Modern methods of mathematical modeling of the development of hydrodynamic instabilities and turbulent mixing

V. F. Tishkin ; V. A. Gasilov ; N. V. Zmitrenko ; P. A. Kuchugov ; M. E. Ladonkina ; Yu. A. Poveschenko

Matematičeskoe modelirovanie, Tome 32 (2020) no. 8, pp. 57-90.

Résumé

The study of the development of perturbations under the influence of various hydrodynamic instabilities, as well as the transition to turbulent mixing and turbulence, has been a subject of considerable interest over the past decades. This is primarily due to the importance of these phenomena for various fields of science and technology. It should be noted, that the final results of the turbulent flows characteristics study have not yet been obtained. This fact stimulates a great interest in this topic, both in sense of physical theory and in sense of approaches to mathematical modeling and numerical methods development. The capabilities of modern computer technology make it possible to carry out numerical experiments in two-dimensional and three-dimensional settings, and to analyze the features of new numerical methods. To date, there are a huge number of methods and their modifications are applied in practice. This review is devoted to the most promising, according to the authors, of them.

Détail

Keywords: mathematical modeling, high accuracy numerical methods, hydrodynamic instabilities, turbulent mixing.

Bibliographie

[1] B. N. Chetverushkin, V. F. Tishkin, “Primenenie vysokoproizvoditelnyx mnogoprocessornyx vychislenij v gazovoj dinamike”, Matematicheskoe modelirovanie: problemy i rezultaty, Nauka, M., 2003, 123–168

[2] A. N. Kolmogorov, “Lokalnaya struktura turbulentnosti v neszhimaemoj vyazkoj zhidkosti pri ochen bolshikh chislakh Rejnoldsa”, Dokl. AN SSSR, 30:4 (1941), 299–303

[3] T. B. Gatski, J. P. Bonnet, Compressibility, Turbulence and High Speed Flow, 2009

[4] D. C. Wilcox, Turbulence Modeling for CFD, 2006

[5] C. Hirsch, “Numerical Computation of Internal and External Flows”, Comp. Methods for Inviscid and Viscous Flows, 2 (1990)

[6] A. V. Rodionov, “On the use of Boussinesq approximation in turbulent supersonic jet modeling”, Int. J. Heat and Mass Transport, 53:5-6 (2010), 889–901 | DOI | Zbl

[7] P. Spalart, S. Allmaras, “A one-equation turbulence model for aerodynamics flows”, 30th Aerospace Sciences Meeting and Exhibit (1992, Reno, NV, USA)

[8] D. C. Wilcox, “Formulation of the $k-\omega$ Turbulence Model Revisited”, AIAA Journal, 46:11 (2008), 2823–2838 | DOI | MR

[9] K. Hanjalic, B. Launder, “A Reynolds stress model of turbulence and its application to thin shear flows”, J. of Fluid Mech., 52:4 (1972), 609–638 | DOI | Zbl

[10] F. R. Menter, “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications”, AIAA Journal, 32:8 (1994), 1598–1605 | DOI

[11] B. E. Morgan, M. E. Wickett, “Three-equation model for the self-similar growth of Rayleigh-Taylor and Richtmyer-Meshkov instabilities”, Phys. Rev. E, 91:4 (2015), 043002 | DOI | MR

[12] B. E. Morgan, O. Schilling, T. A. Hartland, “Two-length-scale turbulence model for self-simi-lar buoyancy-, shock-, and shear-driven mixing”, Phys. Rev. E, 97:1 (2018), 013104 | DOI

[13] P. Y. Chou, “On velocity correlations and the solutions of the equations of turbulent fluctuations”, Quarterly of Appl. Math., 3:1 (1945), 38–54 | DOI | MR | Zbl

[14] J. Rotta, “Statistische Theorie nichthomogener Turbulenz”, Zeitschrift für Physik, 129:6 (1951), 547–572 | DOI | MR | Zbl

[15] J. Smagorinsky, “General circulation experiments with the primitive equations I. The basic experiment”, Mon. Weather Rev., 91:3 (1963), 99–164 | 2.3.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[16] M. Lesieur, O. Metais, “New trends in large-eddy simulations of turbulence”, Ann. Rev. Fluid. Mech., 28 (1996), 45–82 | DOI | MR

[17] S. B. Pope, Turbulent flows, 2000 | MR | Zbl

[18] P. R. Spalart, W. H. Jou, M. Stretlets, S. R. Allmaras, “Comments on the Feasibility of LES for Wings and on the Hybrid RANS/LES Approach”, Advances in DNS/LES, Proceedings of the First AFOSR International Conference on DNS/LES, 1997

