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Bragin, M. D.; Kriksin, Yu. A.; Tishkin, V. F.

Entropic regularization of the discontinuous Galerkin method in conservative variables for two-dimensional Euler equations. (Russian. English summary) Zbl 1484.76038

Mat. Model. 33, No. 12, 49-66 (2021).
Summary: The entropic regularization of the conservative stable discontinuous Galerkin method in conservative variables is constructed on the basis of a special slope limiter for the twodimensional Euler equations. This limiter ensures the fulfillment of the two-dimensional analogs of the monotonicity conditions and a discrete analog of the entropy inequality. The developed method was tested on two-dimensional model gas-dynamic problems.

Cited in 1 Document

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)

Keywords:

Euler equations; discontinuous Galerkin method; finite difference time discretization; conservation law; slope limiter; entropic inequality

Software:

FLEXI
Cite Review PDF
Full Text: DOI MNR

References:

[1] E. Tadmor, “Entropy stable schemes”, Handbook of Numerical Analysis, 17, 2016, 467-493
[2] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Soc. Industr. and Appl. Math., Philadelphia, 1973, 59 pp. · Zbl 0268.35062
[3] S. Osher, “Riemann solvers, the entropy condition, and difference approximations”, SIAM J. Numer. Anal., 21 (1984), 217-235 · Zbl 0592.65069 · doi:10.1137/0721016
[4] F. Bouchut, C. Bourdarias, B. Perthame, “A MUSCL method satisfying all the numerical entropy inequalities”, Math. Comput., 65 (1996), 1439-1461 · Zbl 0853.65091 · doi:10.1090/S0025-5718-96-00752-1
[5] E. Tadmor, “Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems”, Acta Numerica, 2003, 451-512 · Zbl 1046.65078 · doi:10.1017/S0962492902000156
[6] F. Ismail, P. Roe, “Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks”, J. Comput. Phys., 228 (2009), 5410-5436 · Zbl 1280.76015 · doi:10.1016/j.jcp.2009.04.021
[7] 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 · Zbl 1373.76121 · doi:10.4208/cicp.170712.010313a
[8] 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 · Zbl 1252.65150 · doi:10.1137/110836961
[9] X. Cheng, Y. Nie, “A third-order entropy stable scheme for hyperbolic conservation laws”, J. Hyperbolic Differ. Equ., 13:1 (2016), 129-145 · Zbl 1341.65030 · doi:10.1142/S021989161650003X
[10] V. V. Ostapenko, “Symmetric compact schemes with higher order conservative artificial viscosities”, Comput. Math. Math. Phys., 7:1 (2002), 980-999 · Zbl 1077.65507
[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 · Zbl 1457.76129 · doi:10.1134/S0965542517020166
[12] B. Cockburn, “An introduction to the discontinuous Galerkin method for convection-dominated Problems”, Lecture Notes in Mathematics, 1697, 1997, 150-268 · Zbl 0927.65120 · doi:10.1007/BFb0096353
[13] 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 · Zbl 1410.65393
[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 · Zbl 1461.76347 · doi:10.1016/j.camwa.2020.05.004
[15] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Ispolzovanie usrednenii dlia sglazhivaniia reshenii v razryvnom metode Galerkina”, Keldysh Institute preprints, 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 & CS, 5:4 (2013), 346-349 · Zbl 1313.76089
[17] M. E. Ladonkina, V. F. Tishkin, “Godunov method: a generalization using piece-wise polynomial approximations”, Differential Equations, 51:7 (2015), 895-903 · Zbl 1329.76180 · doi:10.1134/S0012266115070083
[18] M. E. Ladonkina, V. F. Tishkin, “On Godunov-type methods of high order of accuracy”, Doklady Mathematics, 91:2 (2015), 189-192 · Zbl 1319.76028 · doi:10.1134/S1064562415020222
[19] V. F. Tishkin, V. T. Zhukov, E. E. Myshetskaya, “Justification of Godunov-s scheme in the multidimensional case”, MM & CS, 8:5 (2016), 548-556 · Zbl 1363.76060
[20] Y. A. Kriksin, V. F. Tishkin, “Entropiinaia reguliarizastiia razryvnogo metoda Galerkina v odnomernykh zadachakh gazovoi dinamiki”, Keldysh Institute preprints, 2018, 100, 22 pp.
[21] M. D. Bragin, Y. A. Kriksin, V. F. Tishkin, “Verification of an entropic regularization method for discontinuous Galerkin schemes applied to hyperbolic equations”, Keldysh Institute preprints, 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”, Mathematical Models and Computer Simulations, 12:5 (2020), 824-833 · Zbl 1444.76068 · doi:10.1134/S2070048220050038
[23] Y. A. Kriksin, V. F. Tishkin, “Chislennoe reshenie zadachi Einfeldta na osnove razryvnogo metoda Galerkina”, Keldysh Institute preprints, 2019, 090, 22 pp.
[24] Y. A. Kriksin, V. F. Tishkin, “Entropy-Stable Discontinuous Galerkin Method for Euler Equations Using Nonconservative Variables”, Mathematical Models and Computer Simulations, 13:3 (2021), 416-425 · Zbl 1456.76073 · doi:10.1134/S2070048221030091
[25] M. D. Bragin, Y. A. Kriksin, V. F. Tishkin, “Entropy-Stable Discontinuous Galerkin Method for Two-Dimensional Euler Equations”, Mathematical Models and Computer Simulations, 13:5 (2021), 897-906 · Zbl 1461.76344 · doi:10.1134/S2070048221050069
[26] S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, A. N. Kraiko, G. P. Prokopov, Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, Nauka, M., 1976, 400 pp.
[27] R. Liska, B. Wendroff, “Comparison of several difference schemes on 1D and 2D test problems for the Euler equations”, SIAM J. Sci. Comput., 25:3, 995-1017 · Zbl 1096.65089 · doi:10.1137/S1064827502402120
[28] M. D. Bragin, B. V. Rogov, “Conservative limiting method for high-order bicompact schemes as applied to systems of hyperbolic equations”, Appl. Numer. Math., 151 (2020), 229-245 · Zbl 1434.65176 · doi:10.1016/j.apnum.2020.01.005
[29] M. D. Bragin, “Entropy stability of bicompact schemes in gas dynamics problems”, Mathematical Models and Computer Simulations, 13:4 (2021), 613-622 · Zbl 1455.76106 · doi:10.1134/S2070048221040086
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