Geodesic
Entropic regularization of the discontinuous Galerkin method in conservative variables for three-dimensional Euler equations
Matematičeskoe modelirovanie, Tome 36 (2024) no. 4, pp. 77-91.

Voir la notice de l'article provenant de la source Math-Net.Ru

The entropic regularization of the conservative stable discontinuous Galerkin method in conservative variables for three-dimensional Euler equations is constructed by the help of a special slope limiter. This limiter ensures the fulfillment of three-dimensional analogues of monotonicity conditions and a discrete analogue of entropic inequality. The developed method was tested on a three-dimensional model problem of a Taylor–Green vortex.
Mots-clés : Euler equations
Keywords: the discontinuous Galerkin method, conservation laws, slope limiter, entropic inequality, Taylor–Green vortex. 
@article{MM_2024_36_4_a4,
     author = {Y. A. Kriksin and V. F. Tishkin},
     title = {Entropic regularization of the discontinuous {Galerkin} method in conservative variables for three-dimensional {Euler} equations},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {77--91},
     publisher = {mathdoc},
     volume = {36},
     number = {4},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2024_36_4_a4/}
}
TY  - JOUR
AU  - Y. A. Kriksin
AU  - V. F. Tishkin
TI  - Entropic regularization of the discontinuous Galerkin method in conservative variables for three-dimensional Euler equations
JO  - Matematičeskoe modelirovanie
PY  - 2024
SP  - 77
EP  - 91
VL  - 36
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2024_36_4_a4/
LA  - ru
ID  - MM_2024_36_4_a4
ER  - 
%0 Journal Article
%A Y. A. Kriksin
%A V. F. Tishkin
%T Entropic regularization of the discontinuous Galerkin method in conservative variables for three-dimensional Euler equations
%J Matematičeskoe modelirovanie
%D 2024
%P 77-91
%V 36
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2024_36_4_a4/
%G ru
%F MM_2024_36_4_a4
Y. A. Kriksin; V. F. Tishkin. Entropic regularization of the discontinuous Galerkin method in conservative variables for three-dimensional Euler equations. Matematičeskoe modelirovanie, Tome 36 (2024) no. 4, pp. 77-91. http://geodesic.mathdoc.fr/item/MM_2024_36_4_a4/

[1] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Soc. Industr. and Appl. Math., Philadelphia, 1973, 59 pp. | MR | Zbl

[2] S. Osher, “Riemann solvers, the entropy condition, and difference approximations”, SIAM J. Numer. Anal., 21 (1984), 217–235 | DOI | MR | Zbl

[3] F. Bouchut, C. Bourdarias, B. Perthame, “A MUSCL method satisfying all the numerical entropy inequalities”, Math. Comput., 65 (1996), 1439–1461 | DOI | MR | Zbl

[4] V. V. Ostapenko, “Symmetric compact schemes with higher order conservative artificial viscosities”, Comput. Math. Math. Phys., 7:1 (2002), 980–999 | MR | Zbl

[5] E. Tadmor, “Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems”, Acta Numerica, 2003, 451–512 | DOI | MR | Zbl

[6] F. Ismail, P. Roe, “Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks”, J. Comput. Phys., 228 (2009), 5410–5436 | DOI | MR | Zbl

[7] U. S. Fjordholm, S. Mishra, E. Tadmor, “Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws”, SIAM J. Numer. Anal., 50:2 (2012), 544–573 | DOI | MR | Zbl

[8] P. Chandrashekar, “Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations”, Commun. Comput. Phys., 14:5 (2013), 1252–1286 | DOI | MR | Zbl

[9] E. Tadmor, “Entropy stable schemes”, Handbook of Num. Analysis, 17 (2016), 467–493 | DOI | MR

[10] X. Cheng, Y. Nie, “A third-order entropy stable scheme for hyperbolic conservation laws”, J. Hyperbolic Differ. Eq., 13:1 (2016), 129–145 | DOI | MR | Zbl

[11] A. A. Zlotnik, “Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations”, Comput. Math. Math. Phys., 57:4 (2017), 706–725 | DOI | DOI | MR | Zbl

[12] G. J. Gassner, A. R. Winters, D. A. Kopriva, “A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations”, Appl. Math. Comput., 272 (2016), 291–308 | MR | Zbl

[13] Cockburn, “An introduction to the discontinuous Galerkin method for convection-dominated Problems”, Lecture Notes in Mathematics, 1697, 1997, 150–268 | DOI | MR

[14] N. Krais, A. Beck, T. Bolemann, H. Frank, D. Flad, G. J. Gassner, F. Hindenlang, M. Hoffmann, T. Kuhn, M. Sonntag, C. D. Munz, “FLEXI: A high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws”, Computers Mathematics with Applications, 81 (2021), 186–219 | DOI | MR | Zbl

[15] M. E. Ladonkina, O. A. Nekliudova, V. F. Tishkin, “Ispolzovanie usrednenii dlia sglazhivaniia reshenii v razryvnom metode Galerkina”, Preprinty IPM im. M.V. Keldysha, 2017, 089, 32 pp.

