Geodesic
An explicit difference scheme for non-linear heat conduction equation
Matematičeskoe modelirovanie, Tome 34 (2022) no. 12, pp. 3-19.

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We develop an algorithm for numerical treatment of non-linear heat conduction problems, which is well adapted to the architecture of high-performance computing systems. The technique is based on hyperbolic heat conduction model and explicit difference scheme. The new scheme provides second order temporal resolution of non-linearity with acceptable time step. An application to plasma dynamic simulations is discussed.
Keywords: non-linear heat conduction equation, hyperbolic heat conduction model, explicit difference scheme, high-performance computing. 
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B. N. Chetverushkin; O. G. Olkhovskaya; V. A. Gasilov. An explicit difference scheme for non-linear heat conduction equation. Matematičeskoe modelirovanie, Tome 34 (2022) no. 12, pp. 3-19. http://geodesic.mathdoc.fr/item/MM_2022_34_12_a0/

[1] B. N. Chetverushkin, M. V. Yakobovskiy, “The Prospects of Development in Russia of High-Performance Computing and Predictive Modeling in Modern Technologies”, Herald of the Russian Academy of Sciences, 91:6 (2021), 661–666 | DOI

[2] Y. B. Zel'dovich, Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Dover Books on Physics, Courier Corporation, USA, 2012, 944 pp.

[3] D. A. Knoll, D. E. Keyes, “Jacobian-free Newton-Krylov methods: a survey of approaches and applications”, J. of Comp. Phys., 193 (2004), 357–397 | DOI | MR | Zbl

[4] M. Viallet et al, “A Jacobian-free Newton-Krylov method for time-implicit multidimensional hydrodynamics. Physics-based preconditioning for sound waves and thermal diffusion”, Astronomy Astrophysics, 586 (2016), A153 | DOI

[5] V. Savcenco, Eu. Savcenco, “Multirate Numerical Integration for Parabolic PDEs”, AIP Conference Proc., 1048, 2008, 470 | DOI

[6] C. Mikidaa, A. Klöcknerb, D. Bodonya, “Multi-rate time integration on overset meshes”, J. of Comp. Phys., 396:2 (2019), 325–346 | DOI | MR

[7] V.T. Zhukov, “Explicit methods of numerical integration for parabolic equations”, MMCS, 2011, no. 3, 311–332 | DOI | MR | Zbl

[8] Yu. B. Radvogin, “Economical algorithms for the numerical solution of a multidimensional heat equation”, Doklady Mathematics, 67:1 (2003), 31–33 | MR | Zbl

[9] M. A. Botchev, “Exponential Time Integrators for Unsteady Advection-Diffusion Problems on Refined Meshes”, Numerical Geometry, Grid Generation and Scientific Computing, Lecture Notes in Computational Science and Engineering, 143, eds. Garanzha V.A., Kamenski L., Si H., Springer, Cham, 2021 | DOI | MR | Zbl

[10] B. N. Chetverushkin, A. V. Gulin, “Explicit schemes and numerical simulation using ultrahigh-performance computer systems”, Doklady Mathematics, 86:2 (2012), 681–683 | DOI | MR | Zbl

[11] B. N. Chetverushkin, O. G. Olkhovskaya, “Modeling of Radiative Heat Conduction on High-Performance Computing Systems”, Doklady Math., 101:2 (2020), 172–175 | DOI | MR | Zbl

[12] D. Mihalas, L. Auer, “On laboratory-frame radiation hydrodynamics”, J. of Quantitative Spectroscopy and Radiative Transfer, 71:1 (2001), 61–97 | DOI

[13] E. E. Myshetskaya, V. F. Tishkin, “Estimates of the hyperbolization effect on the heat equation”, Computational Mathematics and Mathematical Physics, 55 (2015), 1270–1275 | DOI | MR | Zbl

[14] S. I. Repin, B. N. Chetverushkin, “Estimates of the difference between approximate solutions of the Cauchy problems for the parabolic diffusion equation and a hyperbolic equation with a small parameter”, Doklady Mathematics, 88:1 (2013), 417–420 | DOI | MR | Zbl

[15] M. D. Surnachev, V. F. Tishkin, B. N. Chetverushkin, “On conservation laws for hyperbolized equations”, Differential Equations, 52 (2016), 817–823 | DOI | MR | Zbl

[16] B. N. Chetverushkin, A. A. Zlotnik, “On a hyperbolic perturbation of a parabolic initial-boundary value problem”, Applied Mathematics Letters, 83 (2018), 116–122 | DOI | MR | Zbl

[17] B. N. Chetverushkin, O. G. Olkhovskaya, I. P. Tsigvintsev, “Numerical solution of high-temperature gas dynamics problems on high-performance computing systems”, Journal of Computational and Applied Mathematics, 390 (2021), 113374 | DOI | MR

[18] V.P. Krainov, Qualitative Methods in Physical Kinetics and Hydrodynamics, Institute of Physics, American, 1992, 203 pp.

[19] S. V. Lebedev, A. Frank, D. D. Ryutov, “Exploring astrophysics-relevant magneto-hydrodynamics with pulsed-power laboratory facilities”, Reviews of Modern Physics, 91:2 (2019), 025002, 46 pp. | DOI

[20] N. Niassea et al, “3D MHD Simulations of Radial Wire Array Z-pinches”, AIP Conference Proc., 1088, 2009, 125 | DOI

[21] G.A. Bagdasarov, A.S. Boldarev, V.A. Gasilov, S.V. D'yachenko, E.L. Kartashova, O.G. Ol'hovskaya, Svidetel'stvo o gosudarstvennoy registratsii No 2012660911 ot 30.12.2012 Programma dlya EVM “Programmnyi kompleks MARPLE” https://github.com/genabug/mrp-utils

[22] https://www.keldysh.ru/cgi/thermos/navigation.pl?en,home