Error estimates of a finite volume method for the compressible Navier–Stokes–Fourier system

Basarić, Danica; Lukáčová-Medviďová, Mária; Mizerová, Hana; She, Bangwei; Yuan, Yuhuan

doi:10.1090/mcom/3852

Error estimates of a finite volume method for the compressible Navier–Stokes–Fourier system
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by Danica Basarić, Mária Lukáčová-Medviďová, Hana Mizerová, Bangwei She and Yuhuan Yuan

Math. Comp. 92 (2023), 2543-2574

DOI: https://doi.org/10.1090/mcom/3852

Published electronically: May 8, 2023

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Abstract:

In this paper we study the convergence rate of a finite volume approximation of the compressible Navier–Stokes–Fourier system. To this end we first show the local existence of a regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution.

References

Dominic Breit, Eduard Feireisl, and Martina Hofmanová, Local strong solutions to the stochastic compressible Navier-Stokes system, Comm. Partial Differential Equations 43 (2018), no. 2, 313–345. MR 3777877, DOI 10.1080/03605302.2018.1442476
Jan Březina, Eduard Feireisl, and Antonín Novotný, Stability of strong solutions to the Navier-Stokes-Fourier system, SIAM J. Math. Anal. 52 (2020), no. 2, 1761–1785. MR 4083344, DOI 10.1137/18M1223022
C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal. 70 (1979), no. 2, 167–179. MR 546634, DOI 10.1007/BF00250353
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1016/S1570-8659(00)07005-8
Eduard Feireisl, Radim Hošek, David Maltese, and Antonín Novotný, Unconditional convergence and error estimates for bounded numerical solutions of the barotropic Navier-Stokes system, Numer. Methods Partial Differential Equations 33 (2017), no. 4, 1208–1223. MR 3652184, DOI 10.1002/num.22140
E. Feireisl and M. Lukáčová-Medvi\softdová, Statistical solutions for the Navier–Stokes–Fourier system, Preprint, arXiv:2212.06784
Eduard Feireisl, Mária Lukáčová-Medvid’ová, and Hana Mizerová, A finite volume scheme for the Euler system inspired by the two velocities approach, Numer. Math. 144 (2020), no. 1, 89–132. MR 4050088, DOI 10.1007/s00211-019-01078-y
Eduard Feireisl, Mária Lukáčová-Medviďová, Hana Mizerová, and Bangwei She, On the convergence of a finite volume method for the Navier-Stokes-Fourier system, IMA J. Numer. Anal. 41 (2021), no. 4, 2388–2422. MR 4328388, DOI 10.1093/imanum/draa060
Eduard Feireisl, Mária Lukáčová-Medvid’ová, Hana Mizerová, and Bangwei She, Numerical analysis of compressible fluid flows, MS&A. Modeling, Simulation and Applications, vol. 20, Springer, Cham, [2021] ©2021. MR 4390192, DOI 10.1007/978-3-030-73788-7
Eduard Feireisl, Mária Lukáčová-Medvid’ová, and Bangwei She, Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system, Numer. Math. 153 (2023), no. 2-3, 493–529. MR 4557978, DOI 10.1007/s00211-023-01346-y
Eduard Feireisl and Antonín Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system, Arch. Ration. Mech. Anal. 204 (2012), no. 2, 683–706. MR 2909912, DOI 10.1007/s00205-011-0490-3
Eduard Feireisl and Antonin Novotný, Mathematics of open fluid systems, Nečas Center Series, Birkhäuser/Springer, Cham, [2022] ©2022. MR 4434626, DOI 10.1007/978-3-030-94793-4
E. Feireisl, H. Wen and C. Zhu, On Nash’s conjecture for models of viscous, compressible, and heat conducting fluids, Preprint of the Czech Academy of Sciences No. 6-2022.
Thierry Gallouët, David Maltese, and Antonin Novotny, Error estimates for the implicit MAC scheme for the compressible Navier-Stokes equations, Numer. Math. 141 (2019), no. 2, 495–567. MR 3905434, DOI 10.1007/s00211-018-1007-x
Radim Hošek and Bangwei She, Convergent numerical method for the Navier-Stokes-Fourier system: a stabilized scheme, IMA J. Numer. Anal. 39 (2019), no. 4, 2045–2068. MR 4019049, DOI 10.1093/imanum/dry057
Xiangdi Huang and Jing Li, Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows, Comm. Math. Phys. 324 (2013), no. 1, 147–171. MR 3116320, DOI 10.1007/s00220-013-1791-1
Tosio Kato, Linear evolution equations of “hyperbolic” type. II, J. Math. Soc. Japan 25 (1973), 648–666. MR 326483, DOI 10.2969/jmsj/02540648
S. Kawashima, Systems of hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984.
Shuichi Kawashima and Yasushi Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tohoku Math. J. (2) 40 (1988), no. 3, 449–464. MR 957056, DOI 10.2748/tmj/1178227986
M. Lukáčová-Medvi\softdová, H. Mizerová and B. She, New invariant domain preserving finite volume schemes for compressible flows. In: M.L. Muñoz-Ruiz, C. Parés, G. Russo (eds) Recent Advances in Numerical Methods for Hyperbolic PDE Systems. SEMA SIMAI Springer Series, vol 28. Springer, Cham, 131–153, 2021.
Mária Lukáčová-Medvid’ová and Andreas Schömer, Compressible Navier-Stokes equations with potential temperature transport: stability of the strong solution and numerical error estimates, J. Math. Fluid Mech. 25 (2023), no. 1, Paper No. 1, 38. MR 4510551, DOI 10.1007/s00021-022-00733-z
Mária Lukáčová-Medvid’ová, Bangwei She, and Yuhuan Yuan, Error estimates of the Godunov method for the multidimensional compressible Euler system, J. Sci. Comput. 91 (2022), no. 3, Paper No. 71, 27. MR 4414396, DOI 10.1007/s10915-022-01843-6
M. Lukáčová-Medvi\softdová, B. She, and Y. Yuan, Convergence and error estimates of a penalized finite volume method for the compressible Navier–Stokes system with a smooth domain, arXiv:2209.02344.