[16] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Impact of different limiting functions on the order of solution obtained by RKDG”, MM, 5:4 (2013), 346–349 | MR | Zbl

[17] M. E. Ladonkina, V. F. Tishkin, “Godunov method: a generalization using piece-wise polynomial approximations”, Differential Equations, 51:7 (2015), 895–903 | DOI | DOI | MR | Zbl

[18] M. E. Ladonkina, V. F. Tishkin, “On Godunov-type methods of high order of accuracy”, Doklady Mathematics, 91:2 (2015), 189–192 | DOI | DOI | MR | Zbl

[19] V. F. Tishkin, V. T. Zhukov, E. E. Myshetskaya, “Justification of Godunov's scheme in the multidimensional case”, MM, 8:5 (2016), 548–556 | MR | Zbl

[20] Iu. A. Kriksin, V. F. Tishkin, “Entropiinaia reguliarizatsiia razryvnogo metoda Galerkina v odnomernykh zadachakh gazovoi dinamiki”, Preprinty IPM im. M.V. Keldysha, 2018, 100, 22 pp.

[21] M. D. Bragin, Iu. A. Kriksin, V. F. Tishkin, “Verifikatsiia odnogo metoda entropiinoi reguliarizatsii razryvnykh skhem Galerkina dlia uravnenii giperbolicheskogo tipa”, Preprinty IPM im. M.V. Keldysha, 2019, 018, 25 pp.

[22] M. D. Bragin, Y. A. Kriksin, V. F. Tishkin, “Discontinuous Galerkin Method with an Entropic Slope Limiter for Euler Equations”, MM, 12:5 (2020), 824–833 | DOI | MR | Zbl

[23] Iu. A. Kriksin, V. F. Tishkin, “Chislennoe reshenie zadachi Einfeldta na osnove razryvnogo metoda Galerkina”, Preprinty IPM im. M.V. Keldysha, 2019, 090, 22 pp.

[24] Y. A. Kriksin, V. F. Tishkin, “Entropy-Stable Discontinuous Galerkin Method for Euler Equations Using Nonconservative Variables”, MM, 13:3 (2021), 416–425 | DOI | MR | Zbl

[25] M. D. Bragin, Y. A. Kriksin, V. F. Tishkin, “Entropic regularization of the discontinuous Galerkin method in conservative variables for two-dimensional Euler equations”, Mathematical Models and Computer Simulations, 14:4 (2022), 578–589 | DOI | DOI | MR | Zbl

[26] Y. A. Kriksin, V. F. Tishkin, “Entropic Regularization of the Discontinuous Galerkin Method for Two-Dimensional Euler Equations in Triangulated Domains”, Mathematical Models and Computer Simulations, 15:5 (2023), 781–791 | DOI | DOI | MR

[27] D. Drikakis, C. Fureby, F. Grinstein, D. Youngs, “Simulation of transition and turbulence decay in the Taylor-Green vortex”, J. Turbul., 8:20 (2007), 1–12

[28] J. R. Bull, A. Jameson, “Simulation of the Taylor-Green vortex using high-order flux reconstruction schemes”, AIAA J., 53:9 (2015), 2750–2761 | DOI

[29] M. De La Llave Plata, V. Couaillier, M. C. Pape, “On the use of a high-order discontinuous Galerkin method for DNS and LES of wall-bounded turbulence”, Comput. Fluids, 176 (2018), 320–337 | DOI | MR | Zbl

[30] G. Gassner, A. Beck, “On the accuracy of high-order discretizations for underresolved turbulence simulations”, Theor. Comp. Fluid. Dyn., 27 (2013), 221–237 | DOI

[31] V. F. Tishkin, V. A. Gasilov, N. V. Zmitrenko et al, “Modern Methods of Mathematical Modeling of the Development of Hydrodynamic Instabilities and Turbulent Mixing”, Math. Models Comput. Simul., 13 (2021), 311–327 | DOI | DOI | MR | Zbl

[32] M. D. Bragin, “Influence of Monotonization on the Spectral Resolution of Bi-compact Schemes in the Inviscid Taylor-Green Vortex Problem”, Comput. Math. and Math. Phys., 62 (2022), 608–623 | DOI | MR | MR | Zbl

[33] I. P. Mysovskikh, Interpolyatsionnye kubaturnye formuly, Nauka, M., 1981, 336 pp. | MR