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Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 12, Pages 2055–2075 (Mi zvmmf211)  

This article is cited in 25 scientific papers (total in 25 papers)

Piecewise parabolic method on local stencil for gasdynamic simulations

M. V. Popov, S. D. Ustyugov

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

Abstract: A numerical method based on piecewise parabolic difference approximations is proposed for solving hyperbolic systems of equations. The design of its numerical scheme is based on the conservation of Riemann invariants along the characteristic curves of a system of equations, which makes it possible to discard the four-point interpolation procedure used in the standard piecewise parabolic method (PPM) and to use the data from the previous time level in the reconstruction of the solution inside difference cells. As a result, discontinuous solutions can be accurately represented without adding excessive dissipation. A local stencil is also convenient for computations on adaptive meshes. The new method is compared with PPM by solving test problems for the linear advection equation and the inviscid Burgers equation. The efficiency of the methods is compared in terms of errors in various norms. A technique for solving the gas dynamics equations is described and tested for several one-and two-dimensional problems.

Key words: numerical methods in gas dynamics, local stencil, Riemann invariants, numerical methods, hyperbolic systems of equations, PPM, PPML.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:12, 1970–1989

Bibliographic databases:

UDC: 519.633
Received: 30.05.2007
Revised: 05.06.2007

Citation: M. V. Popov, S. D. Ustyugov, “Piecewise parabolic method on local stencil for gasdynamic simulations”, Zh. Vychisl. Mat. Mat. Fiz., 47:12 (2007), 2055–2075; Comput. Math. Math. Phys., 47:12 (2007), 1970–1989

