, V. Ya. Karpov, and M. I. Muhina, “A characteristic-interpolation method of solving transfer equation,” Difffer. Equat.
**22**, 788–794 (1986).zbMATHGoogle Scholar - 17.V. E. Troshchiev, A. V. Nifanova, and Yu. V. Troshchiev, “Characteristic approach to the approximation of conservation laws in radiation transfer kinetic equations,” Dokl. Math.
**69**, 136–140 (2004).zbMATHGoogle Scholar