[19] M. Strelets, “Detached Eddy Simulation of Massively Separated Flows”, AIAA 2001, 2001–0879

[20] K. N. Volkov, V. N. Emelyanov, Modelirovanie krupnykh vikhrej v raschyotakh turbulentnykh techenij, Fizmatlit, M., 2008

[21] F. F. Grinstein, A. A. Gowardhan, J. R. Ristorcelli, “Implicit large-eddy simulation of shock-driven material mixing”, Philos. Trans. A Math. Phys. Eng. Sci., 371 (2003), 20120217 | DOI

[22] F. F. Grinstein, L. G. Margolin, W. J. Rider (eds.), Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics, 2010 | MR

[23] S. M. Bakhrakh i dr., “Raschet gazodinamicheskikh techenij na osnove metoda koncentracij”, DAN SSSR, 257:3 (1981), 566–569 | Zbl

[24] S. M. Bakhrakh, V. F. Spiridonov, “Metod koncentracij rascheta nestacionarnykh techenij sploshnoj sredy”, VANT. Ser. Mat. mod. fiz. processov, 1999, no. 4, 32–36

[25] A. Mignone, “High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates”, J. of Comp. Physics, 2014, no. 270, 784–814 | DOI | MR | Zbl

[26] A. A. Samarskij, V. F. Tishkin, A. P. Favorskij, M. Yu. Shashkov, “Operatornye raznostnye skhemy”, Diff. uravneniya, 17:7 (1981), 1317–1327 | MR

[27] K. Lipnikov, G. Manzini, M. Shashkov, “Mimetic finite difference method”, J. of Comp. Physics, 257 (2014), 1163–1227 | DOI | MR | Zbl

[28] A. Iserles, “Generalized leapfrog methods”, IMAJ. Numer. Anal., 6:3 (1986), 381–392 | DOI | MR | Zbl

[29] V. M. Goloviznin, A. A. Samarskij, “Raznostnaya approksimaciya konvektivnogo perenosa s prostranstvennym rasshchepleniem vremennoj proizvodnoj”, Matem. Mod., 10:1 (1998), 86–100 | MR | Zbl

[30] V. M. Goloviznin, “Balansno-xarakteristicheskij metod chislennogo resheniya uravnenij gazovoj dinamiki”, Dokl. RAN, 403:4 (2005), 459–464 | MR

[31] B. V. Rogov, “High-order accurate monotone compact running scheme for multidimensional hyperbolic equations”, Comp. Math. and Math. Physics, 53:2 (2013), 205–214 | DOI | MR | Zbl

[32] 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 | DOI | MR | Zbl

[33] B. V. Rogov, “Dispersive and dissipative properties of the fully discrete bicompact schemes of the fourth order of spatial approximation for hyperbolic equations”, Appl. Numer. Math., 139 (2019), 136–155 | DOI | MR | Zbl

[34] P. Rouch, Vychislitelnaya gidrodinamika, Mir, M., 1980

[35] A. G. Kulikovskij, N. V. Pogorelov, A. Yu. Semenov, Matematicheskie voprosy chislennogo resheniya giperbolicheskix uravnenij, Fizmatlit, M., 2001

[36] S. K. Godunov, “Raznostnyj metod chislennogo rascheta razryvnykh techenij gidrodinamiki”, Mat. sbornik, 47 (89):3 (1959), 271–306 | Zbl

[37] V. V. Ostapenko, “O monotonnosti raznostnykh sxem”, Sibirskij matematicheskij zhurnal, 39:5 (1998), 1111–1126 | MR | Zbl

[38] V. V. Ostapenko, “O silnoj monotonnosti tryokhtochechnykh raznostnykh sxem”, Sibirskij matematicheskij zhurnal, 39:6 (1998), 1357 | MR | Zbl

[39] V. V. Ostapenko, “On the strong monotonicity of nonlinear difference schemes”, Comp. Math. and Math. Phys., 38:7 (1998), 1119–1133 | MR | Zbl

[40] R. P. Fedorenko, “Primenenie raznostnykh sxem vysokoj tochnosti dlya chislennogo resheniya giperbolicheskikh uravnenij”, ZhVMiMF, 2:6 (1962), 1122–1128 | Zbl

