Variational entropic regularization of discontinuous Galerkin method for gas dynamics equations
Matematičeskoe modelirovanie, Tome 31 (2019) no. 5, pp. 69-84.

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A constructive version of an arbitrary accuracy discontinuous Galerkin (DG) method solving gas dynamics equations is proposed. This DG method is based on the new variational principle of entropic regularization ensuring the implementation of discrete analogs of the conservation laws of mass, momentum, total energy and entropic inequality.
Mots-clés : gasdynamic equations
Keywords: discontinuous Galerkin method, conservation laws, variational principle, entropic inequality.
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Y. A. Kriksin; V. F. Tishkin. Variational entropic regularization of discontinuous Galerkin method for gas dynamics equations. Matematičeskoe modelirovanie, Tome 31 (2019) no. 5, pp. 69-84. http://geodesic.mathdoc.fr/item/MM_2019_31_5_a4/

[1] A.G. Kulikovsky, N.V. Pogorelov, A.Yu. Semenov, Mathematical aspects of numerical solution of hyperbolic systems, Taylor Francis Inc., 2000 | MR

[2] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, v. 6, 2nd ed., Butterworth-Heinemann, 1987, 552 pp.

[3] S.K. Godunov, A.V. Zabrodin, M.Ia. Ivanov, A.N. Kraiko, G.P. Prokopov, Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki, Nauka, M., 1976, 400 pp.

[4] B. Cockburn, “An Introduction to the Discontinuous Galerkin Method for Convection Dominated Problems”, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, 1697, 1998, 151–268 | MR | Zbl

[5] B. Cockburn, G.E. Karniadakis, Ch.-W. Shu, The Development of Discontinuous Galerkin Methods, IMA Preprint Series # 1662, Institute for Mathematics and its Applications, University of Minnisota, December 1999, 51 pp. | MR

[6] B. Cockburn, Ch.-W. Shu, “Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems”, Journal of Scientific Computing, 16:3 (2001), 173–261 | DOI | MR | Zbl

[7] D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, “Unified analysis of discontinuous Galerkin methods for elliptic problems”, SIAM J. on Numer. Anal., 29 (2002), 1749–1779 | DOI | MR

[8] M.E. Ladonkina, O.A. Neklyudova, V.F. Tishkin, “Application of averaging to smooth the solution in DG method”, Keldysh Institute preprints, 2017, 089, 32 pp. | Zbl

[9] M.E. Ladonkina, O.A. Neklyudova, V.F. Tishkin, “Impact of Different Limiting Functions on the Order of Solution Obtained by RKDG”, Math. Mod. Comp. Simul., 5:4 (2013), 346–349 | DOI | MR | Zbl

[10] M.M. Krasnov, P.A. Kuchugov, M.E. Ladonkina, V.F. Tishkin, “Obobshchenie metoda Godunova, ispolzuiushchego kusochno-polinomialnye approksimatsii v mnogomernom sluchae”, Supervychisleniia i matematicheskoe modelirovanie, Trudy XVI mezhdunarodnoi konferentsii (3–7 oktiabria 2016 g.), ed. R.M. Shagaliev, FGUP “RFIATSVNIIEF”, Sarov, 2017, 168–183

[11] 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

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

[13] P.G. Le Floch, J.M. Mercier, C. Rohde, “Fully discrete, entropy conservative schemes of arbitrary order”, SIAM J. Numer. Anal., 40:5 (2002), 1968–1992 | DOI | MR

[14] F. Lagoutiere, C.R. Acad, “A non-dissipative entropic scheme for convex scalar equations via discontinuous cell-reconstruction”, Comp. Rendus Mathematique, 338:7 (2004), 549–554 | DOI | MR | Zbl

[15] H. Zakerzadeh, U.S. Fjordholm, “High-order accurate, fully discrete entropy stable schemes for scalar conservation laws”, IMA J. of Numerical Analysis, 36:2 (2016), 633–654 | DOI | MR | Zbl

[16] U.S. Fjordholm, R. Kappeli, S. Mishra, E. Tadmor, “Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws”, Found. Comput. Math., 17:3 (2017), 763–827 | DOI | MR | Zbl

[17] A.R. Winters, G.J. Gassner, “A Comparison of Two Entropy Stable Discontinuous Galerkin Spectral Element Approximations for the Shallow Water Equations with Non-Constant Topography”, Journal of Computational Physics, 301 (2015), 357–376 | DOI | MR | Zbl

[18] 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”, Applied Mathematics and Computation, 272 (2016), 291–308 | DOI | MR | Zbl

[19] T. Chen, Ch.-W. Shu, “Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws”, J. Comp. Phys., 345 (2017), 427–461 | DOI | MR | Zbl

[20] I.B. Petrov, A.S. Kholodov, “Regularization of discontinuous numerical solutions of equations of hyperbolic type”, USSR Computational Mathematics and Mathematical Physics, 24:4 (1984), 128–138 | DOI | MR | Zbl

[21] Y.A. Kriksin, V.F. Tishkin, “Entropic regularization of discontinuous Galerkin method in one-dimensional problems of gas dynamics”, Keldysh Institute preprints, 2018, 100, 22 pp.