Abstract
The nonlinear transfer of mass, momentum, and energy is the main peculiarity of gas dynamics. A discontinuous particle method is proposed for its efficient numerical modeling. The method is described in detail in application to linear and nonlinear transfer processes. The necessary and sufficient monotonicity and stability condition of the discontinuous particle method for the regularized Hopf equation is obtained. On the simplest example of a discontinuous solution, the method’s advantages, which include the discontinuity widening over only one particle and the selfadaptation of the space resolution to the solution’s peculiarities, are shown.
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References
F. H. Harlow, “The particle-in-cell method for numerical solution of problems in fluid dynamics,” Proc. Symp. Appl. Math. 15, 269(1963).
R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (McGraw-Hill, New York, 1981).
Yu. A. Berezin and V. A. Vshivkov, Particle-in-cell Method in Rarefied Plasma Dynamics (Nauka, Novosibirsk, 1980) [in Russian].
Yu. S. Sigov, Computing Experiment: The Bridge between the Past and Future of Plasma Physics (Fizmatlit, Moscow, 2001) [in Russian].
A. A. Arsen’ev, “On the approximation of the solution of the Boltzmann equation by solutions of the ito stochastic differential equations,” USSR Comput. Math. Math. Phys. 27 (2), 51–59 (1987).
M. F. Ivanov and V. A. Gal’burt, “Stochastic approach to numerical solution of Fokker–Planck equations,” Mat. Model. 20 (11), 3–27 (2008).
S. V. Bogomolov and I. G. Gudich, “Verification of a stochastic diffusion gas model,” Math. Models Comput. Simul. 6, 305–316 (2014).
M. H. Gorji and P. Jenny, “Fokker-Planck-DSMC algorithm for simulations of rarefied gas flows,” J. Comput. Phys. 287, 110–129 (2015).
S. S. Artem’ev and M. A. Yakunin, “Analysis of estimation accuracy of the first moments of a Monte Carlo solution to an SDE with Wiener and Poisson components,” Numer. Anal. Appl. 9, 24–33 (2016).
S. V. Bogomolov, “An entropy consistent particle method for Navier-Stokes equations,” in Proceedings of the ECCOMAS Congress 2004 4th European Congress on Computational Methods in Applied Sciences and Engineering, 2004, Jyvaskyla, Finland.
S. V. Bogomolov, A. A. Zamaraeva, H. Carabelli, and K. V. Kuznetsov, “A conservative particle method for a quasilinear transport equation,” Comput. Math. Math. Phys. 38, 1536–1541 (1998).
R. A. Gingold and J. J. Monaghan, “Smoothed particle hydrodynamics: theory and application to non-spherical stars,” Mon. Not. R. Astron. Soc. 181, 375–389 (1977).
N. M. Evstigneev, O. I. Ryabkov, and F. S. Zaitsev, “High-performance parallel algorithms based on smoothed particles hydrodynamics for solving continuum mechanics problems,” Dokl. Math. 90, 773–777 (2014).
H. Haoyue, P. Eberhard, F. Fetzer, and P. Berger, “Towards multiphysics simulation of deep penetration laser welding using smoothed particle hydrodynamics,” in Proceedings of the ECCOMAS Congress 2016 7th European Congress on Computational Methods in Applied Sciences and Engineering, June 2016.
E. Onate, S. R. Idelsohn, F. del Pin, and R. Aubry, “The particle finite element method–an overview,” Int. J. Comp. Methods 1, 267–307 (2004).
E. Onate, J. Miquel, and P. Nadukandi, “An accurate FIC-FEM formulation for the 1D advection-diffusionreaction equation,” Comp. Methods Appl. Mech. Eng. 298, 373–406 (2016).
Computational Particle Mechanics. http://link.springer.com/journal/40571.
N. N. Kalitkin and P. V. Koryakin, Numerical Methods 2 (Akademiya, Moscow, 2013) [in Russian].
O. A. Oleinik, “Discontinuous solutions of non-linear differential equations,” Usp. Mat. Nauk 12 (3), 3–73 (1957).
A. N. Tikhonov and A. A. Samarsky, “Discontinuous solutions of first order quasilinear equation,” Dokl. Akad. Nauk SSSR 99, 27–30 (1954).
S. N. Kruzhkov, “Generalized solutions of nonlinear first order equations with several independent variables,” Mat. Sb. 72, 108–134 (1967).
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Original Russian Text © A.Zh. Bayev, S.V. Bogomolov, 2017, published in Matematicheskoe Modelirovanie, 2017, Vol. 29, No. 9, pp. 3–18.
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Bayev, A.Z., Bogomolov, S.V. On the Stability of the Discontinuous Particle Method for the Transfer Equation. Math Models Comput Simul 10, 186–197 (2018). https://doi.org/10.1134/S2070048218020023
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DOI: https://doi.org/10.1134/S2070048218020023