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Dispersion Analysis of Smoothed Particle Hydrodynamics to Study Convergence and Numerical Phenomena at Coarse Resolution

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Computational Science and Its Applications – ICCSA 2022 (ICCSA 2022)

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Abstract

The Smoothed Particle Hydrodynamics (SPH) method is a meshless Lagrangian method widely used in continuum mechanics simulation. Despite its wide application, theoretical issues of SPH approximation, stability, and convergence are among the unsolved problems of computational mathematics. In this paper, we present the application of dispersion analysis to the SPH approximation of one-dimensional gas dynamics equations to study numerical phenomena that appeared in practice. We confirmed that SPH converges only if the number of particles per wavelength increases while smoothing length decreases. At the same time, reduction of the smoothing length when keeping the number of particles in the kernel fixed (typical convergence results for finite differences and finite elements) does not guarantee the convergence of the numerical solution to the analytical one. We indicate the particular regimes with pronounced irreducible numerical dispersion. For coarse resolution, our theoretical findings are confirmed in simulations.

V.L. did the dispersion analysis under the support of Russian Science Foundation grant no. 21-71-20003, O.S. performed numerical experiments, and T.M. did the asymptotic analysis; O.S. and T.M. work were founded by Russian Science Foundation grant no 21-19-00429, S.A. visualized the results.

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Authors and Affiliations

  1. Boreskov Institute of Catalysis SB RAS, Lavrentiev Ave. 5, Novosibirsk, 630090, Russia

    Olga Stoyanovskaya & Tamara Markelova

  2. Institute of Mathematics SB RAS, Koptug Ave. 4, Novosibirsk, 630090, Russia

    Vadim Lisitsa

  3. Novosibirsk State University, Pirogova, 2, Novosibirsk, 630090, Russia

    Sergey Anoshin

Authors
  1. Olga Stoyanovskaya
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  2. Vadim Lisitsa
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  3. Sergey Anoshin
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  4. Tamara Markelova
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Corresponding author

Correspondence to Vadim Lisitsa .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Universidad de Málaga, Malaga, Spain

    Eligius M. T. Hendrix

  4. Monash University, Clayton, VIC, Australia

    David Taniar

  5. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

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Stoyanovskaya, O., Lisitsa, V., Anoshin, S., Markelova, T. (2022). Dispersion Analysis of Smoothed Particle Hydrodynamics to Study Convergence and Numerical Phenomena at Coarse Resolution. In: Gervasi, O., Murgante, B., Hendrix, E.M.T., Taniar, D., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2022. ICCSA 2022. Lecture Notes in Computer Science, vol 13375. Springer, Cham. https://doi.org/10.1007/978-3-031-10522-7_14

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  • DOI: https://doi.org/10.1007/978-3-031-10522-7_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-10521-0

  • Online ISBN: 978-3-031-10522-7

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