Geodesic
Application of discontinuous Galerkin method to modeling of two-dimensional flows of a multicomponent ideal gases mixture using local adaptive mesh refinement
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 2, pp. 244-258.

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In this article a numerical algorithm is developed for solving of gas dynamics equations for a mixture of ideal gases on adaptive locally refined grids. The algorithm is based on discontinuous Galerkin method. To avoid the appearance of non-physical oscillations near the discontinuities, the Barth-Jespersen limiter is used. The numerical algorithm is based on the data structure and algorithms of the p4est library. In present work the numerical simulation of one problem of Richtmyer-Meshkov instability development is considered and the triple point problem is solved using the developed numerical algorithm of high accuracy order. The obtained results are in good agreement with the well-known numerical solutions. The pictures plotted basing on the solution describe in detail the dynamics of the complex flows under consideration.
Keywords: discontinuous Galerkin method, gas dynamics equations, multicomponent gas mixture, parallel computing, adaptive mesh refinement
Mots-clés : p4est. 
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R. V. Zhalnin; V. F. Masyagin; E. E. Peskova; V. F. Tishkin. Application of discontinuous Galerkin method to modeling of two-dimensional flows of a multicomponent ideal gases mixture using local adaptive mesh refinement. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 2, pp. 244-258. http://geodesic.mathdoc.fr/item/SVMO_2019_21_2_a6/

[1] R. V. Zhalnin, V. F. Masyagin, M. E. Ladonkina, V. F. Tishkin, “Solving the problem of non-stationary filtration of substance by the discontinuous Galerkin method on unstructured grids”, Computational Mathematics and Mathematical Physics, 56:6 (2016), 989–998 (In Russ.) | DOI | MR | Zbl

[2] M. M. Krasnov, M. E. Ladonkina, V. F. Tishkin, “Implementation of the Galerkin discontinous method in the DGM software package”, Keldysh Institute preprints, 2018, 245, 31 pp. (In Russ.)

[3] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Construction of the limiter based on averaging of solutions for discontinued Galerkin method”, Matematicheskoye Modelirovaniye, 30:5 (2018), 99–116 (In Russ.)

[4] Y. A. Kriksin, V. F. Tishkin, “Variational entropic regularization of discontinuous Galerkin method for gas dynamics equations”, Matematicheskoye Modelirovaniye, 31:5 (2019), 69–84 (In Russ.) | DOI

[5] 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. (In Russ.)

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

[8] C. Burstedde, L. C. Wilcox, O. Ghattas, “p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees”, SIAM Journal on Scientific Computing, 33:3 (2011), 1103–1133 | DOI | MR | Zbl

[9] V. V. Rusanov, “The calculation of the interaction of non-stationary shock waves and obstacles”, USSR Computational Mathematics and Mathematical Physics, 1:2 (1961), 267–279

[10] P. D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation”, Communications on Pure and Applied Mathematics, 7:1 (1954), 159–193 | DOI | MR | Zbl

[11] E. F. Toro, M. Spruce, W. Speares, “Restoration of the Contact Surface in the HLL-Riemann Solver”, Shock Waves, 4 (1994), 25–34 | DOI | Zbl

[12] C.-W. Shu, S. Osher, “Efficient implementation of essentially non-oscillatory shock capturing schemes”, Journal of Computational Physics, 77 (1988), 439–471 | DOI | MR | Zbl

[13] T. J. Barth, D. C. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA Paper 89-0366, 1989

[14] M. J. Berger, P. Colella, “Local adaptive mesh refinement for shock hydrodynamics”, Journal of Computational Physics, 82 (1989), 64–84 | DOI | Zbl

[15] F. Poggi, M.-H. Thorembey, G. Rodrigues, “Velocity measurements in turbulent gaseous mixtures induced by Richtmyer–Meshkov instability”, Phisics of Fluids, 10:11 (1998), 2698–2700 | DOI

[16] R. V. Zhalnin, N. V. Zmitrenko, M. E. Ladonkina, V. F. Tishkin, “Numerical simulation of Richtmyer-Meshkov instability development using the difference schemes of high order of accuracy”, Matematicheskoye Modelirovaniye, 19:10 (2007), 61–66 (In Russ.) | MR | Zbl

[17] M. Kucharik, R. V. Garimella, S. P. Schofield, M. J. Shashkov, “A comparative study of interface reconstruction methods for multi-material ale simulations”, Journal of Computational Physics, 229:7 (2009), 2432–2452 | DOI | MR

[18] M. Kucharik, M. J. Shashkov, “Conservative multi-material remap for staggered multi-material arbitrary lagrangian-eulerian methods”, Journal of Computational Physics, 258 (2014), 268–304 | DOI | MR | Zbl