[41] V. P. Kolgan, “Primenenie principa minimalnykh znachenij proizvodnoj k postroeniyu konechno-raznostnykh sxem dlya rascheta razryvnykh reshenij gazovoj dinamiki”, Uchenye zapiski CzAGI, 3:6 (1972), 68–77

[42] B. Van Leer, “Towards the ultimate conservative difference scheme II: Monotonicity and conservation combined in a second order scheme”, J. of Comp. Physics, 14:361 (1974), 370 | MR | Zbl

[43] B. Van Leer, “Towards the ultimate conservative difference schemes. III. Upstream-cente-red finite-difference schemes for ideal compressible flow”, J. of Comp. Physics, 23:3 (1977), 263–275 | DOI | MR | Zbl

[44] B. Van Leer, “Towards the ultimate conservative difference scheme V: A second order sequel to Godunov s method”, J. of Comp. Physics, 32 (1979), 101–136 | DOI | Zbl

[45] J. P. Boris, D. L. Book, “Flux-corrected transport II: Generalization of the method”, J. of Comp. Phys., 18 (1975), 248–283 | DOI | MR | Zbl

[46] S. T. Zalesak, “Fully multidimensional flux-corrected transport algorithms for fluids”, J. of Comp. Phys., 31 (1979), 335 | DOI | MR | Zbl

[47] A. I. Zhmakin, A. A. Fursenko, “On a monotonic shock-capturing difference scheme”, U.S.S.R. Comp. Math. and Math. Phys., 20:4 (1980), 218–227 | DOI | MR

[48] A. Harten, S. Osher, “Uniformly high-order accurate non-oscillatory schemes. I”, SIAM J. Numer. Anal., 27 (1987), 279–309 | DOI | MR

[49] A. Harten, B. Engquist, S. Osher, S. R. Chakravarthy, “Uniformly high order accuracy essentially non-oscillatory schemes. III”, J. of Comp. Phys., 71 (1987), 231–303 | DOI | MR | Zbl

[50] A. Harten, “On a class of high resolution total-variation-stable finite-difference schemes”, SIAM. J. Numer. Anal., 21 (1984), 1–23 | DOI | MR | Zbl

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

[52] H. G. Yee, R. F. Warming, A. Harten, “Implicit total variation diminishing (TVD) schemes for steady-state calculations”, J. of Comp. Phys., 57 (1985), 327–360 | DOI | MR | Zbl

[53] A. Harten, “ENO schemes with subset resolution”, J. of Comp. Phys., 83:2 (1989), 148–184 | DOI | MR | Zbl

[54] A. Harten, S. Osher, “Uniformly high-order accurate essentially non-oscillatory scheme. I”, SIAM J. Numer. Anal., 24:2 (1987), 279–309 | DOI | MR | Zbl

[55] D. K. Mao, “A treatment of discontinuities in shock-capturing finite difference methods”, J. of Comp. Phys., 92:2 (1991), 422–455 | DOI | MR | Zbl

[56] H. Choi, J. G. Liu, “The reconstruction of upwind fluxes for conservation laws: Its behavior in dynamic and steady state calculations”, J. of Comp. Phys., 144:2 (1998), 237–256 | DOI | MR | Zbl

[57] K. V. Vyaznikov, V. F. Tishkin, A. P. Favorskij, “Postroenie monotonnykh raznostnykh sxem povyshennogo poryadka approksimacii dlya sistem uravnenij giperbolicheskogo tipa”, Matem. Mod., 1:5 (1989), 95–120 | MR | Zbl

[58] A. Harten, B. Engquist, S. Osher, S. R. Chakravarthly, “Uniformly high-order accurate es-sentially non-oscillatory scheme. III”, J. of Comp. Phys., 71:2 (1987), 231–303 | DOI | MR | Zbl

[59] Ch. W. Shu, S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes”, J. of Comp. Phys., 77:2 (1988), 439–471 | DOI | MR | Zbl

[60] Ch. W. Shu, S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes. II”, J. of Comp. Phys., 83:1 (1989), 32–78 | DOI | MR | Zbl

[61] X. D. Liu, S. Osher, T. Chan, “Weighted essentially non-oscillatory schemes”, J. of Comp. Phys., 115 (1994), 200 | DOI | MR | Zbl

[62] G. Jiang, C. W. Shu, “Efficient implementation of weighted ENO schemes”, J. of Comp. Phys., 126 (1996), 202 | DOI | MR | Zbl

[63] M. Dumbser, M. Kaser, J. de la Puente, “Arbitrary High Order Finite Volume Schemes for Seismic Wave Propagation on Unstructured Meshes in 2D and 3D”, Geophys. Journal Int., 6 (2000), 134–165

