Characteristics scheme for the transport equation solving on a tetrahedron grid with barycentrical interpolation

E. N. Aristova ; G. O. Astafurov

Matematičeskoe modelirovanie, Tome 30 (2018) no. 9, pp. 33-50.

Résumé

In this paper, the interpolation-characteristic method of order of approximation is not less than a second to solve the transport equation on an unstructured grid of tetrahedra is constructed. The problem of finding numerical solutions by this method, hereinafter method of short characteristics, is divided into two subtasks. First, there is a resolution of the individual simplicial cell. It is necessary to specify a set of discrete values, the setting of which on the litted faces is mathematically sufficient to find all remaining grid values in the cell. Depending on the location of the cell and the direction of propagation of the radiation there are three different types of illumination. The interpolation in barycentrically coordinates of the cell with 14 free coefficients is proposed. The interpolation takes into account the values of radiation intensity at the nodes, and the average integrated intensity values for edges and faces without adding new points of stensil. This interpolation ensures at least the second order of approximation with some additional members of the third order. The method ensures a conservative redistribution of the outcoming fluxes over cell's faces. The second subtask associated with the choice of order and resolution of the cells and can be solved using graph theory. Numerical calculations confirm the order of convergence is approximately the second.

Détail

Mots-clés : transport equation
Keywords: method of short characteristics, interpolation-characteristic method, second order of approximation, barycentric coordinates.

Bibliographie

[1] B.N. Chetverushkin, Matematicheskoe modelirovanie zadach dinamiki izluchaushego gaza, Nauka, M., 1985, 304 pp.

[2] V.B. Rozanov, D.V. Barishpol'tsev, G.A. Vergunova at al., “Interaction of laser radiation with a low density structured absorber”, J. of Experimental and Theoretical Physics, 122:2 (2016), 256–276

[3] E.N. Aristova, M.N. Gertsev, A.V. Shilkov, “Lebesgue Averaging Method in Serial Computations of Atmospheric Radiation”, Comp. Math. and Math. Physics, 57:6 (2017), 1022–1035

[4] E.N. Aristova, G.O. Astafurov, A.V. Shilkov, “Raschet izluchenia v udarnom sloe spuskaemogo kosmicheskogo apparata s uchetom detalei spektra fotonov”, Kompiuternye issledovania i modelirovanie, 9:4 (2017), 579–594

[5] S.T. Surzhikov, “Radiative-convective heat transfer of a spherically shaped space vehicle in carbon dioxide”, High Temperature, 49:1 (2011), 92–107

[6] D.A. Andrienko, S.T. Surzhikov, “The unstructured two-dimensional grid-based computation of selective thermal radiation in CO2-N2 mixture flows”, High Temperature, 50:4 (2012), 545–555

[7] A.L. Zheleznyakova, S.T. Surzhikov, “Calculation of a Hypersonic Flow over Bodies of Complex Configuration on Unstructured Tetrahedral Meshes Using the AUSM Scheme”, High Temperature, 52:2 (2014), 271–281

[8] K.M. Magomedov, A.S. Kholodov, Setochno-harakeristicheskie chislennye metody, Nauka, M., 1986, 287 pp.

[9] O.V. Nikolaeva, “Nodal scheme to the radiation transport equation on unstructed tetrahedron mesh”, Math. Models Comput. Simul., 7:6 (2015), 581–592

[10] E.N. Aristova, G.O. Astafurov, “Second-Order Short Characteristic Method for Solving the Transport Equation on a Tetrahedron Mesh”, Math. Mod. and Comp. Simul., 9:1 (2017), 40–47

[11] E.P. Sychugova, E.F. Seleznev, Metod konechnyh elementov dlya resheniya uravneniya perenosa na nestructurirovannyh tetraedralnyh setkah, Preprint IBRAE RAN No IBRAE-2014-03, 20 pp.

[12] G.O. Astafurov, “Algoritm obkhoda iacheek v kharakteristicheskikh metodakh resheniia uravneniia perenosa”, Preprint IPM im. M.V. Keldysha

[13] E.I. Yakovlev, L.A. Igumnov, Metodi vichisleniia na super-EVM topologicheskikh kharakteristik triangulirovannykh obiektov, Izdatelstvo Nizhnegorodskogo gosuniversiteta, Nizhniy Novgorod, 2014, 85 pp.

[14] S. Dasgupta, C. Papadimitriou, U. Vazirani, Algorithms, McGraw-Hill Higher Education, 2006, 332 pp.

[15] Y.I. Skalko, R.N. Karasev, A.V. Akopyan, I.V. Tsybulin, M.A. Mendel, “Marshevyi algoritm resheniia zadachi perenosa izlucheniia metodom korotkikh kharakteristik”, Kompiuternye issledovaniia i modelirovanie, 6:2 (2014), 203–215

[16] M.I. Bakirova, V.Ya. Karpov, M.I. Muhina, “A characteristic-interpolation method of solving transfer equation”, Difffer. Equ., 22:7 (1986), 788–794

[17] V.E. Troshchiev, A.V. Nifanova, Yu.V. Troshchiev, “Characteristic Approach to the Approximation of Conservation Laws in Radiation Transfer Kinetic Equations”, Doklady Mathematics, 69:1 (2004), 136–140