Citation in format AMSBIB
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\by M.~V.~Popov, S.~D.~Ustyugov
\paper Piecewise parabolic method on local stencil for gasdynamic simulations
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2007
\vol 47
\issue 12
\pages 2055--2075
\mathnet{http://mi.mathnet.ru/zvmmf211}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2394964}
\transl
\jour Comput. Math. Math. Phys.
\yr 2007
\vol 47
\issue 12
\pages 1970--1989
\crossref{https://doi.org/10.1134/S0965542507120081}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-37649009615}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. V. Popov, S. D. Ustyugov, “Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics”, Comput. Math. Math. Phys., 48:3 (2008), 477–499  mathnet  crossref  mathscinet  zmath  isi
    2. Popov M.V., Ustyugov S.D., “Piecewise Parabolic Method on a Local Stencil for Gas Dynamics and MHD”, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, 1048, 2008, 436–439  crossref  mathscinet  adsnasa  isi  scopus
    3. Ustyugov S.D., “Realistic Simulation of Local Solar Supergranulation”, Exploring the Solar System and the Universe, AIP Conference Proceedings, 1043, 2008, 234–237  crossref  adsnasa  isi  scopus
    4. Ustyugov S.D., Popov M.V., Kritsuk A.G., Norman M.L., “Piecewise parabolic method on a local stencil for magnetized supersonic turbulence simulation”, J. Comput. Phys., 228:20 (2009), 7614–7633  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Popov M.V., Ustyugov S.D., “Piecewise parabolic method on a local stencil for hyperbolic conservation laws”, Hyperbolic Problems: Theory, Numerics and Applications, Part 2, Proceedings of Symposia in Applied Mathematics, 67, no. 2, 2009, 869–878  crossref  mathscinet  zmath  isi
    6. Kritsuk A.G., Ustyugov S.D., Norman M.L., Padoan P., “Simulating Supersonic Turbulence in Magnetized Molecular Clouds”, Scidac 2009: Scientific Discovery Through Advanced Computing, Journal of Physics Conference Series, 180, 2009  crossref  isi  scopus
    7. Kritsuk A.G., Ustyugov S.D., Norman M.L., Padoan P., “Simulations of Supersonic Turbulence in Molecular Clouds: Evidence for a New Universality”, Numerical Modeling of Space Plasma Flows: Astronum-2008, Astronomical Society of the Pacific Conference Series, 406, 2009, 15–22  adsnasa  isi
    8. M. V. Popov, “Piecewise parabolic method on a local stencil in cylindrical coordinates for fluid dynamics simulations”, Comput. Math. Math. Phys., 52:8 (2012), 1186–1201  mathnet  crossref  mathscinet  adsnasa  isi  elib  elib
    9. Baranov A.A., Chardonnet P., Chechetkin V.M., Filina A.A., Popov M.V., “Multidimensional Simulations of Pair-Instability Supernovae”, Astron. Astrophys., 558 (2013), A10  crossref  isi  elib  scopus
    10. Ray M.P., Puranik B.P., Bhandarkar U.V., “Development and Assessment of Several High-Resolution Schemes for Compressible Euler Equations”, Int. J. Comput. Methods, 11:1 (2014), 1350049  crossref  mathscinet  zmath  isi  elib  scopus
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    12. Rider W.J., “Reconsidering Remap Methods”, Int. J. Numer. Methods Fluids, 76:9 (2014), 587–610  crossref  mathscinet  isi  elib  scopus
    13. Chechetkin V.M., Popov M.V., “the Role of Large-Scale Convection in Supernovae Explosions”, Astron. Rep., 59:6 (2015), 457–468  crossref  mathscinet  isi  elib  scopus
    14. Chardonnet P., Filina A., Popov M., Chechetkin V., Baranov A., “on Gamma-Ray Bursts Spectra: a Possible Theoretical Understanding”, Second Icranet Cesar Lattes Meeting: Supernovae, Neutron Stars and Black Holes, AIP Conference Proceedings, 1693, eds. DeAlmeida U., Chardonnet P., Negreiros R., Rueda J., Ruffini R., Vereshchagin G., Vasconcellos C., Amer Inst Physics, 2015, 050002  crossref  isi  scopus
    15. I. M. Kulikov, I. G. Chernykh, E. I. Vorobev, A. V. Snytnikov, D. V. Vins, A. A. Moskovskii, A. B. Shmelev, V. A. Protasov, A. A. Serenko, V. E. Nenashev, V. A. Vshivkov, A. S. Rodionov, B. M. Glinskii, A. V. Tutukov, “Chislennoe gidrodinamicheskoe modelirovanie astrofizicheskikh techenii na gibridnykh superEVM, osnaschennykh uskoritelyami Intel Xeon Phi”, Vestn. YuUrGU. Ser. Vych. matem. inform., 5:4 (2016), 77–97  mathnet  crossref  elib
    16. Kulikov I. Vorobyov E., “Using the PPML approach for constructing a low-dissipation, operator-splitting scheme for numerical simulations of hydrodynamic flows”, J. Comput. Phys., 317 (2016), 318–346  crossref  mathscinet  zmath  isi  elib  scopus
    17. Kulikov I. Chernykh I. Tutukov A., “A New Hydrodynamic Model for Numerical Simulation of Interacting Galaxies on Intel Xeon Phi Supercomputers”, 10Th International Conference on Numerical Modeling of Space Plasma Flows: Astronum-2015, Journal of Physics Conference Series, 719, IOP Publishing Ltd, 2016, 012006  crossref  isi  scopus
    18. Kulikov I. Chernykh I. Protasov V., “Mathematical modeling of formation, evolution and interaction of galaxies in cosmological context”, All-Russian Conference on Nonlinear Waves: Theory and New Applications (Wave16), Journal of Physics Conference Series, 722, IOP Publishing Ltd, 2016, UNSP 012023  crossref  isi  scopus
    19. Kulikov I. Glinsky B. Chernykh I. Nenashev V. Shmelev A., “Numerical Simulations of Astrophysical Problems on Massively Parallel Supercomputer”, 2016 11Th International Forum on Strategic Technology (Ifost), Pts 1 and 2, IEEE, 2016  isi
    20. Yu Ya., Tian B., Mo Z., “Hybrid Monotonicity-Preserving Piecewise Parabolic Method For Compressible Euler Equations”, Comput. Fluids, 159 (2017), 1–8  crossref  mathscinet  zmath  isi  scopus
    21. Roe Ph., “Is Discontinuous Reconstruction Really a Good Idea?”, J. Sci. Comput., 73:2-3, SI (2017), 1094–1114  crossref  mathscinet  zmath  isi  scopus
    22. Kulikov I. Chernykh I. Tutukov A., “A New Hydrodynamic Code With Explicit Vectorization Instructions Optimizations That Is Dedicated to the Numerical Simulation of Astrophysical Gas Flow. i. Numerical Method, Tests, and Model Problems”, Astrophys. J. Suppl. Ser., 243:1 (2019), 4  crossref  isi
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