[64] M. Dumbser, Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains, Shaker Verlag, Aachen, 2005

[65] M. Dumbser, C. D. Munz, “ADER discontinuous Galerkin schemes for aeroacoustics”, Comptes Rendus Mecanique, 333 (2005), 683–687 | DOI | Zbl

[66] M. Dumbser, C. D. Munz, “Arbitrary High Order Discontinuous Galerkin Schemes”, Numerical Methods for Hyperbolic and Kinetic Problems, IRMA series in math. and theoretical physics, eds. S. Cordier, T. Goudon, M. Gutnic, E. Sonnendrucker, EMS Publishing House, 2005, 295–333 | MR | Zbl

[67] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, V. S. Chevanin, “A version of essentially nonoscillatory high-order accurate difference schemes for systems of conservation laws”, Math. Models and Comp. Simul., 2:3 (2010), 304–316 | DOI | MR | Zbl

[68] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2nd Edition, Springer, 1999 | MR | Zbl

[69] E. F. Toro, V. A. Titarev, Solution of the generalized Riemann problem for advection-reaction equations, Roy Proc. Soc., London, 2002, 271–281 | MR | Zbl

[70] E. F. Toro, V. A. Titarev, “ADER: Arbitrary high order Godunov approach”, J. of Scientific Computing, 17:1–4 (2002), 609–618 | MR | Zbl

[71] E. F. Toro, V. A. Titarev, “ADER schemes for three-dimensional nonlinear hyperbolic systems”, J. of Comp. Phys., 204 (2005), 715–736 | DOI | MR | Zbl

[72] E. F. Toro, V. A. Titarev, “ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions”, J. of Comp. Phys., 202 (2005), 196–215 | DOI | MR | Zbl

[73] M. Dumbser, Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains, Shaker Verlag, Aachen, 2005

[74] M. Dumbser, C. D. Munz, “ADER discontinuous Galerkin schemes for aeroacoustics”, Comptes Rendus Mecanique, 333 (2005), 683–687 | DOI | Zbl

[75] M. Dubiner, “Spectral methods on triangles and other domains”, Journal of Sci. Computing, 1991, no. 6, 345–390 | DOI | MR | Zbl

[76] M. Dumbser et al., “Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems”, J. of Comp. Phys., 221:2 (2007), 693–723 | DOI | MR | Zbl

[77] I. Abalakin, P. Bakhvalov, T. Kozubskaya, “Edge-based reconstruction schemes for unstructured tetrahedral meshes”, Intern. J. for numer. methods in fluids, 81:6 (2016), 331–356 | DOI | MR

[78] P. Bakhvalov, T. Kozubskaya, “EBR-WENO scheme for solving gas dynamics problems with discontinuities on unstructured meshes”, Comp. Fluids, 157 (2017), 312–324 | DOI | MR | Zbl

[79] S. K. Godunov, “Raznostnyj metod chislennogo rascheta razryvnykh techenij gidrodinamiki”, Mat. sbornik, 47 (89):3 (1959), 271–306 | Zbl

[80] W. H. Teukolsky, S. A. Vetterling, W. T. Flannery, BP «Section 10.1.2. Lax Method», Numerical Recipes: The Art of Scientific Computing, 3rd ed., Cambridge University Press, New York, 2007 (data obrascheniya 01.03.2015) http://apps.nrbook.com/empanel/index.html#pg=1034

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

[82] V. V. Rusanov, “Raschet vzaimodejstviya nestacionarnykh udarnykh voln s prepyatstviyami”, ZhVMiMF, I:2 (1961), 267–279

[83] A. Kurganov, E. Tadmor, “New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection-Diffusion Equations”, J. of Comp. Phys., 160 (2000), 241–282 | DOI | MR | Zbl

[84] J. Balbas, E. Tadmor, C. C. Wu, “Central non-oscillatory schemes for one- and two-dimensional MHD equations”, J. of Comp. Phys., 201:1 (2004), 261–285 | DOI | MR | Zbl

[85] J. Qiu, C. W. Shu, “On the construction, comparison, and local characteristic decompositions for high order central WENO schemes”, J. of Comp. Phys., 183 (2002), 187–209 | DOI | MR | Zbl

[86] E. Tadmor, J. Tanner, “An adaptive order Godunov type central scheme «Hyperbolic Problems: Theory, Numerics, Applications»”, Proc. of the 9th international conference held at CalTech (2002 Mar.), eds. Hou T., Tadmor E., Springer, 2003, 871–880 | MR | Zbl

[87] P. Arminjon, M. C. Viallon, A. Madrane, “A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservation laws on unstructured grids”, International J. of Comp. Fluid Dynam., 9:1 (1997), 1–22 | MR | Zbl

[88] S. Karni, E. Kirr et al, “Compressible two-phase flows by central and upwind schemes”, Math. Mod. and Num. Anal., 38:3 (2004), 477–493 | DOI | MR | Zbl

[89] P. Woodward, Ph. Colella, “The numerical simulation of two-dimensional fluid flow with strong shocks”, J. of Comp. Phys., 54:1 (1984), 115–173 | DOI | MR | Zbl

[90] Ph. Colella, P. R. Woodward, “The piecewise parabolic method (PPM) for gas-dynamical simulations”, J. of Comp. Phys., 54:1 (1984), 174–201 | DOI | MR | Zbl

[91] R. H. Cohen, A. A. Mirin, ASCI turbulence and instability modeling using the piecewise parabolic method, UCRL-TB-125580, Rev. 4, Lawrence Livermore National Laboratory, 1999 (data obrascheniya 01.03.2015) https://computation.llnl.gov/casc/asciturb/asciturb.html

[92] P. Colella, L. F. Henderson, “The von Neumann paradox for the diffraction of weak shock waves”, Journal Fluid Mechanics, 213 (1990), 71–94 | DOI | MR

[93] M. J. Berger, P. Colella, “Local adaptive mesh refinement for shock hydrodynamics”, J. of Comp. Phys., 82:1 (1989), 64–84 | DOI | MR | Zbl

[94] Yu. V. Yanilkin, V. P. Stacenko, V. I. Kozlov, Matematicheskoe modelirovanie turbulentnogo peremeshivaniya v szhimaemykh sredax, RFYaCz VNIIEF, Sarov, 2009

[95] W. J. Rider, “Reconsidering remap methods”, Intern J. for Numerical Methods in Fluids, 76 (2014), 587 | DOI | MR

[96] M. V. Popov, S. D. Ustyugov, “Piecewise parabolic method on local stencil for gasdynamic simulations”, Comp. Math. and Math. Phys., 47:12 (2007), 1970–1989 | DOI | MR

[97] M. V. Popov, S. D. Ustyugov, “Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics”, Comp. Math. and Math. Phys., 48:3 (2008), 477–499 | DOI | MR | Zbl

[98] S. D. Ustyugov, M. V. Popov, A. G. Kritsuk, M. L. Norman, “Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation”, J. of Comp. Phys., 228 (2009), 7614–7633 | DOI | MR | Zbl

[99] A. Suresh, H. T. Huynh, “Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping”, J. of Comp. Phys., 136 (1997), 83 | DOI | MR | Zbl

[100] W. J. Rider et al., “Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations”, J. of Comp. Phys., 225 (2007), 1827 | DOI | MR | Zbl

[101] (data obrascheniya 01.03.2015) http://www.astro.virginia.edu/VITA/ATHENA/bow-shock.html

[102] W. H. Reed, T. R. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-79, USA, 1973 https://www.osti.gov/servlets/purl/4491151

[103] Proc. ECCOMAS Thematic Conference: European Conf. on High Order Nonlinear Numerical Methods for Evolutionary PDEs: Theory and Appl. HONOM (2017, 27.03–31.03)

[104] B. Cockburn, C.-W. Shu, “The Runge-Kutta discontinuous Galerkin method for conserva-tion laws V: Multidimensional systems”, J. Comp. Phys., 141 (1998), 199–224 | DOI | MR | Zbl

[105] A. V. Volkov, “Osobennosti primeneniya metoda Galerkina k resheniyu prostranstvennykh uravnenij Navie-Stoksa na nestrukturirovannykh geksaedralnykh setkakh”, Uchenye zapiski CzAGI, XL:6 (2009) | Zbl

[106] R. Nastase, D. J. Mavriplis, “High-order discontinuous Galerkin methods using an hp-multigrid approach”, J. Comp. Phys., 213 (2006), 330–357 | DOI | MR | Zbl

[107] H. Luo, J. D. Baum, R. Löhner, “Fast p-multigrid discontinuous Galerkin method for compressibleow at all speeds”, AIAA J., 2008, no. 46, 635–652 | DOI

[108] F. Bassi, S. Rebay, “Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations”, Int. J. Numer. Meth. Fluids, 40 (2002), 197–207 | DOI | MR | Zbl

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

[110] S. M. Bosnyakov, S. V. Mikhailov, V. Yu. Podaruev, A. I. Troshin, “Nestacionarnyj razryvnyj metod Galerkina vysokogo poryadka tochnosti dlya modelirovaniya turbulentnykh techenij”, Matem. Mod., 30:5 (2018), 37–56

[111] M. M. Krasnov et al., “Numerical solution of the Navier-Stokes equations by discontinuous Galerkin method”, J. of Physics: Conf. Ser., 815:1 (2017)

[112] M. M. Krasnov, P. A. Kuchugov, M. E. Ladonkina, V. F. Tishkin, “Discontinuous Galerkin method on three-dimensional tetrahedral grids: Using the operator programming method”, Math. Models and Comp. Simul., 9:5 (2017), 529–543 | DOI | MR | Zbl

[113] M. M. Krasnov, M. E. Ladonkina, “Discontinuous Galerkin method on three-dimensional tetrahedral grids. The use of template metaprogramming of the C++ language”, Programming and Computer Software, 43:3 (2017), 172–183 | DOI | MR

[114] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Application of the RKDG method for gas dynamics problems”, Math. Models and Comp. Simul., 6:4 (2014), 397–407 | DOI | MR | Zbl

[115] K. Yasue, M. Furudate, N. Ohnishi, K. Sawada, “Implicit discontinuous Galerkin method for RANS simulation utilizing pointwise relaxation algorithm”, Communications in Computational Physics, 7:3 (2010), 510–533 | DOI | MR | Zbl

[116] 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

[117] L. Krivodonova, “Limiters for high-order discontinuous Galerkin methods”, J. of Comp. Phys., 226:1 (2007), 276–296 | DOI | MR

[118] C. W. Shu, “High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments”, J. of Comp. Phys., 316 (2016), 598–613 | DOI | MR | Zbl

[119] H. Luo, J. D. Baum, R. A. Lohner, “Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids”, J. of Comp. Phys., 225:1 (2007), 686–713 | DOI | MR | Zbl

[120] J. Zhu, X. Zhong, C. W. Shu, J. Qiu, “Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes”, J. of Comp. Phys., 248 (2013), 200–220 | DOI | MR | Zbl

[121] M. Dumbser, “Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations”, Computers Fluids, 39:1 (2010), 60–76 | DOI | MR | Zbl

[122] J. Peraire, P. O. Persson, “High-Order Discontinuous Galerkin Methods for CFD”, Advances in CFD, chap. 5, v. 2, Adaptive High-Order Methods in Computational Fluid Dynamics, World Scientic Publishing Co, 2011 | MR | Zbl

[123] A. V. Volkov, C. V. Lyapunov, “Monotonizaciya metoda konechnogo elementa v zadachax gazovoj dinamiki”, Uch. zapiski CzAGI, XL:4 (2009), 15–27

[124] M. E. Ladonkina, O. A. Neklyudova, V. F. Tishkin, “Constructing a Limiter Based on Averaging the Solutions for the Discontinuous Galerkin Method”, Math. Models Comp. Simul., 11 (2019), 61–73 | DOI | MR

[125] T. Haga, K. Sawada, An improved slope limiter for high-order spectral volume methods solving the 3D compressible Euler equations, 2009

[126] Cockburn, C. W. Shu, “The local discontinuous Galerkin method for time dependent convection diffusion system”, SIAM J. Numer. Anal., 35:6 (1998), 2440–2463 | DOI | MR | Zbl

[127] V. V. Vlasenko, A. V. Volkov, A. I. Troshin, “Vybor metoda approksimacii vyazkikh chlenov v metode Galerkina s razryvnymi bazisnymi funkciyami”, Uchenye zapiski CzAGI, XLIV:3 (2013)

[128] S. Gottlieb, C. W. Shu, E. Tadmor, “Strong stability-preserving high-order time discretization methods”, SIAM Review, 43:1 (2001), 89–112 | DOI | MR | Zbl

[129] J. Spiteri Raymond, J. Ruuth Steven, “A New Class of Optimal High-Order Strong Stability-Preserving Time Discretization Methods”, SIAM J. Numer. Anal., 40:2 (2002), 469–491 | DOI | MR | Zbl

[130] P. Rasetarinera, M. Y. Hussaini, “An efficient implicit discontinuous spectral Galerkin method”, J. Comp. Phys., 172 (2001), 718–738 | DOI | Zbl

[131] R. Hartmann, “Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations”, Int. J. Numer. Meth. Fluids, 51 (2006), 1131–1156 | DOI | MR | Zbl

[132] R. Hartmann, P. Houston, “Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: Method formulation”, Int. J. Numer. Anal. Model., 2006, no. 3, 1–20 | MR | Zbl

[133] V. Dolešj, “Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows”, Commun. Comp. Phys., 4 (2008), 231–274 | MR | Zbl

[134] K. Yasue, M. Furudate, N. Ohnishi, K. Sawada, “Implicit discontinuous Galerkin method for RANS simulation utilizing pointwise relaxation algorithm”, Commun. Comp. Physics, 7:3 (2010), 510–533 | DOI | MR | Zbl

[135] A. Jameson, S. Yoon, “Lower-upper implicit schemes with multiple grids for the Euler equations”, AIAA J., 25 (1987), 929–935 | DOI

[136] A. Klockner, T. Warburton, J. S. Hesthaven, “Nodal discontinuous Galerkin methods on graphics processors”, J. of Comp. Phys., 228:21 (2009), 7863–7882 | DOI | MR | Zbl

[137] J. Lou, Y. Xia, L. Luo, H. Luo, J. R. Edwards, F. Mueller, “OpenACC-based GPU Acceler-ation of a p-multigrid Discontinuous Galerkin Method for Compressible Flows on 3D Un-structured Grids”, 53rd AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, AIAA 2015-0822

[138] J. Chan et al., “GPU-accelerated discontinuous Galerkin methods on hybrid meshes”, J. of Comp. Phys., 2015, 318 | MR | Zbl

[139] M. M. Krasnov, “Operatornaya biblioteka dlya resheniya trexmernykh setochnykh zadach matematicheskoj fiziki s ispolzovaniem graficheskix plat s arxitekturoj CUDA”, Matem. Mod., 27:3 (2015), 109–120 | Zbl

[140] M. M. Krasnov, “Parallel algorithm for computing points on a computation front hyperplane”, Comp. Math. Math. Phys., 55:1 (2015), 140–147 | DOI | Zbl

[141] Ph. Mocz et al., “A discontinuous Galerkin method for solving theuid and MHD equations in astrophysical simulations”, Mon. Not. R. Astron. Soc., 437 (2014), 397–414 | DOI | MR

[142] M. M. Basko, “The method of artificial viscosity for computing one-dimensional flows”, U.S.S.R. Comp. Math. and Math. Phys., 30:2 (1990), 176–182 | DOI | MR | Zbl

[143] M. P. Arminjon, A. Dervieux, Construction of TVD-like artificial viscosities on 2-dimensional arbitrary FEM grids, INRIA Research Report 1111, 1989 | Zbl

[144] W. D. Schultz, “Two-dimensional Lagrangian hydrodynamic difference equations”, Methods of Computational Physics, v. 3, Academic Press, New York, 1964

[145] I. V. Popov, I. V. Fryazinov, “Adaptive artificial viscosity for multidimensional gas dynamics for Euler variables in Cartesian coordinates”, Math. Models and Comp. Simul., 2:4 (2010), 429–442 | DOI | MR | MR | Zbl

[146] E. V. Stupochenko, S. A. Losev, A. I. Osipov, Relaksacionnye processyv udarnykh volnakh, Nauka, M., 1965

[147] B. N. Chetverushkin, Kineticheski-soglasovannye skhemy v gazovoj dinamike, Izd-vo MGU, M., 1999

[148] A. S. Boldarev, V. A. Gasilov, O. G. Olkhovskaya, “K resheniyu giperbolicheskix uravnenij na nestrukturirovannykh setkakh”, Matem. Mod., 8:3 (1996), 51–78 | MR | Zbl

[149] V. F. Kuropatenko, “Metod postroeniya raznostnykh skhem dlya chislennogo integriro?vaniya uravnenij gazodinamiki”, Izvestiya VUZov. Matematika, 1962, no. 3, 75–83 | MR

[150] R. B. Christensen, Godunov methods on a staggered mesh an improved artificial viscosity, Lawrence Livermore Laboratory report, UCRL-JC-105269, 1991

[151] O. V. Diyankov, I. V. Glazyrin, S. V. Koshelev, “MAG two-dimensional resistive MHD-code using an arbitrary moving coordinate system”, Comp. Phys. Commun., 106 (1997), 76–94 | DOI | Zbl

[152] Yu. A. Poveshhenko i dr., “Ob odnoj dvukhslojnoj polnostyu konservativnoj raznostnoj skheme gazovoj dinamiki v ejlerovykh peremennykh s adaptivnoj regulyarizaciej resheniya”, Preprinty IPM im. M.V. Keldysha, 2019, 014, 23 pp. | DOI

[153] S. R. Kennon, G. S. Dulikravich, A posteriori optimization of computational grid, AIAA-85-0483, 1985 | Zbl

[154] N. A. Darin, V. I. Mazhukin, A. A. Samarskii, “A finite-difference method for solving the equations of gas dynamics using adaptive nets which are dynamically associated with the solution”, U.S.S.R. Comp. Math. and Math. Phys., 28:4 (1988), 164–174 | DOI | MR | MR

[155] G. Beckett, J. A. Mackenzie, A. Ramage, D. M. Sloan, “On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution”, J. Comp. Phys., 2001, no. 167, 372–392 | DOI | MR | Zbl

[156] K. J. Fidkowski, D. L. Darmofal, Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics: Overview and Recent Results, AIAA 2009-1303

[157] W. Huang, R. D. Russell, Adaptive Moving Mesh Methods, Springer Science+Business Media, LLC, 2011 | MR | Zbl

[158] D. F. Hawken, J. J. Gottlieb, J. S. Hansen, “Review of some adaptive node-movement techniques in finite element and finite-difference solutions of partial differential equations”, J. Comp. Phys., 95 (1991), 254–302 | DOI | MR | Zbl

[159] J. F. Dannenhoffer, “A comparison of adaptive-grid redistribution and embedding for steady transonic flows”, Intern. J. for Numerical Methods in Engineering, 1991, no. 32, 653–663 | DOI | Zbl

[160] M. Berger, J. Oliger, “Adaptive mesh refinement for hyperbolic partial differential equations”, J. Comp. Phys., 53 (1984), 484–512 | DOI | MR | Zbl

[161] R. Klein, “Star formation with 3-D adaptive mesh refinement: the collapse and fragmentation of molecular clouds”, J. of Comp. and App. Math., 16:109(1–2) (2011), 123–152 | DOI

[162] Kumar Khattri Sanjay, “Grid Generation and Adaptation by Functionals”, Computational Applied Mathematics, 26:2 (2007), 235–249 | MR | Zbl

[163] A. E. Luczkij, A. V. Plenkin, “Chislennoe modelirovanie na adaptivnykh setkakh vzaimodejstviya udarnykh voln s pogranichnymi sloyami”, Prepr. IPM im. M.V. Keldysha, 2016, 136, 15 pp. | DOI

[164] A. L. Afendikov i dr., “Algoritm mnogourovnevoj adaptacii setok po kriteriyam na osnove vejvlet-analiza dlya zadach gazovoj dinamiki”, Preprinty IPM im. M.V. Keldysha, 2015, 097, 22 pp.

[165] G. Warren, W. K. Anderson, J. Thomas, S. Krist, “Grid convergence for adaptive methods”, AIAA 10th Comp. Fluid Dynamics Conference (1991)

[166] K. G. Powell, P. L. Roe, J. Quirk, “Adaptive-Mesh Algorithms for Computational Fluid Dynamics”, Algorithmic Trends in Computational Fluid Dynamics, ICASE/NASA LaRC Series, eds. Hussaini M.Y., Kumar A., Salas M.D., Springer, New York, NY, 1993 | MR

[167] O. Antepara, O. Lehmkuhl, J. Chiva, R. Borrell, “Parallel Adaptive Mesh Refinement Simulation of the Flow Around a Square Cylinder at Re=22000”, Procedia Eng., 61 (2013), 246–250 | DOI

[168] G. Li, X. Fuamp; F. Wang, “High-resolution multi-code implementation of unsteady Navier–Stokes flow solver based on paralleled overset adaptive mesh refinement and high-order lowdissipation hybrid schemes”, Int. J. of Comp. Fluid Dyn., 2017 | DOI | MR

[169] A. Thornber, J. Griffond, O. Poujade et al., “Late-time growth rate, mixing, and anisotropy in multimode narrowband Richtmyer-Meshkov instability: The $\theta$-group collaboration”, Physics of Fluids, 29:10 (2017), 105107 | DOI

[170] G. Dimonte, D. L. Youngs, A. Dimits et al., “A comparative study of the turbulent Rayleigh-Taylor instability using high resolution three dimensional numerical simulations: The alpha-group collaboration”, Physics of Fluids, 16:5 (2004), 1668–1693 | DOI